The Pennsylvania State University. The Graduate School. Department of Electrical Engineering

Transcription

1 The Pennsylvana State Unversty The Graduate School Department of Electrcal Engneerng BROADBAND ACCESS AND HOME NETWORKING THROUGH POWERLINE NETWORKS A Thess n Electrcal Engneerng by Pouyan Amrshah-Shraz 006 Pouyan Amrshah-Shraz Submtted n Partal Fulfllment of the Requrements for the Degree of Doctor of Phlosophy May 006

2 The thess of Pouyan Amrshah-Shraz was revewed and approved* by the followng: Mohsen Kavehrad W.L.Wess Professor of Electrcal Engneerng Thess Advsor Char of Commttee Davd Mller Assocate Professor of Electrcal Engneerng Lynn Carpenter Assocate Professor of Electrcal Engneerng Osama Awadelkarm Professor of Engneerng Scence and Mechancs Nadne Smth Assocate Professor of Electrcal Engneerng Kenneth Jenkns Professor of Electrcal Engneerng Head of the Department of Electrcal Engneerng *Sgnatures are on fle n the Graduate School

3 ABSTRACT The ncreasng nterest n modern multmeda applcatons, such as broadband Internet, HDTV, etc. requres new access technques for connectng the prvate premses to a communcaton backbone. One promsng technology, Broadband over Powerlnes (BPL), uses electrc powerlnes as a hgh speed dgtal data channel to connect a group of prvate users to a very hgh data rate backbone, such as fber optc. The lnes n power delvery network can be categorzed based on several crtera. Dependng on lne voltage, HV (hgh voltage), MV (medum voltage), and LV (low voltage) grds are typcally defned. Most HV/MV transformers locatons are equpped wth a hgh-speed fber connecton. Therefore, MV lnes can act as the frst ppelne of hgh-speed connecton from backbone to the home users. In ths dssertaton, we explore the potental of ths technology and then examne the system performance enhancement for such channel envronment usng dfferent modulaton and codng technques. Although for nearly a century some elementary transmsson models of these lnes have been avalable, no serous attempt has gone nto a comprehensve BPL channel modelng for hgh frequency spectra. In ths research we use a new modelng of multconductor wave propagaton n overhead lnes, consderng transent effects. The model dentfes ndependent wave modes for overhead lnes and t s useable over a wde range of frequences. The proposed model ncorporates realstc ground admttance, approprate for hgher frequences used by broadband over powerlne communcatons. By calculatng the lossy ground mpedance for mult-conductor lnes, we derve a transfer

4 v functon for these networks. By applyng water fllng n spectral doman, we were able to express the channel capacty of powerlne networks, usng the developed transfer functon. The powerlne channel suffers from multpath fadng and frequency selectvty. Nevertheless, the calculated channel capacty lmt promses very hgh data rates over ths channel. Furthermore, LV powerlne channels suffer from mpulsve bursty nose. In ths dssertaton two models for ths nose are presented and dscussed. One of the major burdens of BPL s the electromagnetc compatblty (EMC) of ths technology to other wreless systems. Snce electrc wres mght radate electromagnetc waves at hgh frequences, precautons need to be employed n order to avod any nterference to other wreless devces. Ths ssue s also nvestgated n ths research brefly. The well-known mult-carrer technque, Orthogonal Frequency Dvson Multplexng (OFDM), s consdered as the modulaton scheme for BPL by most researchers. By the applcaton of OFDM, the most dstnct property of power-lne channel, ts frequency selectvty, can be easly coped wth. Moreover, OFDM can perform better than sngle carrer modulaton n the presence of mpulsve nose. In ths dssertaton the bt error rate (BER) performance of the OFDM system under mpulsve nose and frequency fadng s theoretcally analyzed and closed form expressons for ths performance s derved. In order to make a very effcent use of the allocated bandwdth and energy n OFDM, adaptve allocaton algorthms should be employed. These algorthms are studed comprehensvely n ths thess. Addtonally, an teratve algorthm

5 v s proposed to predct and cancel the mpulsve nose n the OFDM system over powerlne channels. Enhancement technques, such as codng can help an OFDM system to acheve the capacty lmt as close as possble. A theoretcal upper bound on the performance of coded OFDM system s obtaned, gven perfect nterleavng. The effect of the nterleaver length on codng performance s also studed. Smulatons show that the upper bound s qute tght for the case of employng a longer nterleaver. The effect of nterleaver sze on the system performance s studed, as well. In ths dssertaton we also employ Dgtal Fountan codes, whch are a new class of erasure detectng codes. These codes are consdered the state-of-the-art dscovery n codng theory due to ther smplcty and performance. Indoor wreless connectvty s always appealng to consumers because of ts ease of use. For ths reason, ndoor optcal wreless communcatons through lghtng LEDs has been nvestgated n ths research. A sutable channel model s proposed for ths system and the correspondng transmsson capacty values are calculated. It s shown that the combnaton of BPL and whte LED technology makes an effcent method for fulfllng the premse of broadband access for home networkng, whle provdng effcent and low-cost lghtng.

6 TABLE OF CONTENTS v LIST OF FIGURES...x LIST OF TABLES...xv ACKNOWLEDGEMENTS...xv Chapter 1 Introducton Motvaton Dssertaton objectve Dssertaton s outlne...6 Chapter Hgh Frequency Powerlne Channel Modelng and Channel Capacty Wre-lne channel modelng background Sngle conductor over lossy ground Analyss of Mult-conductor Transmsson Lnes Mult-conductor Confguraton and Modal Analyss Usng D Amore and Sarto s Formulaton Propagaton Matrx Dervaton Per-unt-length Seres Impedance and shunt-admttance Matrces Dervatons...6. Powerlne network channel modelng MTL crcut model Multpath model Channel capacty concept Shannon s theorem Water-fllng concept Numercal results MV lne networks LV lne networks...43 Chapter 3 Nose and nterference Nose n MV lnes Background nose Corona Nose Narrowband Nose Nose n LV lne Colored background nose Synchronous perodc mpulsve nose Asynchronous perodc mpulsve nose...57

7 3..4 Burst mpulsve nose Impulse rate Relatve dsturbng tme Impulse ampltude Impulse wdth and spacng LV lne nose modelng Statstcal nose model Tme-based nose model Electromagnetc nterference Radaton theory bascs Smulaton results Load mbalance effect on radaton pattern...75 Chapter 4 Modulaton Technques for BPL systems Sngle-carrer modulaton Spread spectrum Drect Sequence Spread Spectrum Frequency Hoppng Comparson of FHSS and DSSS Orthogonal Frequency Dvson Multplexng (OFDM) OFDM system concept OFDM and orthogonalty concept OFDM system n non-deal channel condtons Effect of OFDM system on BPL nose Performance of OFDM system n BPL channel condtons Adaptve loadng for OFDM Adaptve OFDM algorthm for ncreasng data rate Error probablty crteron MSSS crteron Iteratve algorthm Adaptve OFDM algorthm for mprovng system performance Impulsve nose cancellaton n OFDM Decson drected mpulsve nose cancellaton The teratve mpulsve nose cancellaton Mult-carrer CDMA (MC-CDMA) MC-CDMA Analyss...15 Chapter 5 Codng Convolutonal codng OFDM nterleavng technques to tackle the effect of burst nose on convolutonal codes performance v

8 5. Concatenated codng Chapter 6 Indoor Whte LED communcatons Cellular whte LED communcatons plannng Indoor optcal channel modelng Chapter 7 Concluson and future work Summary of results Future work Publcatons Bblography v

9 LIST OF FIGURES x Fgure 1-1: A typcal power lne access network archtecture...3 Fgure -1: A thn wre over ground...11 Fgure -: (a) Real and (b) Imagnary part of propagaton constant of an overhead wre at heght of 10 meter obtaned by three dfferent methods...16 Fgure -3: Attenuaton constant of an overhead wre at heght of 10 centmeter obtaned by three dfferent methods...17 Fgure -4: Modes of three-phase power lnes...19 Fgure -5: Couplng methods of BPL: a) wre-to ground and b) wre-to-wre...0 Fgure -6: A mult-conductor confguraton...1 Fgure -7: Frequency spectra of (a) Attenuaton constants, and (b) Phase constants of MTL system shown n Fgure Fgure -8: Frequency spectra of the (a) real and (b) magnary parts of characterstc mpedance of a mult-conductor lne wth common mode njecton....9 Fgure -9: Multpath sgnal propagaton; cable wth one tap Fgure -10: Frequency response of a matched MTL system for 1 Km span. (a) ampltude, (b) phase and ts assocated capacty lmts (c)...38 Fgure -11: Frequency response of a msmatched MTL system for 1 Km span. (a) ampltude, (b) phase, (c) Channel mpulse response and (d) ts assocated capacty lmts Fgure -1: The smulated complex network...41 Fgure -13: Frequency response of complex network shown n Fgure -1: (a) Ampltude, (b) Phase, (c) Channel mpulse response and (d) Lnes transmsson capacty bounds...4 Fgure -14: The transfer functon of network shown n Fgure 7 of [31]. (a) Measurement, (b) Crcut theory smulaton...44

10 Fgure -15: (a) Smulated frequency and (b) mpulse response of a LV powerlne network depcted n [31] by multpath method and (c) ts assocated capacty lmts...45 x Fgure 3-1: Comparson of nose levels between MV and LV lnes...48 Fgure 3-: Corona dscharges modeled wth shunt current sources...49 Fgure 3-3: Corona nose power spectrum n poor weather...51 Fgure 3-4: Total nose power spectrum n an arbtrary powerlne system, operatng n poor weather...5 Fgure 3-5: Background nose n LV lnes...53 Fgure 3-6: Synchronous perodc nose modelng...55 Fgure 3-7: Envelope curve of perodc mpulsve nose...55 Fgure 3-8: Rectangular envelope curve and ts Fourer transform of perodc mpulsve nose...56 Fgure 3-9: Fourer transform of perodcally contnued envelope curve of perodc mpulsve nose...56 Fgure 3-10: Envelope curve of Asynchronous mpulsve nose...58 Fgure 3-11: Markov model for burst nose: (a) Modelng burst groups (b) Modelng sngle mpulses wthn a burst group...64 Fgure 3-1: Geometrcal confguraton of the smulated MV Powerlne...70 Fgure 3-13: Ampltude frequency spectra of radated felds from the MV confguraton of Fgure 3-1 computed by GNEC and establshed mathematcal method for wre -to-ground couplng: (a) E y, (b) H z, (c) E z, (d) H y, (e) E x, (f) H x...71 Fgure 3-14: Ampltude frequency spectra of radated felds from the MV confguraton of Fgure 3-1 computed by GNEC and establshed mathematcal method for wre 1-to-wre couplng: (a) E y, (b) H z, (c) E z, (d) H y, (e) E x, (f) H x...73 Fgure 3-15: Lateral profle of the feld components at 10 MHz for the (a) and (b) Wre -to-ground (c) and (d) wre 1-to-wre couplngs...74 Fgure 3-16: The powerlne confguraton for radaton pattern smulaton...75

11 Fgure 3-17: Far feld radaton from powerlne confguraton n Fgure 5.1 for (a) central common mode, (b) sde common mode and (c) dfferental mode njectons...77 Fgure 3-18: Radaton pattern from an unbalanced powerlne wth (a) 10 mh n seres dfference, (b) 1 mf n parallel dfference and (c) 1 μf n parallel dfference Fgure 4-1: System performance of a sngle-carrer system wth 50-tap DFE...8 Fgure 4-: Prncple of bandwdth spreadng...84 Fgure 4-3: A DSSS transmttser...87 Fgure 4-4: A DSSS recever wth a matched flter...88 Fgure 4-5: System performance of a perfectly synchronzed DSSS system wth a complcated Rake recever...89 Fgure 4-6: Transmtter for FHSS...91 Fgure 4-7: Recever for FHSS...91 Fgure 4-8: General block dagram realzaton of an OFDM system...96 Fgure 4-9: OFDM and orthogonalty prncpal...97 Fgure 4-10: Guard nterval nserton Fgure 4-11: The performance of uncoded OFDM system n a fadng channel wth mpulsve nose analytcally and by smulaton Fgure 4-1: Transmsson rate for dfferent adaptve algorthms wth error probablty crteron operatng n channel of Fgure Fgure 4-13: Transmsson rate for dfferent adaptve algorthms wth MSSS probablty crteron operatng n channel of Fgure Fgure 4-14: The effect of adaptve OFDM loadng to the performance of a communcaton system Fgure 4-15: Decson drected mpulsve nose cancellaton OFDM recever block dagram Fgure 4-16: Effect of M on the performance of mpulse cancellaton...11 x

12 Fgure 4-17: Performance of decson drected mpulsve nose cancellaton recever wth M= Fgure 4-18: Performance of proposed teratve algorthm n a Markov-based nose model envronment...14 Fgure 4-19: MC-CDMA transmtter block dagram...16 Fgure 4-0: MC-CDMA trecever block dagram...17 Fgure 4-1: The performance of uncoded MC-CDMA system n a fadng channel wth mpulsve nose analytcally and by smulaton...19 Fgure 5-1: A half rate convolutonal code...13 Fgure 5-: Effect of convlutonal codng on the system performance and ts upper bound Fgure 5-3: Sequence of 16 OFDM symbols wth 8 subcarrers each, affected by mpulsve nose (symbol 11) and a narrowband nterferer (subcarrer 4) Fgure 5-4: Coded MC-CDMA smulaton results wth dfferent nterleaver szes and analytcal upper bound assumng nfnte nterleaver Fgure 5-5: FER performance versus SNR comparson of codes wth dfferent rates Fgure 5-6: FER performance of the system versus nverse of the code rate for dfferent SNR values Fgure 6-1: Vsble lght communcatons usng whte LED Fgure 6-: An optcal communcatons scenaro...15 Fgure 6-3: A model room usng whte LED communcaton system confguraton Fgure 6-4: Illustraton of sgnal propagaton. The room surface s composed of 3 elements: a, b, and c Fgure 6-5: Illustraton of (a) drect path, (b) sngle reflecton, and (c) two reflectons. Reflectons are counted from dffusng spots to a recever Fgure 6-6: New representaton of drect, frst, and second reflecton for a mutple source and a sngle recever x

13 Fgure 6-7: (a) Ampltude of the G vector for a small FOV (7 o ), the recever s located at (.5,4.5,0.9). (b) Ampltude of G at the same locaton wth a larger FOV (90 o ). Room dmensons are 6mx6mx3m Fgure 6-8: Impulse response of (a) pont A at (3,.5,0.9) and (b) pont B at (0.5,0.5,0.9) Fgure 6-9: Calculaton of optcal path dfference x

14 LIST OF TABLES xv Table -1: Sgnal Propagaton paths of the examned network n Fgure Table 5-1: Mnmum dstances for an 8 16 nterleaver Table 5-: Matrx of 8 16 nterleaver for k 1 =113, k = Table 5-3: Comparson between decodng tme for RS codes and Tornado codes...146

15 ACKNOWLEDGEMENTS xv I wsh to express my deepest apprecaton to my professor and thess advsor, Professor Mohsen Kavehrad for hs gudance, encouragement, and creatve deas. Much grattude and thanks are also extended to my commttee members: Professors D. Mller, L. Carpenter, O. Awadelkarm, and N. Smth for ther tme, encouragement and help. I had the pleasure of knowng the faculty, students and staff of the Center for Informaton and Communcaton Technology Research at Penn State Unversty over the past few years, many of whom have helped me mmeasurably. I would lke to dedcate ths thess to my parents and brother for ther neverendng love throughout my lfe, whch made all of ths possble.

16 Chapter 1 Introducton 1.1 Motvaton Fast Internet access s growng from a convenence nto a necessty n all aspects of our daly lves. Unfortunately, ths has been held back by the hgh expenses of wrng nfrastructure essental to delver such hgh-speed nternet access especally to prvate homes, small offces and rural areas, where the nstallaton of any knd of new wres tlts the scales of the economc feasblty to a non-proftable state. Ths problem s known as the last mle problem whch has been an actve area of research throughout research communty. One promsng technology, Broadband over Powerlnes (BPL), uses electrc powerlnes as a hgh-speed dgtal data channel to connect a group of prvate users to a very hgh data rate backbone, such as fber optc. The lnes n power delvery network can be categorzed based on several crtera. Dependng on lne voltage, HV (hgh voltage), MV (medum voltage), and LV (low voltage) are typcally defned. Wthn a dstrbuton grd, dependng on the topologcal confguraton, ether overhead lnes or underground cables are used. Overhead MV lnes dffer consderable n structure and physcal propertes compared to other wre-lnes as twsted-par, coaxal and fber-optc cables. The power-lnes n USA hang overhead at a heght of ~10 meters above ground, for more than 85% of locatons, smply because over ground wrng s 10 tmes cheaper than underground. Typcally, 4 wres;

17 three phases and a neutral (sometmes a grounded neutral) wth ~ 1 meter spacng between wres are used over the earth. Wres are made of alumnum wth conductvty; 61% of standard annealed copper. Wres are unshelded; thus, quas-guded (guded plus unguded) modes exst. Each power-lne acts as a BUS; thus the avalable band has to be shared among users; that s, half dozen homes per transformer, n the USA. Whle sgnfcant research and development have been done on BPL n Europe, where underground cables are typcally used n ths applcaton, there has not been much done on MV overhead lnes n USA. The power dstrbuton grd resembles an omnpresent, wdely branched herarchcal structure. Each home s equpped wth electrcty by means of LV powerlne grd. LV lnes are dstrbuted to each power plug n every room n a house. More than 99% of the homes n the USA have access to electrcty, whereas connectvty level s far less for cable and phone lnes. Thus, a combnaton of MV and LV powerlnes can be an approprate canddate for provdng broadband access to every home n the country, offerng both last mle and last meter solutons. In Europe, there have been sgnfcant efforts to utlze the LV dstrbuton grd consstng mostly of ground cables and nbuldng LV network as a medum for broadband servce delvery [1][][3][4][5][6][7]. Unlke the European system, a typcal power-lne access network n the US s composed of MV dstrbuton grd and LV lnes to and from houses. It s usually composed of 3 phases and a neutral wre or 3 phases and a grounded neutral wre. Measurements over 3- phase overhead MV lnes were performed n South Korea [8][9]. A typcal scenaro for such an access network s shown n Fgure 1-1. Currently, many vendors have proposed use of frequences between MHz to 50 MHz for BPL applcaton [10].

18 3 Fgure 1-1: A typcal power lne access network archtecture Orgnally desgned for power delvery rather than sgnal transmsson, power lne has many non-deal propertes as a communcatons medum. Impedance msmatches at jonts cause reflectons that generate condtons smlar to those created by multpath fadng n wreless communcatons. Whle there have been lots of research efforts to characterze the European ground cables, there has not been avalable a proper theoretcal model for mult-conductor overhead MV lnes, a typcal stuaton n USA. It has to be kept n mnd that the use of electrc network to carry RF waves s not unconstraned, because by occupyng a frequency band of about to over 50 MHz, a frequency overlap s created wth the exstng servces such as; the entre HF servces, long-wave, medum-wave, and short-wave rado and amateur rado bands, as ar-borne

19 4 power lnes also act as antennas, transmttng and recevng nterference to and from the surroundng envronments. Indoor wreless connectvty s always appealng to consumers because of ts ease of use. One of the conventonal wreless access systems s W-F. These systems and smlar other wreless schemes suffer from so many shortages, such as nterference, not provdng qualty-of- servce (QoS), adequate coverage, etc. A better canddate for wreless home networkng s optcal wreless. Use of conventonal lasers for optcal ndoor communcatons has not been feasble as yet, because of the hgh cost of laser sources. Instead of lasers, LEDs can be used as communcatons transmtters connected to electrc grd, recevng hgh data rate sgnals va BPL. Recently, whte LEDs emerged n the market and are consdered as future lamps. Apparently, n the near future, the ncandescent and fluorescent lamps wll be replaced by the low cost, effcent and mnature whte LEDs. Researchers pledge that by 01, these devces wll reach 7W and 1000lm. Ths s brghter than a 60-w bulb and yet draws a current provded by 4 D-sze batteres [11]. A Japanese research team suggested the use of the same whte LEDs not only for lghtng the homes, but also as lght sources for wreless n-house communcatons [1]. Usng ths new and developng technology along wth MV/LV powerlne communcatons can create a revoluton n the area of consumer networkng due to ts effcency and affordablty.

20 1. Dssertaton objectve 5 In order to be able to desgn a system that can communcate effectvely n a certan envronment, three man concepts need to be consdered. Frst, an accurate model of the communcaton channel s requred. Secondly, a modulaton scheme approprate to the channel condtons and system requrements need to be consdered. Fnally, a recever archtecture that s approprate to the modulaton scheme need to be selected. Thus, n order for BPL technology to succeed, several hurdles have to be overcome, such as fndng a sutable model for the power lne channel that ncorporates sgnal degradaton through the lne and nterference sources, determnng the approprate frequency allocaton scheme and acceptable transmsson power levels to mnmze nterference nto exstng servces, and fnally selectng sutable modulaton, codng and detecton schemes and measures to mnmze the effect of external nterference on the proposed system. Based on the above observatons wth respect to the channel and Electromagnetc Compatblty (EMC), one has to be careful, when choosng an approprate modulaton scheme. Selecton of a modulaton scheme for PLC must account for three major factors: Presence of nose and mpulse dsturbances causng a relatvely low SNR. Tme-varyng frequency-selectve nature of channel. Regulatory constrants wth regards to electromagnetc compatblty that lmts the transmtted power. Reducton of power spectral densty of PLC sgnals n order to mnmze radaton.

21 6 A choce should be made of ether a robust soluton (.e., one provdng suffcent qualty for a wde range of varatons of the model parameters) or an adaptve one. The followng modulaton schemes are of hgh nterest as canddates for robust solutons: sngle-carrer modulaton, spread spectrum, mult-carrer modulaton, or a combnaton of spread spectrum and mult-carrer modulaton. Adaptve transmsson schemes maxmze the use of channel at the cost of an ncrease n complexty; adaptvty can be mplemented through adaptaton of the transmtted power level, constellaton sze, and symbol rate. One objectve of ths work s to research BPL communcaton systems, through methodcal examnaton of the aforementoned three communcaton system components, n order to propose a system archtecture that can provde a relable performance for BPL system condtons and requrements n addton to provdng enough transmsson capacty for the system to be deployed n commercal applcatons. Moreover, another goal of ths research s to provde an nsght about feasblty of ndoor whte LED broadband communcatons systems and ntroduce the promsng characterstcs of ths novel technque. 1.3 Dssertaton s outlne Ths dssertaton s organzed as follows; chapter ntroduces the bascs of channel modelng for ths system and compares dfferent methods. In ths chapter, we dscuss about accuracy of each method and based upon the most accurate method, we buld up our network channel model. Addtonally, ths chapter provdes a bref revew on

22 7 channel capacty concepts and based on ths theory, t presents channel capacty bounds for our modeled channels. Chapter 3 ntroduces the nose ssue and ts source for the powerlne communcaton system. Moreover, t ntroduces the electromagnetc nterference caused by deployment of broadband powerlne systems and t provdes possble methods to mtgate ths problem. Chapter 4 provdes a revew of dfferent modulaton schemes sutable for BPL and then t studes the performance characterstcs of each of these schemes for dfferent powerlne channel stuatons. The codng schemes and ther performance mprovements are dscussed n Chapter 5. Furthermore, dfferent codng methods are compared to one another and ther complexty and achevable mprovements are dscussed n ths chapter. In chapter 6, we present a bref revew of ndoor broadband whte LED communcatons channel modelng and we demonstrate the potental of ths novel scheme for provdng broadband connectvty to the home users. Chapter 7 concludes the dssertaton and descrbes the future tasks needs to be consdered for further developments n ths research.

23 Chapter Hgh Frequency Powerlne Channel Modelng and Channel Capacty.1 Wre-lne channel modelng background Powerlne wres are transmsson lnes (TL) for the electromagnetc waves smlar to any electromagnetc TL. Frequency response, H (f), of a matched transmsson lne can be expressed by means of a propagaton constant, γ. The voltage along wre at a dstance l from the source, V (l), s obtaned by [13]: V ( l) = H ( f ) V ( 0).1 H ( f ) γ ( f ) l α ( f ) l jβ ( f ) l = e = e e. where V(0) s the voltage at the source. α s the real part of propagaton constant and s called attenuaton constant, β s the magnary part of propagaton constant and s called phase constant. By havng the propagaton constant, one may easly fnd the transfer functon for TL at a desred pont on the lne. Therefore, the major attenton n channel modelng of Transmsson Lnes s focused on fndng the respectve propagaton constant for that specfc lne..1.1 Sngle conductor over lossy ground Hstorcally, fndng the propagaton constant of a thn wre over earth was of nterest to the researchers snce early 0 th century because of ts applcaton n power

24 9 transmsson and telephone communcatons. These systems operate at very low frequences. At these frequences, the heght of wre s a small fracton of wavelength and all the coupled energy nto the wre propagates n quas-tem mode. Thus, the early works n ths feld were focused on fndng the dstrbuton characterstcs of ths propagaton mode n transmsson lne. Carson reported the earlest soluton for ths problem n 196 [14]. In hs work, he calculated values for dstrbuton parameters of a quas-tem mode n transmsson lne. In dong so, he made three assumptons: 1) The propagaton constant s not sgnfcantly dfferent than that n delectrc; therefore, Laplace s equaton can be substtuted for two-dmensonal wave equaton n ar. ) The dsplacement currents n earth surface are neglgble. 3) The effect of earth conductvty on the per unt length parallel admttance s neglgble. These assumptons restrct the soluton to very low frequences and/or perfectly conductng earth. To fnd the exact soluton for ths problem at hgh frequences wth lossy ground return, we need to derve the modal equatons. Kkuch [15] [16] n 1956 derved the exact modal equatons for very thn wres above the earth. In ths work, he used quasstatc and asymptotc expanson of the exact modal equaton to nvestgate the transton from quas-tem to surface wave propagaton. Carson s method s essentally a low frequency approxmaton of the transmsson lne mode. On the contrary, Kkuch s result s assocated wth the entre frequency spectrum of the same mode. Kkuch showed,

25 10 expermentally and theoretcally, that as frequency ncreases, the transmsson lne quas- TEM mode reverts to a TM mode. Accordng to Kkuch, as frequency ncreases there exst a hgh feld concentraton around the wre and a large longtudnal dsplacement currents that act as the return currents over ar, thus mnmzng the role of the earth as the return current path. Therefore, after certan frequency, the pathloss of transmsson lne dmnshes by ncreasng frequency. In 197, Wat [17] extended Kkuch s work and could derve and solve the exact modal equaton for a thn wre above the earth. A summary of hs work s descrbed here. Consder the problem shown n Fgure -1.Here, a perfectly conductng wre wth radus a s located n free-space. The wre s parallel to the surface of homogenous conductve earth at heght h. Earth s characterzed wth a relatve permttvty, ε g, and a permeablty, μ g =μ 0, and a conductvty σ g. Wat showed that the propagaton constant, γ, for the wave on the wre can be expressed n terms of the standard transmsson modal lne equaton: γ = ZY = α + j β.3 where Z s the equvalent seres lne mpedance and Y s the equvalent shunt lne admttance.

26 11 Fgure -1: A thn wre over ground Both Z and Y are functons of γ: Z 1 = jωμ 0 ( Λ + ( Q jp) ).4 π Y = πjωε 0 Λ + ( N jm ).5 Λ = ( ja k0 γ ) K0( jh k0 ) K 0 γ.6 Q jp = u 0 h e λa) d u u g cos( λ.7 N jm = 0 k g u e u 0 h λa) d 0 + k 0 u g cos( λ.8 0 = + γ k0 u λ and u = λ +γ.9 g k g jσg k o = ω μ 0 ε 0 and k g =ω μg( εg ).10 ω

27 1 As Wat n [17] ponted out, the earler work of Carson represents a specal case of soluton to.3. If a k0 γ << 1, h k0 γ << 1, h>>a, k 0 h << 1, and k 0 k g << 1, then the small argument approxmaton of Bessel functon can be nvoked, Q-jP can be smplfed and N-jM can be neglected. Wth these approxmatons, the fnal result n.3 s dentcal to Carson s answer for the propagaton constant: γ = k 1 0 J c h ln( ) a.11 where J c s the approxmaton of (Q-jP) and s gven by: J λ h c = ( λ k λ e d λ g ).1 k g 0 Carson s ntegral n.1 can be expressed as a seres that was presented n [14]. Unfortunately, even wth these assumptons and usng the seres expanson, the soluton appears to be rather complcated. However, Carson noted that the leadng terms n the seres are of mportance n many practcal cases, whch could lead to a smple closed form answer [14]. The earler result of Kkuch [15] s also embedded n the Wat s full wave soluton. If a s vanshngly small compared to the wavelength, the small argument approxmaton for Bessel functon can be substtuted and the cos(aλ) term may then be set to unty n Q-jP and N-jM. Based on these assumptons, the results of Kkuch and Wat are dentcal.

28 13 All the aforementoned modal equatons were derved by assumng that wres are perfect conductors. In real powerlne systems, the wres are made from near-perfectly conductng materals. In MV overhead powerlnes, the conductng materal n wres s Alumnum wth 30% less conductvty than the Copper. To deduct a more generalzed soluton for ths problem several nvestgatons were conducted n ths feld snce early 1970 s. For example: Wat et al. [18] n 1975, Olsen et al. [19] n 1978 and a more recent one, D amore et al. [0] n The basc concepts of these exploratons are the same; the dfference s n the assumptons and estmatons, whch are advsable n dfferent cases for convertng a complcated result to a smple one. Bascally, for analyzng ths problem, we need to satsfy the boundary condtons on the nterface between the surface of wre and the medum surroundng the wre. Suppose we characterze wre wth a conductvty σ w and a permttvty ε w and a permeablty μ w =μ 0. From [0], for boundary condtons we wll have: E ( x, h, a) = E ( x, h, a) x wre x ar.13 The x- component of electrc feld on the wre s gven by: E ( x, h, a) = z ( γ) I( x).14 x wre where z (γ) s wre nner mpedance and from[19], t s formulated as: ωμωu ( γ) = πak z k k ε I0( juωa) I ( ju a) ω ω 1 ω jσ w w w = 0 ε0 ωε u ω k = ω γ.17

29 obtaned by: Followng the procedure n [0], the x-component of electrc feld n ar s 14 jωμ0 EX ( x, h, a) ar= M( γ, h, a) I( x) π k0 γ γ M ( γ, h, a) = Λ( γ, h, a) + S1g ( γ, h, a) S g ( γ, h, a) k k 0 Λ( γ, h, a) = K 0 ( ja k0 γ ) K 0( j 4h + a k0 + γ 1 exp( u h ) S g ( γ, h, a ) = 0 1 exp( jzλ ) dλ u 0 + u g 1 exp( u0 h) S g ( γ, h, a) = exp( jzλ) dλ k g k0 u0 + u g 0 ) u u 0 = + γ k 0 g λ.3 = γ λ + k g.4 we wll get: S 1g and S g are called Sommerfeld ntegrals. By combnng.18 and.14 n.13 ( γ ) z = jωμ π 0 M ( γ, h, a).5 By the help of equatons.15 to.1, equaton.5 can be solved for γ. Equaton.5 s composed of Bessel and Sommerfeld ntegrals, whch are functons of γ; therefore, the general explct answer to.5 cannot be obtaned and t needs numercal methods to solve.5 for propagaton constant, γ. Solvng ths equaton numercally s not an easy procedure because there s no certanty about startng pont for γ. An unsutable startng pont can cause answers to dverge. D amore et al. n [0] assumed k w >>k 0 and k 0 -γ 0,

30 15 whch are admssble for those cases when the wavelength s smaller than ten tmes the heght. For example, for the system wth a 10 meters heght above earth, these estmatons are vald, roughly up to 100 MHz frequences. Wth these assumptons n mnd, the propagaton constant can be expressed as: 1 h πz ˆ + + ˆ ( jωμ0 ) ln( ) S1g ( h) γ = k a 0.6 h ln( ) + Sˆ ( h) g a ωμ w I 0 ( jk w a ) zˆ =.7 πak I ( jk a ) w ˆ w S g ( h) = 0.5ln(1 + α r ).8 k0 S ( ) ln(1 g h = + β r ).9 k + k ˆ 1 g 0 k0 + k g α = and β =.30 k k g k k k 0 r g = 4h a.31 For better understandng of these formulatons we consder two cases: frst, when the wre s heght s large enough compared to operatng wavelength, smlar to MV powerlne stuaton and second, when ths heght s smaller or comparable to the wavelength as n the case n LV powerlnes. In the frst case the confguraton used n our smulatons s composed of one wre wth σ w = S/m and ε w = ε 0 at 10 meter above the earth. The wre has a radus of 1 cm. Earth s characterzed by σ g =0.005 S/m and ε g =13ε 0. The magnary and real parts of propagaton constant for ths wre are depcted n Fgure - for three dfferent formulatons. As t s shown n ths fgure, for attenuaton constant, the three formulatons

31 16 agree at low frequences and ncrease wth frequency ncrements, but after some frequency, D amore s formulaton shows a decrease n attenuaton constant, Kkuch s formulaton goes to a saturated stuaton and Carson s result ncreases monotoncally wth frequency. On the contrary, phase constants of all three methods agree almost over the whole frequency band. 8 x Attenuaton constant[neper/m] D'amore's method Kkuch's method Carson's method Frequency[Hz] (a) Phase constant[rad/m] D'amoes's method Kkuch's method Carson's method Frequency[Hz] x 10 7 (b) Fgure -: (a) Real and (b) Imagnary part of propagaton constant of an overhead wre at heght of 10 meter obtaned by three dfferent methods

32 17 For smulaton of the second case, the same wre and earth characterstcs are used. The heght of wre above the ground s 10 cm. In ths stuaton our desred wavelengths (frequences between 1 to 100 MHz) are not fracton of heght anymore; therefore, the D amore s assumptons are not applcable. On the other hand, at ths heght and frequency range, Carson s formulaton of.11 can be used. Furthermore, Kkuch s λ assumpton n [15] ( >> 1 ) s vald for almost all the frequency range; thereby the h approxmate seres expanson for ntegrals of.4 and.5 can be appled. Fgure -3 llustrates the three dfferent results for the attenuaton constant of ths wre obtaned from the mentoned methods. Kkuch and Carson method s results are approxmately n agreement, whereas D amore approach offers sgnfcantly dfferent values for attenuaton constant n ths frequency range. Fgure -3: Attenuaton constant of an overhead wre at heght of 10 centmeter obtaned by three dfferent methods

33 18 These two examples showed the applcablty of the studed methods n certan condtons. The results from these two cases can suggest whch method s more relable to use n specfc stuatons of powerlne communcaton channel modelng. Consequently, for overhead MV powerlne networks, where the lnes are 10 meter or more above ground, D amore s method s the most accurate model whereas for ndoor LV powerlnes ether Carson or Kkuch s approach s more preferred because the wres are close to the ground return path..1. Analyss of Mult-conductor Transmsson Lnes Analyss of transmsson lnes consstng of two parallel conductors has been a well-understood topc. Ths understandng can be further extended to matrx notatons to cover mult-conductor transmsson lnes (MTL), nvolvng more than conductors [1]. For a two-conductor lne, we have forward- and reverse- travelng waves. For an MTL wth (n+1) conductors placed parallel to the x-axs, there are n forward- and n reversetravelng waves wth respectve veloctes. These waves can be descrbed by a coupled set of n, frst-order, matrx partal dfferental equatons whch relate the lne voltage V (x,t), =1,, n, and lne current I (z,t), =1,, n. Each par of forward- and reverse-travelng waves s referred to as a mode. For example, n the case nvolvng 3 conductors and a ground return, we can defne 3 modes as gven n Fgure -4 []. Usng these ndependent modes, we can decompose currents I 1 through I 3 as a lnear combnaton of 3 modal currents. Common mode (also called ground mode) s characterzed by the hghest attenuaton among the modes, and s propagaton through 3 phases and a return va the

34 19 earth. Involvng sgnal propagaton and return only through wres, dfferental mode (also called aeral mode) 1 and show somewhat lower attenuaton than common mode. Whle the common mode current I c s the same n magntude and n drecton for 3 lnes, the dfferental mode currents I D1 and I D are the same n magntude but dffer n drecton for 3 lnes. Common mode currents are much smaller n magntude than dfferental mode currents, but are sgnfcant snce whle the radated E-feld from the dfferental mode currents subtract, those from common mode currents tend to add [1]. Ths s an mportant ssue n terms of Electromagnetc Compatblty (EMC) of BPL systems and potental nterference nto exstng local communcatons systems n the shared bands. Fgure -4: Modes of three-phase power lnes In BPL, dependng on the way sgnal s coupled to the lnes, as llustrated n Fgure -5, ether wre-to-wre (WTW) or wre-to-ground (WTG) njecton s feasble. For WTW njectons, dfferental modes are mostly excted. For a WTG njecton, n a case of couplng to the mddle phase, the common mode and the dfferental mode are excted. Generally, these modes are not orthogonal unless the wavelength of electromagnetc wave nsde the conductors s a small fracton of the heght of wres and the spacng between the wres s a small fracton of wavelength [3]. Ths condton s

35 satsfed for practcal MV power-lne systems up to 100 MHz. Beyond ths frequency, the dscrete modes lose ther orthogonalty and contnuous modes start to appear. 0 Fgure -5: Couplng methods of BPL: a) wre-to ground and b) wre-to-wre The prevalent mode of propagaton n an MTL s the transverse electromagnetc (TEM) wave. An MTL s capable of gudng waves whose frequency values vary from DC to a pont where the lne s cross-sectonal dmensons such as lne separatons become a sgnfcant fracton of wavelength. At hgher frequences, hgher-order modes coexst wth the TEM mode, so other gudng structures such as wavegudes and antennas are more practcal. Addtonally, mperfectons n the lne conductors,.e. presence of nearby conductors, and asymmetres n the physcal termnal exctaton such as offset source postons may also create non-tem currents [0]. In a mult-conductor geometry, whch s depcted n Fgure -6, n wres are placed parallel to the x-axs. Parameters h and a refer to the heght above the earth and the radus of th wre, respectvely. Permttvty, permeablty, and conductvty of ground are respectvely represented by ε g, μ 0 and σ g. These parameters for wre are expressed by ε w, μ 0 and σ w. ε 0 and μ 0 are permttvty and permeablty of free space. Parameter Δ j s

36 defned as the dstance between the th wre and the j th wre along the z-axs. Dstance parameter d s defned as the shortest dstance between the th and the j th wres and can be j descrbed by: 1 d j =.3 ( h h j ) + Δj Fgure -6: A mult-conductor confguraton Bascally, ths problem s analyzed by solvng the so-called curl-maxwell equatons and satsfyng the boundary condtons on each and every wre [4][5]. By dong so, the result for fndng the transmsson constant s the answer to the matrx equatons wth Bessel and Sommerfeld ntegrals. The procedure s very smlar to what we followed for a sngle wre, but usng matrces nstead of vectors. D amore et al. have dscussed ths subject wth detals n [6]. Takng a number of steps can solve the equatons for n lne voltages and n lne currents, descrbng MTL. Frst, per-unt-length parameters such as nductance, capactance, conductance and resstance are determned for the consdered lne. Secondly,

37 the MTL equatons are solved n the form of a sum of n forward- and n reverse- travelng wave equatons, wth n unknown coeffcents. Thrdly, termnaton condtons such as ndependent voltage/current sources, load and source mpedance values are ncorporated n the MTL equatons n order to determne the n unknown coeffcents [1]. As stated earler, the frst step n solvng the MTL equatons s to obtan per-untlength parameters for the conductors. For ths, Carson [14] suggested ncorporatng ground mpedance. However, ths model, wthout consderng the ground admttance, s only sutable for low frequency values and/or under good conductve ground plane condtons. Next, as an effort to fnd a new ground return path model for hgher frequences and/or under poor ground conductvty condtons, a new procedure was suggested. Ths methodology by D Amore et al [7] ncorporates per-unt-length seres-mpedance and shunt-admttance matrces, usng the curl-maxwell feld equatons. The detals of ths formulaton are mentoned n the comng subsectons Mult-conductor Confguraton and Modal Analyss Usng D Amore and Sarto s Formulaton The well-known second-order dfferental equatons descrbng the propagaton on a conductor can be extended to matrx form as n.33, where V and I are n-by-1 column vectors of voltage and current n each wre, P s an n-by-n propagaton matrx, and t represents the matrx transpose. d dx V = PV d I t, = P I.33 dx

38 3 Denotng the n-by-n per-unt-length seres mpedance matrx by Z and the n-byn per-unt-length shunt-admttance matrx by Y, the propagaton matrx P s expressed as: P = Z Y.34 Often, smlarty transformaton s adopted n the analyss of mult-conductors, where a change of varables s defned as n.35, such that actual phase voltages V and current I can be related wth mode voltages m MV V = and m V and mode currents m I [3]. m I = NI.35 Now,.35 substtuted nto.36 wll result n.37. d d V = ZI, I = YV dx dx m 1 m d V 0 M ZN V = m 1 m dx I N YM 0 I Suppose we can somehow use M and N to dagonalze Z and Y as n.38 M 1 Z N = z = dag { z 1 K z n },.38 N 1 Y M = y = dag y 1 K y } { 4 Thus, we can obtan n uncoupled pars of frst-order dfferental equatons for MTL as gven by.39. The solutons to.39 are well known. Therefore, we can use ths decouplng technque to solve MTL equatons. d dx d dx V V m 1 m n d = z I dx M m d = z nin, I dx = m m m 1 1, I1 y1v1 m n = y V, m n n.39

39 4 Also, combnng the two equatons n.38, we obtan.40,.41, and.4. Wth M and N beng the th column of M and N, λ represents the egen-value assocated wth the egenvector M and N. The set } { 1 n λ λ λ K s the set of all egen-values of matrx P. Now, the propagaton constant of the th mode can be represented by.43, wth α and β beng the attenuaton and phase constant of the th mode, respectvely. Then, the general equaton for modal currents s expressed by.44, where the vectors of undetermned constants stll need to be determned. Modal voltages can be determned, lkewse. Actual voltages and currents of phases are verfed by.35. Characterstcs mpedance and admttance matrces are also defned by equaton.45. y z PM M Z Y M M Y M Z NN M = = =.40 z y N P N N Z Y N N Z MM Y N = = = t.41 } { 1 dag λ n λ λ γ K = = = y z z y.4 n j... 1, = + = = β α λ γ.43 = + + m n m x x m n m x x m n m I I e e I I e e I I n n M K M O M L M K M O M L M γ γ γ γ = = = N Z N N Y N Y Z 1 c c γ γ.45

40 .1.. Propagaton Matrx Dervaton 5 Wth the dervatons n secton.1..1, now we need to further characterze P. Ths can be done by enforcng the contnuty condton of the x-component of the E-feld at each ar-wre nterface [7]. Followng the reference, P can be expressed by nternal, external, and ground mpedance and admttance, as s gven by.46. Each of the mpedance and admttance terms are represented by.47,.49,.51,.5, and.53 where defntons n.48,.50,.54,.55,.56 and.57 are used. Z ( ˆ 1 1 Z + Z + Z )( Y + Yˆ ) 1 = e g e g P.46 μ w f I 0 ( jk wa ) = dag { Z 1... Z j... Z n}, Z j = ak wi1( jk wa ).47 I o : frst knd Bessel functon of zero order I 1 : frst knd Bessel functon of frst order 1/ ε σ k.48 jωμ 0 Z e = A.49 π h D j A ln, ln, [( ) ] 1/ = Aj = Dj = h + hj + Δj.50 a d w w k = 0 w k j 0 = ω μ0ε0 ε 0 ωε0 j Y = jωε πa 0 1 e.51 ˆ jωμ 0 Z g = F π Yˆ g = j 0π F.5 1 g 1 g ωε.53

41 0 F F 1 h + h j + jδ j + ξ 1 = h + h + jδ 1 g j ln = h + h j + jδ j + ξ 6.54 j j 3 g j ξ ln.55 h + h j + jδ j g k0 0 g, ξ = 3 =.56 k 0 + k g k0 k0 k g ξ 1 =, ξ k k k g 0 0 k + k ε g σ g = k 0 j.57 ε ωε.1..3 Per-unt-length Seres Impedance and shunt-admttance Matrces Dervatons Dervatons n sub-secton.1.. can be used to obtan propagaton matrx. However, to derve Z and Y themselves, more rgorous formulas should be appled for several reasons outlned n [7]. For ths reason, we need to evaluate t P expressed by.46 and Z by substtutng g F represented by.59 nto.58. Equaton.58 can be further 3g substtuted nto.46 to obtan Z. Also, Y can be calculated by.60, yeldng.61. g Y jωμ 0 1 t Z g = F1 g F3 g P.58 π jωε 0π h + jδ j + ξ 3 F 3 g j = ξ ln.59 h + jδ Y 1 g = j 0 π ( F g F3 g ) For numercal computaton purposes, we used a four-wre confguraton wth 10 meters above the earth and 70 centmeters spacng between wres; each wre s assumed j ωε.60 1 ωε.61 1 = j 0 π ( A + F g F3 g )

42 to have a cm dameter. The ground plane s characterzed by relatve permttvty of ε g =13 and conductvty of σ g =5 ms/m. The frequency spectra of four propagaton constants are computed by usng the method descrbed earler and the attenuaton constants are represented n Fgure (a) (b) Fgure -7: Frequency spectra of (a) Attenuaton constants, and (b) Phase constants of MTL system shown n Fgure-.7. Phase constants overlap on all the entre frequency range. On the contrary, attenuaton constants show dfferent behavor and values. Common mode shows hgher attenuaton over the frequency range and the attenuaton factors for the three aeral modes are close to one another. Common mode attenuaton factor ncreases up to some frequency and decays beyond. Ths ncdent s due to resonance phenomenon n ground medum, whch s ntally capactve and by ncreasng frequency t exhbts an nductve behavor. Ths phenomenon can be seen from Fgure -8, whch llustrates real and magnary parts of characterstc mpedance of a lne wth common mode njecton. The real part of characterstc mpedance s always characterzed by postve values, because

43 8 the lne s a passve system, but magnary part of characterstc mpedance s negatve up to one frequency and t becomes postve beyond that frequency. Ths transton n the magnary part of the characterstc mpedance s justfed by the followng argument: at low frequences, the dsplacement currents are nsgnfcant and ground plane acts as a good conductor. Therefore, the proper terms of the nput mpedance matrx of the lne are capactve and the magnary part of characterstc mpedance shows negatve values. On the contrary, at hgher frequences the earth can be consdered as a good delectrc and the proper terms of the nput equvalent mpedance matrx become nductve, whch makes the magnary part of characterstc mpedance postve. The null value s reached at the resonance frequency, correspondng to the mnmum of the real part. The change of behavor n attenuaton constant can also be explaned by means of skn depth. The skn depth of wre s expressed by.6 n [8]. The skn depth, δ, s related to the nverse of frequency, ω. Therefore, as frequency ncreases, the current s forced to the surface of wre, causng resstance of wre to ncrease and generate more loss. The aeral mode s just nvolved wth wres and loss n ths mode orgnates from the loss n wres, so the skn depth follows.6. Ths s the reason of ncreasng attenuaton factors n Fgure -7 for Aeral modes. δ c =.6 ωμ σ 0 c

44 9 (a) (b) Fgure -8: Frequency spectra of the (a) real and (b) magnary parts of characterstc mpedance of a mult-conductor lne wth common mode njecton. On the other hand, common mode propagaton s nvolved wth the ground, and skn depth of a delectrc lke ground s gven n [8] as: δ g = ω μ ε 0 g ( 1 σ g 1+ ( ) ωε g 1).63

45 At hgher frequences the skn depth s gven by: 30 σ μ.64 g 0 δ g ε g It s notced from equaton.64 that at hgh frequences, the skn depth s ndependent of frequency. Thus, the transmsson loss due to skn depth effect wll not ncrease beyond approxmately f t =σ g /πε g. There s however one addtonal effect whch s responsble for the decrease of attenuaton constant beyond f t. The axal electrc feld of a lne current s a Hankel functon of second order, H 0 (εr) where ε ncreases wth frequency and r s the dstance from wre and constant. At low frequences, εr s small and the decay of Hankel functon s slow, however, at hgh frequences the argument of Hankel functon becomes larger and makes t decay, more rapdly. The result s that electrc feld s more confned around the wre at hgh frequences and weaker feld strength reaches the earth, causng the earth s loss to decrease. Therefore, the total loss curve ncreases monotoncally pausng near the peak of the earth loss curve. It then decays at hgher frequences, as shown n Fgure -7 for the common mode attenuaton factor.. Powerlne network channel modelng..1 MTL crcut model Wth MTL theory from [1], t s possble to express the voltage and current at any node of gven network. The matrx equaton relatng the nput and output quanttes

46 of the l j -long j th lne secton between nodes j and k n an n-wre MTL network, s expressed n the followng form: 31 V I k k ( ω ) Φ 11 ( ω, l = ( ω ) Φ 1 ( ω, l j j ) ) Φ ( ω, l 1 Φ ( ω, l j j ) V ) I j j ( ω ) ( ω ).65 In whch V k, V j are the vectors of the wre-to-ground voltages and I k, I j are the vectors of the lne currents. In the prevous expresson Φ 11, Φ 1, Φ 1 and Φ are the n n matrx coeffcents of the transmsson matrx of the lne secton: t 1 φ 11 ( ω, l j ) = φ ( ω, l j ) = M cosh (ml j ) M.66 φ ( ω, l ) = M snh (m l )m M 1 Z.67 1 j j φ 1 ( ω, l j ) = Y Mm snh (m l j ) M.68 n whch: m = dag { m k }, m k = λ k k = 1,... n.69 snh (m l j ) dag { snh ( m l j )} = 1,... n =.70 cosh (m l j ) dag { cosh ( ml j )} = 1,... n =.71 Wth help of transmsson matrx coeffcents, the j th lne secton s then represented by means of a PI-type equvalent crcut, n whch the seres admttance matrces Y 1 and the two shunt admttance matrces Y a =Y b are, respectvely: Y 1 1 = φ1, Ya = Yb = φ1 (U φ ) wth U beng the n n unt dagonal matrx. Wth ths approach, each lne between two nodes s presented by a two-port PItype crcut model wth ther respectve transfer matrces. By dong so, the whole network

47 3 s consdered as a composton of several of such models and the analyss s straghtforward thereafter. For the source and load n the network more attenton needs to be pad because of the source s admttance nature (capactve or nductve) and load s mpedance. Further dscusson of ths matter s completely presented n [1]. Ths approach of modelng needs an n-depth knowledge of network crcuts theory and power engneerng prncpals... Multpath model Ths method of modelng s smlar to the wreless channels modelng and mostly used by researchers n communcatons feld whereas the complcated MTL method s more favorable for researchers n crcut theory. Earler, the voltage propagaton along the wre follows.1. Therefore, by havng the propagaton constant, one may easly fnd the frequency response of power lne wre at a desred pont on the conductor. As dscussed earler, each mode of couplng has a dfferent propagaton constant. Hence, there s a dfferent frequency response for each mode. Part of a propagatng sgnal reflects back to transmtter at the branch junctons due to the mpedance msmatch and the remander travels through [5]. Reflecton coeffcent s defned for each node as the rato of reflected sgnal power to the total receved sgnal power at the node. In the same way, the transmsson coeffcent s defned as the rato of transferred sgnal power to the total receved sgnal power at the

48 33 node. Obvously, the reflecton and transmsson coeffcents are equal or less than unty and the sum of all the transmsson and reflecton coeffcents at each node s unty. Reflectons cause sgnal propagaton not to take place along a sngle straght path from a transmtter to a recever n the power-lne network. Addtonal paths may also exst due to reflectons at the network junctons. Ths creates a multpath scheme wth frequency selectvty, smlar to a rado channel. When sgnal passes through a juncton, t wll be multpled by transmsson coeffcent of the juncton and when t reflects back from a juncton, t wll be weghted by reflecton coeffcent of that juncton. Therefore, each arrved path at a recever s weghted by a factor, g, whch s the product of reflecton and transmsson coeffcents of nodes along the path. As reflecton and transmsson coeffcents are equal or less than one, the weghtng factors are equal or less than unty, as well. For better explanaton, Multpath sgnal propagaton can be studed by a smple example, whch can be easly analyzed n Fgure -9 wth the help of Table -1. The lnk has only one branch and conssts of the segments (1), (), and (3) wth the lengths l 1,l,l 3 and the characterstc mpedances Z l1, Z l and Z l3. In order to smplfy the consderatons, A and B are assumed to be matched, whch means Z A =Z l1 and Z B =Z l. The remanng ponts for reflectons are C and D, wth the reflecton factors denoted as r 1C, r 3C, r 3D, and the transmsson factors denoted as t 1C, t 3C. These values are gven by: r 1 C = ( Z ( Z l l Z Z l 3 l 3 ) ) + Z Z l 1 l 1.73

49 34 r 3 C r 3 D = Z D l 3 =.74 ( Z ( Z Z l l D Z Z + l 1 l 1 Z Z l 3 ) ) + t 1 C 1 r1 C Z Z l 3 l 3.75 =.76 t 3 C 1 r 3 C =.77 Wth these assumptons, an nfnte number of propagaton paths s possble n prncple, due to the multple reflectons (.e. A C B, A C D C B, A C D C D C B, and so on). Each path has a weghtng factor, g, representng the product of the reflecton and transmsson factors along the path. The more transtons and reflectons occur along a path, the smaller the weghtng factor wll be, because, as mentoned earler, transmsson and reflecton coeffcents are less than one. Furthermore, longer paths exhbt hgher attenuaton, so that they contrbute less to the overall sgnal at the recevng pont. Due to these facts, t s reasonable to approxmate the bascally nfnte number of paths by fnte number of domnant paths. Fgure -9: Multpath sgnal propagaton; cable wth one tap.

50 35 Table -1: Sgnal Propagaton paths of the examned network n Fgure -9. Path No. Sgnal Path Attenuaton factor (g ) Path Length (d ) 1 A C B t 1C l 1 +l A C D C B t 1C.r 3D.t 3C l 1 +l 3 +l. N A C (D C) N-1 B t 1C.r 3D.(r 3C.r 3D ) (N-).t 3C l 1 +(N-1)l 3 +l Wth these weghtng coeffcents, we may express the network as a summaton of multple paths wth dfferent length and weghtng factors. The propagaton along a wre follows.1, so one can easly express the multpath network channel model as: N H ( f ) = = 1 g e α ( f ) d e jβ ( f ) d.78 where N s the number of sgnfcant arrved paths at the recever, d s the length of th path and g s the weght factor of the th path. Ths formulaton s bascally smlar to what has been mentoned n [9], however, wth a dfferent model for propagaton constant..3 Channel capacty concept Suppose a source sends r messages per second, and the entropy of a message s H bts per message. The nformaton rate s R = r H bts/second.

51 36 One can ntutvely reason that, for a gven communcaton system, as the nformaton rate ncreases the number of errors per second wll also ncrease. Surprsngly, however, ths s not the case..3.1 Shannon s theorem A gven communcaton system has a maxmum rate of nformaton C known as the channel capacty. If the nformaton rate R s less than C, then one can approach arbtrarly small error probabltes by usng ntellgent codng technques. To get lower error probabltes, the encoder has to work on longer blocks of sgnal data. Ths entals longer delays and hgher computatonal requrements. Thus, f R< C then transmsson may be accomplshed wthout error n the presence of nose. Unfortunately, Shannon s theorem s not a constructve proof t merely states that such a codng method exsts. The proof can therefore not be used to develop a codng method that reaches the channel capacty. The negaton of ths theorem s also true: f R > C, then errors cannot be avoded regardless of the codng technque used.

52 .3. Water-fllng 37 Accordng to [30] and by usng water-fllng n spectral doman process, we can express capacty lmt of a channel wth addtve Gaussan nose as: c = 1 log 1 + p N 0 ( f ) H ( f ) N 0 ( f ) H ( f ) + df.79 where p s the sgnal power at the specfc frequency and s chosen such that + N f 0 ( ) p df = P.. The notaton [X] + means Max {X,0} and P s the average H f ( ) transmtted power. In.79, N 0 (f) s the nose spectral densty n the system..4 Numercal results.4.1 MV lne networks For numercal computaton purposes, we used the four-wre confguraton employed before n subsecton Fgure -10 (a, b) depcts frequency response of a matched transmsson channel over a 1 km span MTL system wth the mentoned confguraton. As the system s matched, sgnal does not reflect at the recever-end and sgnal path s one straght pont-to-pont path. In ths case, the only loss comes from MTL path loss.

53 38 (a) (b) (c) Fgure -10: Frequency response of a matched MTL system for 1 Km span. (a) ampltude, (b) phase and ts assocated capacty lmts (c). Fgure -10(a) represents ampltude of frequency response for two couplng methods: common mode and dfferental mode 1. Common mode exhbts more loss than dfferental mode, especally at low frequences. As frequency ncreases, losses of the two

54 39 confguratons become comparable. Also, one may notce that both systems show a very low loss at hgh frequences over a 1 km repeater span. The fact that MV overhead power lnes resemble a low loss transmsson system shows promse for data delvery at hgh rates. Also, ths s a cause for concern, regardng potental nterference nto exstng servces, as elaborated on extensvely n NTIA reports [10]. Interference problem must be remeded, as stated earler. Fgure -10(c) llustrates the water fllng channel capacty lmts of.79 for ths system at dfferent transmtted power levels. For evaluatng channel capacty, we chose a unform -105 dbm/hz as a representatve of average nose spectral densty heght. Referrng to [8], ths value s a conservatve average estmate of practcal background nose level for MV power lnes n Korea. It s nterestng to see both dfferental and common modes couplng systems show almost the same capacty characterstcs, especally at hgh frequences. Ths s due the fact that both systems are approachng the same loss level at hgher frequences. Accordng to Fgure -10(c), wth an deal matched MTL system, over 50 MHz of channel band, we can delver almost 600 Mbps by launchng 10dBm transmt power. In realty, ths low loss nature of MTL systems degrades extensvely by several mparments. As mentoned earler, msmatch at junctons causes the travelng sgnal to reflect back creatng multpath. Ths can decrease the average channel capacty value. As an example, Fgure -11 (a) and (b) show frequency responses of a 1 km span MTL system wth msmatches at transmtter and recever, when coupled dfferentally. Reflecton factor at both ends s 0.3.

55 40 (a) (b) (c) Fgure -11: Frequency response of a msmatched MTL system for 1 Km span. (a) ampltude, (b) phase, (c) Channel mpulse response and (d) ts assocated capacty lmts. (d) Reflectons at junctons create resonance n frequency response. Ths s smlar to a cavty wth partal reflectng ends appled n oscllators. Also, due to the reflecton, not all the transmtted sgnal energy s receved by the recever, causng an average loss

56 41 ncrease by 3 db, compared to the deal case. Fgure -11(c) represents channel mpulse response and as t s seen, there are domnant paths from transmtter to recever. Fgure -11(d) shows channel capacty bounds of such system at dfferent transmt powers. The average channel capacty for 10 dbm transmt launch power at 50 MHz channel band, s now less than 500 Mbps. Fgure -1: The smulated complex network. Over an actual power-lne network, there always exst several branches and junctons between a transmtter and a recever. These branches cause nulls n transmsson channel frequency response due to multpath. To nvestgate ths phenomenon, we smulated the complex network shown n Fgure -1. In ths network we have three branches between transmtter and recever. Each end of these branches s an open-crcut, so reflecton factor at each end s one. Also, we have assumed that transmtter and recever mpedance s matched to that of the lne. Impedance of each branch s related to length of that branch and accordng to those mpedances each juncton has an assocated reflecton and transmsson coeffcent.

57 4 (a) (b) (c) Fgure -13: Frequency response of complex network shown n Fgure -1: (a) Ampltude, (b) Phase, (c) Channel mpulse response and (d) Lnes transmsson capacty bounds. (d) Our smulaton program performs and exhaustve search for all possble paths from the transmtter to the recever and elmnates those paths that have power less than 1% of straght path power. Fgure -13 (a) and (b) show ampltude and phase of complex

58 43 network frequency response. Reflectons create deep nulls n the frequency response. Smulaton shows there are 11 domnant paths and from Fgure -13 (c), 11 pulses wth dfferent arrval tmes are dstngushed. Delay spread n ths network s almost 3 mcroseconds. Fgure -13 (d) s the llustraton of the channel capacty lmts for ths complex network. The average capacty n ths network wth a 10 dbm launched transmt power level at 50 MHz band s about 400 Mbps. Obvously, the junctons and branches between transmtter and recever degrade the system performance extensvely compared to the deal pont-to-pont case..4. LV lne networks As t was dscussed before, for MV lne confguraton D amore s formulaton s not applcable wthn the desred frequency spectrum. In ths case, Carson s method [14] s more approprate. Power cables used for sngle-phase ndoor wrng are comprsed of three conductors whch one of them s the ubqutous earth ground; therefore, there are two modes of propagaton n such systems, common and dfferental. Smlar to MV lnes, the modelng of these lnes can be done ether by crcut theory approach or multpath method. The characterstcs of LV power-lne networks are very well known and there are a varety of research actvtes n ths area to explot dfferent features of LV grd. One of the most recent and comprehensve efforts of ths knd s done by Gall and Banwell [31][3]. Ths research uses MTL theory along wth the crcut theory method, to characterze the ndoor LV power-lne networks. In

59 44 mentoned references authors provde the smulaton result for a specfc channel and compare ther smulated channel model to the channel model gven by experment. These results are shown n Fgure -14. Although the smlarty of the two results s strkng and, more mportant, all of the frequency notches are accounted for, there are some dfferences n the two transfer functons, whch are caused by the dfference between the assumed characterstcs of wres n smulaton and the real characterstcs of wres n the experment. (a) (b) Fgure -14: The transfer functon of network shown n Fgure 7 of [31]. (a) Measurement, (b) Crcut theory smulaton

60 45 (a) (b) (c) Fgure -15: (a) Smulated frequency and (b) mpulse response of a LV powerlne network depcted n [31] by multpath method and (c) ts assocated capacty lmts Usng methods and algorthms of multpath method along wth Crason s formulaton, we smulated the same network confguraton used n [3]. The result of frequency response and mpulse response of such channel s llustrated n Fgure -15 (a) and (b). Our result and results n [3] are n agreement, as t s seen from Fgure -14 and

61 46 Fgure -15. It s seen from mpulse response of Fgure -15(b) that the maxmum delay spread s less than 1 mcrosecond and there are 4 sgnfcant paths from the transmtter to the recever. The capacty lmts of ths channel are depcted n Fgure -15(c). For evaluaton of these lmts we assumed an addtve unform background nose, wth 10 dbm/hz as spectral densty level. Accordng to [8], the background nose n LV networks has a smaller PSD level than n MV and has the average value around 10 dbm/hz. It s seen from Fgure -15(c) that the average capacty n ths network wth 10 dbm launched transmt power can reach 600 Mbps at 60 MHz.

62 Chapter 3 Nose and nterference Besdes the ampltude and phase dstorton, nose s the crucal factor nfluencng hgher data rates achevng over power lnes. The desgn of approprate codng and modulaton technques, by means of computer smulaton, requres detaled dscrete model of powerlne nose. Nose spectral densty level s also mportant parameter for evaluatng network performance and capacty. Moreover, Electromagnetc Compatblty (EMC) problem s an ssue that should be consdered n deployment of BPL systems. The man EMC problem s the emsson of electromagnetc nose, whch can nterfere wth publc rado. In ths chapter frst we dscuss the nature of nose both n MV and LV lne networks, then the EMC ssues are analyzed thoroughly. 3.1 Nose n MV lnes Over MV lnes, two types of nose are domnant; colored background nose and narrowband nose [8] Background nose Background nose s the envronmental nose, whch s hghly dependant on the weather, geography, above ground heght and etc. Fgure 3-1 s from [8], whch depcts a

63 48 measured background nose level n a MV powerlne test bed, and t s compared wth other nose levels, whch are measured from apartments and houses. Accordng to ths comparson, one can conclude that the nose level n medum voltage lnes s hgher than that n low voltage, approxmately by 0-30dBm/Hz n the frequency from 1MHz to 0MHz. Fgure 3-1: Comparson of nose levels between MV and LV lnes Background nose s always present at the communcaton channel and has dfferent level values n dfferent frequency bands Corona Nose Corona dscharge s a major cause of background nose, especally under humd and severe weather condtons [33]. When MV powerlne s n operaton, a strong electrc feld exsts n the vcnty of wres. Ths feld accelerates free electron charges n the ar surroundng the conductors. These electrons nteract wth the molecules n the ar and produce free electrons and postve ons. Ths process causes an avalanche, called corona

64 49 dscharge. The dscharge nduces current pulses n conductors wth random varatons of ampltude and separaton ntervals. Induced currents can be modeled wth current sources n powerlne system, as n Fgure 3-. Dscharges on the three dfferent phase conductors occur at dfferent tmes. Each tme when the voltage on a partcular phase s hgh enough, a corona burst occurs and nose s generated. Fgure 3-: Corona dscharges modeled wth shunt current sources Corona nose s permanent, wth ampltude dependng on supply voltage, geometrcal confguraton and bundle conductor composton of the lne and on weather condtons. In the carrer frequency range, corona nose s quas constant wth a slow decay. In short frequency ranges, t can be consdered as whte nose. CIGRE group study has nvestgated the nature of corona nose n [34]. In ths work, a new formulaton s gven for modelng of corona nose. Let I be the resultant current reachng the end of the th conductor, where t s measured and expressed as a voltage across a reference resstance R 0. As a consequence, the corona nose s defned as a voltage nose n db above 1μV and gven by 3.1. CN μ = I + 0 log R 0 3.1

65 n whch I s expressed n db above 1μA and R 0 n ohm. 50 Corona current, I, s evaluated by dstngushng the contrbutons dervng from generaton and propagaton phenomena: I = Γ + U 3. whch Γ, n db above 1 μa m, s the exctaton functon and U, n 1 db above 1 m, s the propagaton factor. Exctaton functon, Γ, havng the meanng descrbed n [35], depends on quanttes such as the maxmum electrc gradent g on the conductor surface, the radus r and the number n of conductors n the bundle, as well as the weather condton represented by factor K n. Many dfferent expressons derved for ths functon n lterature. In ths thess we use the expresson obtaned n [34] and s gven by 3.3. Γ = Γ o + Δb 1dB r Γ o = log + K n 3.4 g 3. 8 BW Δ b = 10 log 5kHz 3.5 BW s the consdered bandwdth for nose modelng. A random unform dstrbuton of corona sources njects currents, whch travel from the njecton pont along the excted th conductor. The propagaton factor of the th conductor of length L has the followng expresson:

66 where N s the number of conductors, ε 0 s the ar permttvty, G k and C k are the generc coeffcents of the current propagaton matrx and lne capactance matrx, respectvely. The procedure for calculatng U has been descrbed n [36] and [37]. Followng the procedures descrbed n [34], we obtaned the corona nose spectral densty of the MV lne network confguraton mentoned n chapter n a poor weather condton of 80% humdty. The result s depcted n Fgure 3-3. L U = 0 log σ dx 3.6 N 0 k = 1 1 σ = G k C k 3.7 πε 0 51 Fgure 3-3: Corona nose power spectrum n poor weather 3.1. Narrowband Nose Narrowband nose s the nterference from other narrowband wreless devces and servces n the frequency range of BPL systems, lke HAM or short-wave rados.

67 5 Narrowband nose dffers from tme to tme and place to place because the BPL frequency range s not occuped n all places by rado devces, unformly. Also, narrowband nose s tme dependant. In [8], ths nose s modeled as: NBN ( f ) = A B ( f f ) f B ( f ) = s n c ( ) 6 for -0. MHz< f <0. MHz where f s the th band centre frequency and A s the th band nose ampltude. These parameters needed to be measured for dfferent places n dfferent tmes. Fgure 3-4: Total nose power spectrum n an arbtrary powerlne system, operatng n poor weather Total nose s the sum of narrowband nose and background nose. For example, Fgure 3-4 shows the total nose spectrum for an arbtrary case, operatng n poor weather condton wth lmted amount of narrowband nterferer n the frequency bandwdth.

68 3. Nose n LV lne 53 Contrary to MV lnes, nose n LV lnes s more complcated and unpredctable. Accordng to [38] the addtve nose n LV lnes s categorzed to fve groups: Narrowband nose, Colored background nose, Synchronous perodc mpulsve nose, Asynchronous perodc mpulsve nose and Burst mpulsve nose. The frst one, Narrowband nose, s the same as of the MV lnes, whch was dscussed earler Colored background nose Background nose s caused by supermposng of multple sources of nose wth relatvely low power. Generally, power densty of background nose s between -10 db (V /Hz) and -140 db (V /Hz) wth an ncreasng power densty towards lower frequences (e.g. below 1 MHz). A typcal measurement result of background nose wth low power densty along wth the establshed model from [39] s llustrated n Fgure 3-5. Fgure 3-5: Background nose n LV lnes

69 54 Generally, we can see that power densty of background nose decreases by ncreasng the frequency. Results of multple measurements of nose n [40] showed that ths behavor can be approxmated by an exponental decayng curve n logarthmc scale. The same approach s found n [39]. Such a curve s also drawn n grey color nto Fgure 3-5. Power densty of background nose can be gven by A ( f ) = A + A 0 e f f Wth A, power densty s gven for, whle A 0 s gvng the dfference between A( ) and A(0). Wth the thrd parameter f 0 the decay rate of rse can be modeled. For the gven example, we have A = -136 db (V /Hz), A 0 = -38 db (V /Hz) and f 0 = 0.7 MHz. Ths model enables modelng background nose as a whte nose process, whch gets a spectral colorng by a flter. 3.. Synchronous perodc mpulsve nose Synchronous mpulsve nose occurs n 50 Hz-alternatng voltage current frequences of 50 Hz or 100 Hz. They are caused by net-synchronous power converters occurrng n dmmers and by all knds of rectfers usng dodes. Synchronous mpulsve nose can be characterzed by an envelope curve, whch can be gven by a perodcally repeated rectangular sgnal. Ths envelope gves ampltude, cycle duraton, and the wdth of an mpulse, as t s shown n Fgure 3-6.

70 55 Fgure 3-6: Synchronous perodc nose modelng Spectral characterstcs can be assumed as constant for a certan nose source. Wth ths assumpton, the former varaton n tme gets lost, but for examnng the mpact of nose on a communcaton system ths approxmaton s acceptable. The rectangular envelope of perodc mpulsve nose s depcted n Fgure 3-7. Ths envelope s characterzed by three parameters: ampltude of mpulse A, duraton of mpulse or mpulse wdth t B, and perod t P, whch corresponds to the recprocal value of repetton frequency. In practcal cases usually, we have t p >>t B. Fgure 3-7: Envelope curve of perodc mpulsve nose Imagne a rectangular sgnal r(t) wth the duraton t B and ampltude A. Then ts Fourer transform, R(f), s expressed by 3.11 and ther relaton s shown n Fgure 3-8. R sn( π ft ) B ( f ) = A t B 3.11 π ft B

71 56 Fgure 3-8: Rectangular envelope curve and ts Fourer transform of perodc mpulsve nose In rectangular envelope sgnal, r(t) s multpled by a tran of mpulses wth duraton t P resultng r p (t). From [41]the Fourer transform of ths perodc envelope, R p (f), s gven by 3.1 and ts frequency spectrum representaton s llustrated n Fgure 3-9. Fgure 3-9: Fourer transform of perodcally contnued envelope curve of perodc mpulsve nose t B sn( πft B ) n R p ( f ) = A δ ( f ) t ( πft ) t P B n= P 3.1 The synchronous perodc nose, n sp (t), results from multplcaton of colored background nose, n(t), by r p (t). n sp F ( t) = n( t). r ( t) N ( f ) = N ( f ) R ( f ) 3.13 p sp p

72 57 The spectrum of perodc mpulsve nose s no more dentcal wth the spectrum of the fundamental nose sgnal n(t), but t s convoluted wth the spectrum of the perodc rectangular sequence. However, as bandwdth of nose sgnal s very hgh compared to 1/t B and addtonally, the spectrum N(f) s very flat, therefore the falsfcaton of the perodc mpulsve nose s neglgble n practce Asynchronous perodc mpulsve nose Asynchronous mpulsve noses are manly caused by swtchng power supples. Generally, frequences are between 50 khz and MHz. As these noses occur only for a short tme and have relatvely low ampltude, t s very dffcult to measure and analyze them. Ther mpact on the measured nose spectrum and on communcaton systems can be best descrbed n ncreasng power densty of background nose. Thus, these knds of noses are consdered as a part of colored background nose [4] Burst mpulsve nose Ths knd of nose occurs tme-randomly. In lterature, ths nose s also known as asynchronous mpulsve nose [38]. Ths type of nose s caused by all knds of swtchng operatons, for example by household applances, electrc motors, or condenser dscharge lamps. Asynchronous mpulsve nose very often occurs n bursts, whch ncreases ther dsturbng mpact. Wth many dfferent sources ths nose has very dfferent propertes regardng tme response and spectral propertes.

73 58 We can state that t s suffcent for nose modelng to reproduce the spectral behavor. Ths can be acheved by rasng colored background nose durng tme of occurrence of the mpulsve nose. Thereby, the ampltude of nose determnes the extent of ths rasng. The model of asynchronous mpulsve nose s smlar to that of the perodc nose. Instead of a sequence of rectangles, we now use a complex tme sequence control, whch determnes occurrences of tme and duraton of sngle noses. Lke perodc mpulsve nose, mpulsve burst nose can also be characterzed by ts envelope wth three parameters as t s shown n Fgure These are ampltude A, pulse wdth t B, and pulse dstance t A,. Instead of pulse dstance, sometmes the tme dfference between the begnnngs of two consecutve mpulses s also used: nter arrval tme, t IAT. The man dfference to perodc nose s that the parameters ampltude, pulse wdth and dstance are dfferent for every sngle nose. Thus, t s decsve to set up a model, whch comprehends these parameters as realstc as possble. Fgure 3-10: Envelope curve of Asynchronous mpulsve nose For data transmsson va power lne networks mpulsve burst nose s especally mportant as ts ampltude and power densty generally exceeds those of the other nose by far. Ths could lead to loss of bg data packets.

74 59 Burst mpulsve nose can be characterzed by ampltude, pulse wdth, pulse dstance and ther spectral propertes. In contrast to perodc nose, these propertes are not constant, but dfferent for every sngle nose. Now, goal of modelng s to descrbe regulartes n those propertes wth statstcal methods. The dffculty s to cover the varety of characterstc parameters n a consstent and untary model. For modelng and for generaton of nose t makes sense to regard the envelope of the nose and ts spectral behavor separately. Extensve analyss of propertes of mpulsve nose has been done n [38]. Research for pulse wdth also can be found n [43] Impulse rate An mportant crteron for characterzaton of ntensty of mpulsve nose s the mpulse rate. The dffculty at evaluaton of measurements s the fact that mpulse rate s manly determned by perodc mpulsve nose. However, they should not be consdered for modelng burst mpulsve nose. Addtonally, mpulse rate essentally depends on detecton senstvty for mpulsve nose. Detecton senstvty specfes the voltage level, whch must be exceeded by the maxmum absolute value of an mpulse to be recognzed as nose. When level s rased, mpulse rate decreased dramatcally. Typcal values for average mpulse rate are between 0.1 mpulse per second for low dsturbed and 100 mpulses per second for hghly dsturbed envronments wth a detecton voltage level of 100mV. However, the mpulse rate of burst nose s temporarly much hgher [38].

75 3..4. Relatve dsturbng tme 60 Relatve dsturbng tme descrbes the fracton of tme, whch s dsturbed n average. Lke mpulse rate, the determnaton of ths parameter essentally depends on the detecton voltage level. Typcal values are specfed n [38] dependng on the envronment wth 0,001% up to 1% usng a detecton voltage level of 100mV. Thereby, perodc nose was also covered Impulse ampltude The dstrbuton of mpulse ampltudes follows an exponental dstrbuton wth good approxmaton. Dscrepances occur when few sources of nose are domnatng leadng to accumulaton at certan ampltudes. Impulse ampltudes above 1V are very rare Impulse wdth and spacng Impulse duraton (mpulse wdth) and dstance are also exponentally dstrbuted. Thereby, we can yet see that multple dstrbutons are overlappng. Typcal values for mpulse duraton are 100µs, typcal values for mpulse spacng are between 0.01s and 1s. Impulse duraton, mpulse dstance, relatve dsturbng tme and mpulse rate are not ndependent, but are nfluencng each other, makng generaton of a reference nose scenaro for channel smulaton more dffcult.

76 3..5 LV lne nose modelng 61 Modelng of mpulsve nose over broadband LV power-lne channels has been a challenge for researchers snce early 1980s. Several dfferent modelng methods are avalable. Zmmerman and Dostert n [44] propose one of the frst modelng methods for ths knd of nose at hgh frequences. In ther method, they use parttoned Markov models to characterze the nature of the mpulsve nose. Generally, we can establsh two dfferent methods for modelng LV powerlne nose: statstcal or tme-based model Statstcal nose model The statstcal modelng of mpulsve nose has been of nterest to researchers for a long tme. Mddleton n [45] and [46] categorzes mpulsve nose n two classes of A and B. The nose n BPL s consdered as class A Mddleton nose by most researchers n ths feld. Based on ths model, the nose, mpulsve plus background nose, s a sequence of..d random complex varables wth the probablty dstrbuton functon (PDF) of: α m z p Z ( z ) = exp( ) π σ σ m = 0 m m 3.14 wth The varance σ m s defned as: m A A α m = e 3.15 m!

77 6 where σ g and σ are the power of background nose and mpulsve nose, respectvely. The parameter A s called the mpulsve ndex, whch s the product of average rate of mpulsve nose and mean duraton of a typcal mpulse. For small A, we get hghly structured mpulsve nose whereas for large values of A, the nose PDF becomes Gaussan [45]. The parameter Γ s called background-to-mpulsve nose rato. By combnng equatons 3.16 and 3.17, the varance σ m can be expressed as: Equaton 3.14 shows that the PDF of nose s a weghted sum of Gaussan PDFs wth zero mean, therefore the mean and varance of nose can be acqured by the followng equatons: ( ) Γ + Γ + + = 1 ) ( A m g m σ σ σ 3.16 g σ σ = Γ 3.17 Γ + Γ = ) ( A m m σ g σ ).exp( 1. ). (. } { 0 = = = = = m m m m Z z z z dz z P z z E σ πσ α μ 3.19 ) (! ).exp( 1. } { 0 0 = = + Γ Γ = = = m m g A m m m m z A m m A e z z z E σ σ πσ α σ 3.0

78 63 Ths model gves a good and precse nsght about the PDF and Power Spectral Densty (PSD) of nose. However, t cannot be used for the tme representaton of mpulsve nose unless an nfnte-sze nterleaver s assumed. Ths s due to the fact that Mddleton model assumes the mpulsve nose as an..d process, whereas n practce mpulsve nose s a random sequence wth memory. An nfnte-sze nterleaver can guarantee the ndependency of the nose sgnals n tme Tme-based nose model In ths model, same as n [4], we use two smplfed Markov models to represent the burst errors caused by mpulsve nose. Ths model has two layers. The frst layer (hgher layer) descrbes the ncdence of the burst groups and the second layer (lower layer) artculates the sngle mpulse wthn the burst group. Each layer has ts own probablty transton matrx descrbng the Markov process. Our model s a realstc way of representng mpulsve nose n ths envronment. For hgher layer, the Markov model has two states of dsturbed and undsturbed. In the dsturbed state, a burst group of noses happens, whle n undsturbed state there s no mpulsve nose. Wthn the dsturbed state, we defne two other Markovan states wth lower tme resoluton: nose and no nose. Fgure 3-11 llustrates these two Markov models along wth ther transton probabltes.

79 64 (a) Fgure 3-11: Markov model for burst nose: (a) Modelng burst groups (b) Modelng sngle mpulses wthn a burst group (b) It's the most mportant to know the probabltes for the sngle states and especally for the sum of two states. Ths s an mportant crteron for desgnng transton probablty matrces for Markov models. The needed probabltes can be determned usng the statonary state probabltes. In an rreducble Markov-Chan, there s a statonary state probablty vector π to whch the state probabltes of the Markov Chan are convergng ndependently from the start state. In a two-state Markov chan, as n our case, ths matrx s gven by 3.1 p1 p1 π = [ π 1 π ] = [ ] 3.1 p + p p + p 1 A measurement n [47] shows that the average tme of the dsturbed state n the hgher level s 5ms and the average tme of undsturbed state s 1 second for a specfc LV lne network. The expected tme of the system n dsturbed state s gven as: E k 1 1 { t } t k.( 1 p ). p = t ms whch t r1 s the tme resoluton for the frst layer. If one consders t r1 to be 1ms, from 3., P 1, wll be equal 0.8, whch makes P 1,1 equvalent to 0.. Based on the average 1 = r 1. 1, 1, r p k = 1 1, 1,. = 1 1

80 65 tme of dsturbed and undsturbed states, the statonary state dstrbuton of the frst layer model s expressed as [47]: π 1 =[π 1,1 π 1, ]=[ ]. Thus, the transton probablty matrx for the frst level s: P 1 = The tme resoluton for the second layer s selected as t r =1 mcrosecond. The statonary state probabltes n the second layer are ndependent random varables wth equal probabltes, whch makes π =[π,1 π, ]=[ ]. We assume the average duraton of the dsturbed and undsturbed states are 50 mcroseconds. Ths wll gve the transton matrx for the lower layer as: P = Now that we have transton probabltes, t s nterestng to nvestgate the behavor of burst and mpulse rates. By the parameters n P 1 the burst rate s equal to 1 burst per second accordng to 3.5. n π 1,1 1 p E { t1,1} t r 1 1,11 1 = = 1 per second 3.5 The mpulse rate durng a burst nose s also calculated by equaton 3.6 as well as the parameters n P. n π,1 π, = + E{ t,1} E{ t, } (. p + π. ) = = π,1,1, p,1 per second t r 3.6 The total average mpulse rate s:

81 n B = π n 100 per second 3.7 1,. = 66 The two dscussed nose models are two dfferent representatons of the mpulsve nose. The margnal probablty statstcs of Markov-based model can be evaluated by Mddleton model, f approprate A and Γ correspondng to the Markov model are chosen. Mddleton model s a good model for the nose statstcs whereas the Markov-based models are good representatve of nose n tme doman. 3.3 Electromagnetc nterference Characterzaton of electromagnetc emssons assocated wth PLC systems s a complex problem due essentally to: Complcated and huge varety of possble confguratons of MV network n power transmsson system; Varaton of load for dfferent networks at dfferent tme; The defnton of an adequate measurement method for emsson; The defnton of adequate lmts n order to acheve a reasonable compromse between avodng nterference wth other equpment and not forbddng the use of ths new technque. Recently, NTIA, n ther extensve reports [10], made recommendatons to FCC to devse regulatory methods for measurements, deployment and smulaton of BPL systems. Accordng to ths report, there are some frequency ntervals that are dedcated to emergency servces and all BPL systems have to avod occupyng these frequency

82 67 ntervals. It s also noted n ths report that for evaluatng radaton patterns from powerlnes, both far and near feld should be consdered and nether can exceed FCC regulatory lmts Radaton theory bascs The electromagnetc radaton pattern of a sngle wre over ground s studed under the context of recevng (wave) antennas by H. Beverage n early 190 [48]. Afterward, several research attempts are conducted to fnd a mathematcal expresson for radaton pattern of ths knd of antennas. Wat n [17] and [49] provdes a comprehensve mathematcal nsght. In [49], Wat ntroduces two functons as Hertz Potental Functons to descrbe the radated electrc and magnetc felds. As, t s mentoned earler, lnes n powerlne networks are composed of three or more wres. Therefore, by usng MTL theory of [1] and Wat s method, we can develop mathematcal expressons for radated felds. Followng equaton.35, the current of the th -wre at dstance x from the source can be wrtten as a functon of modal currents n the followng form: m m I ( x) = N 1 I1 ( x) +... N n I n ( x) 3.8 Assumng the matched condton, the k th modal currents s descrbed by: where m I ( 0 ) s the value at x=0. k I k m k ( x) = I ( 0 )exp( γ x) 3.9 m k

83 68 Followng the steps mentoned n [49], the Hertz Potental Functons generated by the k th mode current from the th wre are the followng: The defntons of ( j) k S and ( j) k Λ are gven n [49]. In [49] the feld components are gven as functons of Hertz Potental Functons. By takng the dervatves and mathematcal procedures the radated feld components of k th modal current over the th wre are expressed as: ) ( )] ( [ ),, ( ) ( ) ( ) ( x I N S k S k k j z y x m k k k k k k k Ek γ γ Λ π ωε Π = 3.30 ) ( ) ( ) ( ),, ( ) ( ) ( x I N S S k j z y x m k k k k k k Hk = γ π γ Π 3.31 ) ( ) ( ) ( ),, ( ) ( ) ( ) ( x I N S S k j k z y x E m k k k k k k Ek k k x γ Λ ωπε Π γ + + = + = 3.3 ) ( )] ( [ ) ( ),, ( ) ( ) ( ) ( ) ( x I N S S S k j z j x y z y x E m k k k k k k k k Hk Ek k y = = γ Λ ωπε γ Π ωμ Π 3.33 ) ( )] [ ) ( ),, ( ) ( ) ( x I N S y j x z z y x E m k k k k k Hk Ek k z = + = Λ ωπε γ Π ωμ Π 3.34 ) ( ) ( ) ( ),, ( ) ( ) ( x I N S S j k z y x H m k k k k k Hk k k x = + = π γ Π γ 3.35

84 69 H z k H y k Π ( x, y, z ) = ( y x j ( 3) = [ Λk S π Π Hk ( x, y, z ) = ( jωε z x j ( 4 = [ Λk π ) + j( k 0 S Hk ( 5 ) k ( 9 ) k 0 Π + jωε 0 z )] N Π y k + γ S k Ek I m k ) ( 6 ) k ( x) S Ek ( 7 ) k ) )] N k I m k ( x) All the aforementoned equatons are based on the assumpton that the wavelength of electromagnetc wave nsde the conductors s a small fracton of the heght of wres and the spacng between the wres s a small fracton of wavelength. Wth ths assumpton, we can consder that the modes are orthogonal and equaton.35 s vald. For hgher frequences contnuous modes appear and assumpton of dscrete modes s not suffcent anymore. The x-component of the electrc feld due to th wre current s obtaned as sum of the n contrbutons n 3.3 related to n propagaton modes. The total x-component of the electrc feld radated by n-conductor system s the followng: x n n E ( x, y, z) = E ( x, y, z ) 3.38 = 1 k = 1 Smlar expressons are defned for all the components of the E and H felds. xk 3.3. Smulaton results The mentoned mathematcal method s appled to the calculaton of the hgh frequency electromagnetc feld radated from carrer channels on the MV powerlne

85 havng the geometrcal confguratons shown n Fgure 3-1. The ground return path s characterzed by conductvty σ g =5 ms/m and relatve permttvty ε rg = Fgure 3-1: Geometrcal confguraton of the smulated MV Powerlne The same confguraton s also used by GNEC [50] to evaluate the radated electromagnetc felds from the system. GNEC uses numercal algorthm to solve the ntegrals wthout any assumptons or constrants. The ampltude of the electrc and magnetc feld components are computed at pont x=500m along the lne from the exctng pont, z=10m and y=m above the ground at dfferent frequences. At frst, the conductor-to-ground couplng s consdered. The channel s excted by a voltage source V(0) havng the rms value of 1V n the frequency range of 1 to 100 MHz.

86 71 (a) (b) (c) (d) (e) Fgure 3-13: Ampltude frequency spectra of radated felds from the MV confguraton of Fgure 3-1 computed by GNEC and establshed mathematcal method for wre -to-ground couplng: (a) E y, (b) H z, (c) E z, (d) H y, (e) E x, (f) H x (f)

87 7 The obtaned results are shown n Fgure From these fgures, t s seen that the domnant components are E y and H z. The peak value of these domnant components occurs at 3MHz, whch corresponds to resonance frequency of ground medum, as t s dscussed n Chapter. In ths frequency, the behavor of ground medum changes to nductve, whch was ntally capactve at low frequences. It s also seen that the numercal results of GNEC and the developed method are n agreement, specally at low frequences. As frequency ncreases, the dfference between the results of two methods becomes consderable. Ths s to due to the fact that the assumptons of mathematcal method are not applcable at hgher frequences wth ths confguraton; therefore the numercal method provdes more accurate results. The ampltude spectra of the radated electromagnetc feld components, generated by the wre1-to-wre couplng wth V 1 (0)=0.5 V and V = -0.5V, are represented n Fgure In ths confguraton the dfferental mode s excted and as t s shown, the ampltude of radated components are much less than the case of the common mode couplng. Ths ncdent s due to the cancellaton of radated electromagnetc feld caused by two opposte current flows of forward and return paths. In ths fgure the GNEC results are depcted, as well. The same argument, mentoned for last results s applcable here: as frequency ncreases, the mathematcal method s assumptons are not vald, thereby the dfference between two results s sgnfcant at hgh frequences.

88 73 (a) (b) (c) (d) (e) Fgure 3-14: Ampltude frequency spectra of radated felds from the MV confguraton of Fgure 3-1 computed by GNEC and establshed mathematcal method for wre 1-to-wre couplng: (a) E y, (b) H z, (c) E z, (d) H y, (e) E x, (f) H x (f)

89 74 The lateral profles of the feld components for conductor -to ground and wre 1- to-wre couplngs are computed at x=500m and heght y= m above the ground surface. Fgure 3-15 llustrates these profles at frequency 10 MHz for the confguraton of Fgure 3-1. (a) (b) (c) Fgure 3-15: Lateral profle of the feld components at 10 MHz for the (a) and (b) Wre -to-ground (c) and (d) wre 1-to-wre couplngs (d)

90 3.3.3 Load mbalance effect on radaton pattern 75 In the subject range of frequences, MHz, BPL devces and the powerlnes that carry BPL sgnals have the potental to act as unntentonal radators. The amount of radaton depends on the symmetry of the network at rado frequences. Symmetry s defned n terms of mpedance between conductors and ground. If for a twowre lne, the mpedance between each conductor and ground s equal, the lne s symmetrcal or balanced. A lack of symmetry leads to an unwanted, common mode sgnal. Common mode currents flow n parallel n both conductors, whle return portons flow through ground. Balanced lnes are necessary for dfferental mode transmsson n whch currents are equal n magntude and flow n opposte drectons on the sgnal conductors. The felds radatng from these conductors tend to cancel each other n the far feld area. On parallel or nearly parallel, non-concentrc conductors, common mode currents at rado frequences produce more radaton than dfferental mode currents. Fgure 3-16: The powerlne confguraton for radaton pattern smulaton

91 76 For smulatng the mpact of loads on radaton pattern, we used GNEC to evaluate far feld radaton patterns along the wre for the powerlne system confguraton shown n Fgure In our smulatons, we examned both common and dfferental mode njectons. In common mode, sgnal can be njected ether on mddle conductor or on sde conductors. The far feld radaton patterns for dfferental and two common modes at 10 MHz are depcted n Fgure Horzontal axs unt s μv/m. The far feld radaton pattern has hgher values n both common mode njectons than n dfferental mode. In dfferental mode njecton, reverse and forward paths are two current flows wth the same ampltude n opposte drectons, as t s shown n Fgure -4. These currents are travelng n two parallel dentcal conductors wth relatvely small separaton dstance. Due to ther opposte drecton and small dstance, ther radated electrc feld tend to cancel out one another at far feld. On the other hand, n common mode njecton, forward path s a current flow n one phase conductor and the return path s earth s surface, whch s a lossy delectrc. The dstance between one phase and earth s at least ten tmes of the separaton of two phases. Also, n common mode njecton, forward current s travelng n a near-perfectly conductng materal, whereas return current transmts through a lossy materal. Because of these facts, t s expected that the cancellaton of electrc felds, emtted from forward and return paths, at far feld, n common mode njecton, s much less than those n dfferental mode njecton. We should keep n mnd that ths cancellaton degrades n dfferental mode f the load mpedances between each wre and earth are not equal. As t s seen from Fgure 3-17 (a) and (b), the radaton pattern from central common mode njecton s symmetrc but t s

92 77 not true for radaton pattern of the sde njecton. It s due to the fact that the envronment around the njected wre n central njecton s symmetrcal, whch s not the case for sde njecton. (a) (b) (c) Fgure 3-17: Far feld radaton from powerlne confguraton n Fgure 5.1 for (a) central common mode, (b) sde common mode and (c) dfferental mode njectons.

93 78 (a) (b) (c) Fgure 3-18: Radaton pattern from an unbalanced powerlne wth (a) 10 mh n seres dfference, (b) 1 mf n parallel dfference and (c) 1 μf n parallel dfference. Fgure 3-18 represents far feld radaton patterns from powerlne confguraton shown n Fgure 3-16 for dfferental modes and three dfferent load msmatches. From these fgures and Fgure 3-17(c), one can conclude that load msmatch between two lnes

94 79 degraded the far feld cancellaton n dfferental mode njecton. Therefore, the radaton pattern from the dfferentally njected powerlne wth load msmatch has hgher ampltude than the radaton pattern of dfferentally njected powerlne wth balanced loads. The more the msmatch ncreases, the more radaton s generated by powerlne system.

95 Chapter 4 Modulaton Technques for BPL systems The choce of the modulaton technque for a gven communcatons system strongly depends on the nature and the characterstcs of the medum n whch t has to operate. The powerlne channel presents hostle propertes for communcatons sgnal transmsson, such as nose, multpath and strong channel selectvty. The selecton of a modulaton scheme for BPL must account for three major factors: 1- The presence of nose and mpulse dsturbances causng a relatvely low Sgnal-to-Nose Rato (SNR) - The tme-varyng frequency selectve nature of the channel 3- Regulatory constrants wth regard to EMC lmt the transmtted power. A choce should be made of ether a robust soluton,.e. one provdng suffcent qualty for a wde range of varatons of the model parameters, or an adaptve one. The problem s more complcated for the home networkng by the need to make powerlnebased home networkng cost-compettve wth other wred or wreless solutons.

96 4.1 Sngle-carrer modulaton 81 Sngle-carrer modulaton can be an attractve proposton from the complexty pont of vew. However, snce the powerlne channel s hghly nonlnear, strong Inter Symbol Interference (ISI) s nevtable. Thereby, powerful detecton and equalzaton technques needed to be deployed for ths technque to perform effectvely. The deep frequency notches n the powerlne channel transfer functon prevent the use of lnear equalzers, snce the nose enhancement that these notches cause can be a serous drawback on a hghly nosy channel lke powerlne channel. Instead, for these channel nonlnear Maxmum A Posteror (MAP) or sequence Vterb detectors can be used. However, the computatonal complexty of these detectors ncreases exponentally wth the length of the mpulse response, whch s consderable n the powerlne channel. Decson Feedback Equalzaton (DFE) mght also be attractve. The problem of complexty stll exsts wth DFE n ths hghly nonlnear channel. Moreover, burst nose can cause catastrophc error propagaton n DFE, whch makes the equalzer s performance degrade sgnfcantly. To counteract ths, DFE scheme mght be drven by soft decson feedback algorthm, whch adds to the equalzer complexty. For comparson purposes, we smulated a sngle carrer communcaton system for the channel of Fgure -13 wth the bt rate of 90 Mbts/sec. For ths system, we used a hard decson DFE equalzer wth 50 taps to mtgate the ISI. The performance of such a system s llustrated n Fgure 4-1. By ncreasng the SNR, after some pont, the sngle carrer system no longer mproves the overall performance. It s due to deep notches n

97 the channel transfer functon causng the lnear equalzer to enhance the nose more than elmnatng ISI [51]. 8 Fgure 4-1: System performance of a sngle-carrer system wth 50-tap DFE Based on ths dscusson and result, we can conclude that conventonal snglecarrer modulaton s not a good canddate for BPL communcatons systems. 4. Spread spectrum Spread spectrum s a type of modulaton that spreads data to be transmtted across the entre avalable frequency band, n excess of the mnmum bandwdth requred to send the nformaton. The frst spread-spectrum systems were desgned for wreless dgtal communcatons, specfcally n order to overcome the jammng stuaton, that s, when an ntruder ntends to dsrupt the communcaton. To dsrupt the communcaton, the ntruder needs to detect that a transmsson s takng place and then transmt a jammng sgnal that s desgned to confuse the recever. Therefore, to mtgate the

98 83 jammng communcatons system must be able to make these tasks as dffcult as possble. Frstly, the transmtted sgnal should be dffcult to detect by the ntruder, and for ths reason the transmtted spread-spectrum sgnal s mostly called nose-lke sgnal. Secondly, the sgnal should be dffcult to dsturb wth a jammng sgnal. Spread spectrum orgnates from mltary needs and fnds most applcatons n hostle communcatons envronments; such s the case n the BPL envronments. Its typcal applcatons are the wreless cellular phones, wreless LANs, and Bluetooth. In some cases, there s no central control over the rado resources, and the systems have to operate even n the presence of strong nterferences from other communcaton systems and other electrcal and electronc devces. In ths case, the jammng s not ntentonal, but the electromagnetc nterferences from the communcatons system may be strong enough to dsturb the communcaton of the other systems operatng n the same spectrum. The prncple of the spread spectrum s llustrated n Fgure 4-, where the orgnal nformaton sgnal, havng a bandwdth B and duraton Ts, s converted through a pseudo-nose sgnal nto a sgnal wth a spectrum occupaton W, wth W>> B. The multplcatve bandwdth expanson can be measured by a spread-spectrum parameter called Spreadng Factor (SF). For mltary applcatons, the SF s between 100 to 1000, and n the UMTS/W-CDMA system the SF les between 4 and 56. Ths parameter s also known as spreadng gan or processng gan and s defned by: W G = = W. T S 4.1 B

99 84 Fgure 4-: Prncple of bandwdth spreadng Ths spectral spreadng benefts the communcatons system n the followng aspects: 1- Low spectral power densty of the transmtted sgnal. Ths feature s advantageous wth respect to applcatons, where narrowband systems workng n the same frequency band as the spread spectrum system shall not be severely dsturbed by the latter, or where t s mportant to hde the own sgnals n the envronmental nose n order to acheve a low probablty of ntercepton (LPI). Ths feature s partcularly attractve for BPL deployment because several regulatory ssues lmt the transmtted sgnal. - Reducton of band lmted nterference. Informaton theory tells us that the mpact of nterference s not so much determned by the total power of the nterferng sgnal, but rather by the spectral power densty

100 85 generated by the nterferer n the frequency band occuped by the dsturbed sgnal. For a gven lmted power of the nterferer, the processng of the spectrally spread desred sgnals at the recever has the effect, wth respect to the nterferer, of a smultaneous spectral spreadng of the nterferer power, whch benefcally reduces the effectve spectral power densty of the nterferer. 3- As addtonal advantages of spectral spreadng, one should menton the nherent frequency dversty, whch helps to combat the detrmental effects of frequency selectve channels n nformaton transmsson systems, and an mproved temporal resoluton n the case of channel dentfcaton or tme estmaton systems. Unfortunately, the mechansm of nterference reducton does not work f the bandwdth of the band lmted nterferer s much larger than the bandwdth of the spectrally spread desred sgnal, or f the nterferer would have a spectral power densty non-vanshng over all frequences. Ths case s very common n BPL wth mpulsve nose due to the lmted avalable bandwdth for transmsson and broad frequency spectra of mpulsve nose. In ths case, a burst of nose happens and spread spectrum system fals to operate whle the mpulsve nose exsts. To take full advantage of the nterference suppresson of spread spectrum technques, a szable bandwdth expanson s needed, whch may severely lmt the maxmum data rate for a gven transmsson bandwdth. Moreover, channel equalzaton s necessary n ths system, whch from the last subsecton we know t s hard to mplement.

101 86 There are dfferent methods of spreadng technques used n lterature. Here we ntroduce two common technques: Drect Sequence Spread Spectrum (DSSS) and Frequency Hoppng (FH) Drect Sequence Spread Spectrum Drect Sequence Spread Spectrum (DSSS) s the most commonly appled form of the spread spectrum n communcatons systems. To spread the spectrum of the transmtted sgnal, DSSS modulates the data sgnal by a hgh rate pseudorandom sequence of phase-modulated pulses before mxng the sgnal up to the carrer frequency of the communcaton lnk. In the DSSS transmtter demonstrated n Fgure 4-3, the nformaton bt stream a[n], whch has symbol rate of 1/T b and an ampltude between +1 and -1, s converted nto a contnuous sgnal a(t) through a smple Pulse Modulaton Ampltude (PAM). To spread the spectrum of the nformaton sgnal a(t), t s then multpled by an unque hgh rate dgtal spreadng code c(t) that has many zero crossngs per symbol nterval wth perod T c. For the generaton of the spreadng sgnal c(t), frst a code sequence c[m] s generated by a Pseudo-Nose Sequence (PNS) generator wth a frequency 1/T c and then modulated through PAM.

102 87 a[n] PAM a(t) s(t) c[m] PAM c(t) E T b b Cos ( πf t ) c Fgure 4-3: A DSSS transmttser Dfferent sngle-carrer modulatons can be used to push the spread sgnal to the hgh frequency, such as BPSK, QPSK, or the M-PSK. By consderng the DSSS transmtter based on BPSK modulaton n Fgure 4-3, the sgnal carrer has a peak ampltude of E T b b, where E b s the energy per nformaton bt. Then the transmtted sgnal s(t) can be wrtten as [5]: where E b s( t ) = cos( πf ct ) a ( t ) c( t ) 4. T b a( t ) = a[ n] T ( t nt b ) n = c( t ) = c[ m ] T ( t mt c ) m = T denotes a unt ampltude rectangular pulse wth duraton T. By takng T c =T b /N, after modulaton, the transmtted sgnal has a bandwdth of N/ T b. Ths means that the bandwdth of the transmtted sgnal s N tmes wder than the bandwdth of the orgnal nformaton sgnal. Then, the spreadng factor s equal to N. b c

103 88 At the recever sde, demodulaton and a de-spreadng operaton are utlzed to recover the orgnal sgnal. In the case of deal channel, the recever s just a downmxng stage followed by a flter whch s matched to consecutve T b -segments of c(t), a so-called code matched flter. The multplcaton by the demodulatng sgnal wth frequency f c turns the sgnal back to ts baseband form. Then a code sequence c(t) dentcal to the one generated n transmtter have to be generated at the recever and multpled wth the baseband sgnal. If a good synchronzaton between the two codes sequences s provded, ther correlaton wll be equal to one. In ths case, after submttng the baseband sgnal to the correlator, we get, at ts output, a sgnal ă(t), whch normally s smlar to a(t). The obtaned sgnal s then sampled at a rate 1 / Tb and a decson or estmaton about the orgnal ampltude of sample, ether +1 or -1, s made n order to buld the orgnal bt stream a[n]. Ths scheme of the recever where a matched flter s mplemented wth a correlator s llustrated n Fgure 4-4. Code-matched flter r(t) ă(t) (.) dt sgn{.} t=nt b ă[n] Cos( πf ct) c(t) Fgure 4-4: A DSSS recever wth a matched flter For a non-deal channel, an estmaton of channel s needed to resolve the dfferent arrved paths of the multpath lnk. Ths knd of recever s called Rake recever. Rake recevers can resolve a lmted number of paths. The complexty of Rake recevers

104 89 ncreases wth the number of paths that they can resolve. Therefore, we can expect for the hostle channel of BPL system a very complcated Rake recever s needed to acheve acceptable performance. In [53] DSSS was tested for transmsson n the band 4-0 MHz. In that study t s mentoned that a Rake recever was experenced to fal f proper equalzaton was not provded. Fgure 4-5 shows the performance of a DSSS system under the channel condton of Fgure -13 wthout mpulsve nose. In ths system the spreadng factor s 8 and 8- PSK modulaton s used to acheve the aggregate bt-rate of 48 Mb/s n the bandwdth of MHz. A very complcated Rake recever s used wth 100 taps to resolve 9 paths. The synchronzaton s assumed to be perfect. Fgure 4-5: System performance of a perfectly synchronzed DSSS system wth a complcated Rake recever

105 4.. Frequency Hoppng 90 In a Frequency Hoppng Spread-Spectrum system (FHSS) the sgnal frequency s constant for specfed tme duraton, referred to as a tme chp T h. The transmsson frequences are then changed perodcally. Usually, the avalable band s dvded nto non-overlappng frequency bns. The data sgnal occupes one and only one bn for a duraton T h and hops to another bn afterward. It s frequently convenent to categorze FH system as ether fast-hop or slow-hop, snce there s a consderable dfference n the performance for these two types of systems. A fast-hop system s a system n whch the hoppng frequency 1/T h, s greater than the message bt rate l/t b. In a slow-hop system, the hop rate s less than the message bt rate. The block dagram of a FHSS transmtter and ts correspondng recever are presented n Fgure 4-6 and Fgure 4-7, respectvely. In the FHSS system, FSK modulaton schemes, whch allow non-coherent detecton, are usually employed, because t s practcally dffcult to buld coherent frequency syntheszers [5]. Accordng to the generated pseudorandom sequence code, the frequency syntheszer generates a sgnal wth a frequency among a predefned set of possble frequences, whch has to carry the baseband sgnal over the transmsson channel. For M-ary FSK, the data sgnal can be expressed as n = a( t ) = P ( t nt ) cos( πf t + φ ) 4.5 Tb b n n where f n Є {f s0, f s1, f sm-1 } and P s the avenge transmtted power.

106 91 a[n] Hghpass Data Modulator a(t) s(t) flter h(t) Frequency Syntheszer FH code clock Code generator Fgure 4-6: Transmtter for FHSS r(t) Hghpass flter Hghpass flter h(t) Frequency Syntheszer FH code clock Code generator Data demodulator ǎ[n] Fgure 4-7: Recever for FHSS The frequency syntheszer outputs a hoppng sgnal s gven by: m = h( t ) = ( t mt ) cos( πf t + φ ) 4.6 Th h m m where f' m Є {f' h0, f' h1, f' hl-1 }. In ths case, there are L frequency bns n that FHSS system. Let T b = N T h be the constrant for fast hoppng, whch becomes T h = N T b n case of slow hoppng. The transmtted sgnal n a fast hoppng FHSS s gven by:

107 s t) = P T ( t mth Δ )cos[( πf )( ) ] m + πf m t Δ + φ h m + φ 4.7 ( m N N m= And n the case of slow hoppng, the transmtted sgnal s: 9 s t) = P Tb ( t ntb Δ )cos[( πf )( ) ] n + πf n t Δ + φ n + φ 4.8 ( m N N n= where x s the largest nteger whch s smaller than or equal to x and Δ s a unform random varable on [0, T b ). The requrement of orthogonalty for the FSK sgnals forces the separaton between the adjacent FSK symbol frequences to be at least 1/T h for fast hoppng, and 1/T b for slow hoppng. Therefore, the mnmum separaton between adjacent hoppng frequences s M/T h for fast hoppng, and M/T b for slow hoppng. At the FHSS recever sde, the man task s to regenerate a pseudorandom sequence that must be smlar to the one generated at the transmtter, and accordng to whch the modulaton of the sgnal n the hgh frequency was acheved. Ths should allow a correct demodulaton of the transmtted sgnal. However, t s mportant to note that another demodulaton has to follow, n our example, accordng to the M-ary FSK. Then the recovered sgnal has one of the M possble frequences and ths should allow a correct estmaton of the value of a[n] at each tme perod T b Comparson of FHSS and DSSS The comparson can be evaluated by dfferent parameters, such as the spectral densty reducton, nterference susceptblty or capacty. Both DSSS and FHSS reduce the average power spectral densty of a sgnal. For an optmal system realzaton, the

108 93 objectves are to reduce both transmtted power and power spectral densty, to keep them from nterferng wth other users n the frequency band. DSSS spreads ts energy by tmespreadng the sgnal. Therefore, nstead of havng all the transmtted energy concentrated n the data bandwdth, t s spread out over the spreadng bandwdth. The total power s the same, but the spectral densty s lower. Of course, more channels are nterfered wth than before, but at a much lower level. Furthermore, f the spread sgnal comes n under the nose level of most other users, t wll not be notced. On the other hand, FHSS sgnals lower only ther average power spectral densty by hoppng over many channels. But durng one hop, a FHSS sgnal appears to be a narrow band sgnal, wth a hgher power spectral densty. The nterference susceptblty s another mportant parameter, whch allows BPL system to operate approprately. In DSSS recevers, the de-spreadng operaton conssts n multplyng the receved sgnal by a local replca of the spreadng code. Ths correlates wth the desred sgnal to push t back to the data bandwdth, whle spreadng all other non-correlated sgnals. After the de-spread sgnal s fltered to the data bandwdth, most of the nose s outsde ths new narrower bandwdth and s rejected. Ths helps only wth narrowband and uncorrelated nterference, and t has no advantage for wdeband nterference snce spread nose s stll nose and the percentage that falls wthn the data bandwdth s unchanged, whch s the case for mpulsve nose n LV networks. The FHSS sgnal s agle and does not spend much tme on any one frequency. When t hts a frequency that has too much nterference, the desred sgnal s lost. In a packet swtched system, ths results n a retransmsson, usually over a clearer channel. In a fast enough FHSS system, the porton of lost sgnal may be recovered by usng a FEC.

109 94 Based on above dscusson, we can conclude that DSSS s more approprate than FHSS n hostle channel condtons of BPL. However, some practcal ssues wth spreadspectrum, such as low bt rate, complexty and synchronzaton n BPL system reman unsolved. Furthermore, LV networks are hghly polluted wth mpulsve nose, whch makes the spread-spectrum performance degrade severely. The man advantage of spread-spectrum s ts EMC, by the radaton of very weak electromagnetc felds n the envronment. Mentoned reasons caused researchers to lose ther nterest for usng spread-spectrum n BPL. 4.3 Orthogonal Frequency Dvson Multplexng (OFDM) Frequency Dvson Multplexng (FDM) has been a wdely-used technque for sgnal transmsson n frequency selectve channels. Essentally, FDM dvdes the channel bandwdth nto sub-channels and transmts multple relatvely low rate sgnals by carryng each sgnal on a separate carrer frequency so that on each carrer ISI s elmnated or at least mnmzed. To facltate separaton of the sgnals at the recever, the carrer frequences are spaced suffcently far apart so that sgnal spectra do not overlap. Further, n order to separate the sgnals wth readly szeable flters, empty spectral regons are placed between the sgnals. As such, the resultng spectral effcency of the system s qute low. In order to solve the bandwdth effcency problem, orthogonal frequency dvson multplexng (OFDM) was proposed, whch employs orthogonal tones to modulate the sgnals [54]. The tones are spaced at frequency ntervals equal to the symbol rate and are

110 95 capable of separaton at the recever. Ths carrer spacng provdes optmum spectral effcency. Although OFDM was proposed n the 1960 s t was not wdely employed untl the 1990 s, largely because of sgnfcant crcut desgn ssues. Today, OFDM s a major contender for 4G wreless applcatons wth sgnfcant potental performance enhancements over exstng wreless technology. OFDM s now well-proven technque n applcatons such as audo broadcastng (DAB) and asymmetrc dgtal subscrber lne (ADSL). Furthermore, t s a canddate for dgtal vdeo broadcastng (DVB). OFDM proves to be robust aganst channel delay spread, because t s strongly related to FH technque OFDM system concept The basc dea of OFDM s to splt a hgh rate data stream nto a number of lower rate streams and transmt these streams smultaneously, and n parallel over a number of orthogonal subcarrers. The orthogonalty of subcarrers guarantees that the streams do not nterfere wth one another. It s possble that subcarrers lose ther orthogonalty due to multpath or channel non-statonary behavor. In ths case, sub-carrers nterfere wth one another and cause nter-carrer nterference (ICI). A very general block dagram realzaton of an OFDM system s depcted n Fgure 4-8. Each data stream s sent n one subchannel, so that each has ts own coherence bandwdth. If the bandwdth of transmtted sgnal s less than coherence bandwdth of the channel, nter-symbol nterference (ISI) s elmnated on that channel. Lower data rate streams have lower bandwdth and therefore, f we choose enough subcarrers, we wll be

111 able to have very low-rate parallel data streams n each such that each subchannel wll be ISI free. 96 d 0 d[n] Mappng S/P d 1 IFFT Channel FFT P/S De- Mappng ~ d[n] d N-1 Fgure 4-8: General block dagram realzaton of an OFDM system The number of bts assgned to each sub-carrer s varable based on the varablty of sgnal to nose rato across the frequency range. Optmzaton of ths bt assgnment wll be detaled n further sectons. The number of sub-carrers, N, used n an OFDM system s chosen as a trade-off between the frequency offset of adjacent carrers and the adjacent channel nterference. A greater number of sub-carrers mples less adjacent channel nterference, but ncreased susceptblty to frequency offset, and vceversa OFDM and orthogonalty concept The dvson of data stream to several sub-carrers can be mplemented easly usng Inverse Fast Fourer Transform (IFFT). The match flterng of each sub-channel also s done by Fast Fourer transform (FFT). These operatons perform reversble lnear mappngs between N complex data symbols and N complex OFDM symbols. An N-pont

112 FFT requres only on the order of N log N multplcatons rather than N 97 as n a straghtforward computaton [54]. Due to ths fact, an OFDM system typcally requres fewer computatons per unt tme than an equvalent system wth equalzaton. In OFDM the sub-carrer pulse used for transmsson s chosen to be rectangular. Ths has the advantage that a smple Inverse Dscrete Fourer Transform (IDFT), whch can be mplemented very effcently as Inverse Fast Fourer Transform (IFFT), can perform the task of pulse formng and modulaton. Accordngly, n the recever we only need a FFT to reverse ths operaton. Accordng to the theorems of the Fourer Transform the rectangular pulse shape wll lead to a sn(x)/x type of spectrum of the sub-carrers as t s shown n Fgure 4-9. Fgure 4-9: OFDM and orthogonalty prncpal Obvously the spectrums of the sub-carrers are not separated but overlapped. The reason why the nformaton transmtted over the carrers can stll be separated s the socalled orthogonalty relaton. By usng an IFFT for modulaton we mplctly chose the spacng of the sub-carrers n such a way that at the frequency where we evaluate the receved sgnal (ndcated as arrows) all other sgnals are zero.

113 Transmsson of data n the frequency doman usng an FFT, as a computatonally effcent orthogonal lnear transformaton, results n robustness aganst ISI n the tme doman. In ths system, the nput data wth rate R s dvded nto N parallel nformaton sequence wth rate R/N where N s the number of carrers. Each sequence modulates a carrer. The frequency of m th carrer s: where f 0 s the lowest frequency of the carrers and T s the OFDM symbol duraton. The transmtted sgnal n OFDM s then wrtten as: m f m = f T 98 s( t ) = N 1 j π fm ( t T ) dm( ) e N m= 1 = 4.10 where d m () s the complex symbol of the m th subchannel at the tme nterval T, keepng n mnd that an arbtrary modulaton type can be used. The tme dscrete equvalent of the transmtted sgnal at k th subchannel s gven by s k N 1 = d N m = 1 m e j πm k N 1 k N 4.11 In the case of deal channel wth addtve nose, the receved sgnal after flterng and samplng s: r k = s + z 1 k N 4.1 k k where z k s the addtve nose at the k th subchannel. The transmtted symbols d m are recovered from the receved sequence by performng an N pont DFT, as follows:

114 99 R k 1 = N N 1 = N = d + Z k N = 1 N k s = 1 l = 1 e d j π l e k N + j π ( l k ) N 1 N + Z k N = 1 1 k z e j π N k N 4.13 where Z k s DFT of nose at k th subchannel OFDM system n non-deal channel condtons The results of equaton 4.13 are drven by assumng two condtons: 1- The recever and the transmtter must be perfectly synchronzed. Ths means they both must assume exactly the same modulaton frequency and the same tme-scale for transmsson. - The channel s deal and there s no multpath and delay spread. The frst condton s satsfed n practcal applcatons by nsertng some known data (plot) sgnal to the OFDM symbol. Ths plot s mportant to dentfy the ampltude and phase reference of the mappng constellaton n each subcarrer so that the complex data symbols can be demodulated, correctly. The second condton cannot be obtaned n practce. Wth multpath n the transmsson channel, the orthognoalty of subchannels s destroyed, due to the ISI ntroduced to the system. The spectra of an OFDM sgnal are not strctly band-lmted and the multpath fadng causes each subchannel spread the power nto adjacent channels. Moreover, the delay tme larger than symbol tme contamnates the next symbol.

115 100 One way to tackle ths problem s to create a cyclcally extended guard nterval tme n each OFDM symbol [54], where each OFDM symbol s preceded by a perodc extenson of the sgnal tself. The total symbol duraton s T total =T g +T s, where T g s the guard nterval tme and T s s the OFDM symbol tme. Fgure 4-10 shows a typcal guard nterval tme. Each symbol s made of two parts. The whole sgnal s contaned n the actve symbol, the last part of whch s also repeated at the start of the symbol and s called guard nterval. The reason for ths s to convert the lnear convoluton of the sgnal and channel to a crcular convoluton and thereby causng the DFT of the crcularly convolved sgnal and channel to smply be the product of ther respectve DFTs. If the guard nterval tme s longer than the multpath delay of channel, the effect of ISI can be elmnated. The rato of the guard nterval to the useful symbol duraton s applcatondependant because the nserton of guard nterval wll reduce the data throughput. At the recever-end ths guard nterval should be removed before takng FFT from the receved sgnal. Symbol M-1 Guard Interval Guard Symbol M Interval Symbol M+1 Fgure 4-10: Guard nterval nserton Consderng a non-deal channel, the receved sgnal n k th subchannel s expressed by 4.14.

116 101 r k l L = = 1 h s l + z k l k 1 k N 4.14 where h l s the channel mpulse response and L s the length of channel mpulse response. Equaton 4.14 s a tme convoluton between channel coeffcents and the transmtted data. After takng FFT at the recever, assumng a long enough guard nterval to avod ISI, the receved sgnal can be expressed as: R k = H d + Z 1 k N 4.15 k k k where H k s the channel transfer functon at k th subchannel Effect of OFDM system on BPL nose s gven by: From equaton 4.13, the nose at the recever s the DFT of the channel nose and N 1 N Z k = ze, k = 01,,... N 1 N = 1 jπ k 4.16 As t s mentoned earler n chapter 3, z s are..d random varables wth a dstrbuton functon as n Accordng to Central Lmt Theorem, f a sample mean x s obtaned from samples that are taken from a large populaton, and the samples are of suffcently large ensemble sze, the dstrbuton of x s well approxmated by a Gaussan dstrbuton. Ths Gaussan dstrbuton s characterzed by a mean μ x = μ and a standard devaton σ x = σ / K, where μ and σ are the mean and standard devaton of the populaton and K s the sample sze. By consderng 4.16, we can rephrase Z k as:

117 10 Z = N 1 k z n N n = 0 N 4.17 n whch z n = z n e j πkn N 4.18 Therefore, based on Central Lmt Theorem, Z k s Gaussan wth mean μ Z = μ z = 0 and standard devaton σ Z = N.( σ z / N ) = σ z, n whch σ z s expressed by 3.0. DFT procedure spreads the effect of mpulsve nose over multple subcarrers n a way that nose on each subband exhbts a Gaussan behavor. Ths s one of the major benefts of OFDM system n an mpulsve nose envronment Performance of OFDM system n BPL channel condtons By assumng long enough guard nterval to avod ISI, each subband can be consdered as a regular ISI-free addtve Gaussan nose (AGN) channel and the analyss becomes a straght forward procedure. We know from [51] that the Bt Error Rate (BER) of a QAM scheme under AGN regme follows P M M BER = 1 (1 P ) E av = (1 ) Q( ) 4.0 M ( M 1) N 0 where M s the modulaton level and M= k when k s even. E av s the average symbol power and N 0 s the nose power.

118 103 Thus, f QAM modulaton s used for each subcarrer, we can express the average BER of an OFDM system wth N subcarrers (assumng no ISI) as: P BER k, M avg = 1 N k = 1 (1 P For our analyss and smulaton purposes, we desgned an OFDM system for the channel ntroduced n Fgure -13. The occuped bandwdth of the system s chosen to be 50 MHz. The capacty lmt of ths channel at 50 MHz wth 10-dBm launched power from Fgure -13 s around 400 Mbts/sec. The delay spread of the channel s 3 mcroseconds. To avod ISI and ICI, whle losng less than a 0.5 db due to guard nterval nserton, we chose an OFDM symbol nterval equal to 9 tmes the delay spread, whch s equal to 7 mcroseconds. The subcarrer spacng s now the nverse of 7-3=4 mcroseconds, provdng 4 KHz. By consderng 50 MHz bandwdth, at most we can use 100 subcarrers. We desgned a system wth 104 subcarrers and one known plot channel for estmaton. On each subchannel QAM modulaton wth approprate modulaton level M s chosen. If on each subchannel QPSK modulaton s used, the overall bt rate wll be equal to 104**4 Kbts/sec= 90 Mbts/sec. Fgure 4-11 shows the performance of such a system wthout codng, analytcally and by computer smulatons, under the channel condtons shown earler n Fgure -13, for two dfferent levels of M. For smulaton results, two nose models were consdered: the tme-markov model of Fgure 3-11 and the statstcal model of equaton Both models are utlzed by the parameters found n chapter 3. It s shown that results by both these models are n N k, M ) H k E av = ( 1 ) Q( ) 4. M ( M 1) σ Z

119 104 agreement and also wth analytcal results. One should notce that Markov based model s a tme representaton of the nose, whereas Mddleton model s a statstcal llustraton, therefore the statstcal characterstcs of the tme doman model over a very long tme wll be the same as Mddleton s expresson. Our smulatons usng Markov model are run for an extensvely long perod of tme. Consequently, as t s confrmed n Fgure 4-11, both models results are qute smlar. Fgure 4-11: The performance of uncoded OFDM system n a fadng channel wth mpulsve nose analytcally and by smulaton Adaptve loadng for OFDM Adaptve modulaton and bt allocaton s an mportant technque that yelds ncreased data rates over non-adaptve uncoded schemes. An nherent assumpton n channel adaptaton s some form of channel knowledge at both the transmtter and the

120 105 recever. Gven ths knowledge, both the transmtter and recever can have an agreed upon modulaton scheme for ncreased performance. The advantage of OFDM s that each sub-channel s relatvely narrow-band and s assumed to have flat fadng. However, t s entrely possble that a gven sub-channel has a low gan, resultng n a large BER. Thus, t s desrable to take advantage of subchannels havng relatvely good performance; ths s the motvaton for adaptve modulaton. In the context of tme-varyng channels, there s a decorrelaton tme assocated wth each frequency-selectve channel nstance. Thus, a new adaptaton must be mplemented each tme the channel decorrelates. Two crtera can be set for adaptaton technques: data rate and error probablty Adaptve OFDM algorthm for ncreasng data rate The objectve ths optmzaton s to maxmze the data rate, R b for a gven total transmsson power P total such that the receved data acheves a specfed performance. The performance crtera are applcaton-dependant. For hard-decson codng systems or systems wthout codng probablty of error s chosen crteron. As soft-decson, mnmum dstance decodng s wdespread n data transmsson, the mean squared sgnal separaton (MSSS) s the performance crteron for such systems. For now, we assume a general crteron, whch has value c on each subchannel and average C per bt over all the subchannels.

121 The objectve s to maxmze data rate N R b = b = by optmzng the allocaton of power, P, and data rates b on each of N subchannels, such that the constrant 4.3 s satsfed. C = N c =1 4.3 = C max R b where the parameter c s assumed to be zero for any subchannel on whch b =0. Defne b to be the vector of b s and P to be the vector of correspondng P s. Then, the optmzaton problem s stated as follows: Maxmze N R b = b = 1 wth respect to b and P Є R N 4.4 N Subject to h 1 = P Ptotal = = 1 h =C-C max =0 4.6 Usng the Lagrange multpler method [55], the Lagrangan functon s defned: L( b, P, λ b + λ 4.7 1, λ ) = R + λ1h1 h Ths s a convex problem, therefore the vectors b * and P * are optmum when: Equaton 4.8 yelds: L( b *, P *, λ, λ ) = R b + λ h + λ h = - R b b * * { b, P } L Pj C + λ 1 * * + λ * * = 0 { b, P } { b, P } 4.9 b b j = 1

122 107 - R b P * * { b, P } L Pj C + λ 1 * * + λ * * = 0 { b, P } { b, P } 4.30 P P j = 1 for all such that b 0. Snce R b s ndependent of power dstrbuton and also P s ndependent of data allocaton equatons 4.30 and 4.9 gve: C -1 + λ * = 0 {, } b P * 4.31 b C λ 1 + λ * = 0 {, } b P * 4.3 P Therefore, the optmum soluton s found by solvng the gradent condtons C b { b *, P * } = ξ C P { b *, P * } = ξ 4.34 for all such that b 0, where ξ 1 and ξ are constants such that N R b = b = 1, N P = =1 P total and C=C max Error probablty crteron Ths crteron s consdered by most researchers for adaptve loadng algorthms n OFDM systems, such as n [56]. Wth ths crteron C s the average bt error probablty, p e. Thus the receved data must acheve a specfed bt error probablty p e =p max =C max. If

123 108 the bt error probablty on the th subchannel s denoted by p then c, whch s the symbol error probablty on the th subchannel wll be equal to b p. Based on the defnton of bt error probablty, the crteron C s gven by By applyng 4.35 to 4.33 and 4.34 the optmum condtons become: If we assume an ICI-free OFDM system then the symbol error on each subchannel depends only on the bt allocaton on the same subchannel. Therefore, the optmum condtons wll be: If M-ary QAM modulaton s utlzed for each subchannel wth M = k when k s even, then the symbol error probablty n th subchannel, c s gven by: b N N N e R c b p b p C = = = = = = = ξ = = }, { max }, { * * * * P b P b p b c R b C N j j b = ξ = = }, { }, { * * * * P b P b N j j b b c R P C 4.37 = ψ 1 }, { * P * b b c 4.38 = ψ }, { * P * b P c ) (, M P c = 4.40

124 109 where P, M s gven by 4.. Now, equaton 4.40 can be appled to optmum condtons of 4.38 and 4.39 to fnd an optmum soluton. Unfortunately, by dong so an explct expresson for b and P cannot be obtaned, hence R b cannot be calculated drectly [57]. However, by help of teratve algorthm, the optmum dstrbuton of power and bt allocatons can be calculated. Ths algorthm wll be dscussed later n ths secton MSSS crteron MSSS s defned to be the mean squared error of the receved sgnal dvded by the average dstance of ponts n the sgnalng constellaton. Suppose that on subchannel, adjacent ponts x and y n the receved sgnal constellaton at the sampled output of the matched flter for a nose-free transmsson are separated by a dstance r ( x, y) = x y Over the M constellaton ponts, the average of ponts dstance squared s then defned as { r ( x, y } d E ) = 4.41 The mean squared error of the receved sgnal at subchannel s gven by { a a } σ = E 4.4 where a s the transmtted data symbol on subchannel and ā s the correspondng receved symbol. The MSSS crteron for each subchannel, c s defned by c σ = ε = 4.43 d

125 Therefore, the average MSS per bt crteron, C, s wrtten as 110 C = σ ζ = N N ε = 1 d = 1 = N R b b 4.44 = 1 By applyng 4.44 to 4.33 and 4.34 and assumng no ICI, the optmum condtons become: c * = ψ {, } 1 b P * 4.45 b c * = ψ {, } b P * 4.46 P To utlze the optmum condtons more specfcally, we choose our system to have an M-ary QAM on each subchannel. The system s ISI and ICI free, as well. The receved sgnals pass through a matched flter on each subchannel. Then they multply by a gan, specfc for each subband, to mnmze the MSE between the transmtted and receved symbol. If g s the gan on the th subchannel, the receved symbol at ths subband s a = H g a + H g n 4.47 n whch n s the addtve nose at subband and H s the ampltude of transfer functon of the same subband. H s assumed to be constant over the band of that subchannel. Thereby, from 4.4 the MSE between the transmtted and receved symbol s equal to σ 4.48 = ( 1 g H ) E + H N 0

126 where N 0 s the power addtve nose and s the energy of transmtted symbol on th subchannel, whch s equal to P dvded by the bandwdth of subband. The optmum recever gan to mnmze σ s equal to 111 g E = 4.49 N 0 + E H and consequently the MSE s gven by = E N 0 σ 4.50 N 0 + E H From [51] and by consderng the matched flter and gan at the recever, the average of ponts dstance squared n a nose free transmsson s equal to E ( H g ) d = 4.51 ( M 1) By combnng 4.51 and 4.50 and 4.49, the expresson 4.43 for c =ε can be utlzed. Subsequently, the optmum condtons of 4.45 and 4.46 are exploted. Unfortunately, smlar to the other case, these equatons are not explctly solvable [57]. However, the teratve algorthm presented n next secton can fnd the soluton for both crtera Iteratve algorthm The optmzaton condton of, 4.33 and 4.34, often are not explctly solvable but requre an teratve soluton. The optmzaton of the transmsson s performed usng a steepest descent algorthm to assgn the data and power among the subchannels.

127 11 After the ntalzaton, the optmzaton can be broken nto two sub-problems,.e., mnmze C wth respect to: 1) the subchannel bt allocaton and ) the power dstrbuton. The overall data rate s ncreased at each step to ensure that the soluton s feasble,.e., acheves the constrants. For better understandng of the algorthm, we defne for varables for each subchannel as follows: The optmzaton procedure can then be descrbed wth followng steps: 1) Intalzaton: system ntalzed at a condton on whch constrants 4.5 and 4.6 are satsfed ) Mnmze C wth respect to b : Select the mnmum allowed ncrement or decrement of bt allocaton, Δb. In QAM system ths parameter has to be a multplcand of. Afterward for each subchannel calculate the terms + b c, Δ and b c, Δ. Fnd subchannels I and J such that 0 > = + + }, { }, {, P b P b b b c c c Δ Δ < = }, { }, {, P b P b b b c c c Δ Δ < = + + }, { }, {, P b P P b P c c c Δ Δ > = }, { }, {, P b P P b P c c c Δ Δ 4.55 { } + + = b b I c c,, mn Δ Δ and { } = b b J c c,, mn Δ Δ 4.56

128 113 where Δ + c I, b >0 and Δ c J, b <0 and I J. Then, f + Δ c I, b < Δ c J, b, transferrng Δb from subchannel J to I wll result n the largest decrease n C. Thereby, ncrease b I by Δb and decrease b J by the same amount. Repeat ths procedure wth the new bt allocaton untl Δ c J, b + Δ c I, b. At ths pont, no further transfer wll reduce C. 3) Fnd the hghest feasble b : At step the constrant 4.6 s no longer vald. Therefore, t s possble to ncrease the bt rate to meat ths constrant. Fnd the subchannel I such that + { c } Δ c I, b = mn Δ, b Then, f C max C Δ + c I, b, ncreasng the number of bts n subchannel I by Δb wll result n th smallest ncrease n C. Increase b I by Δb. Repeat ths process untl Δ > C max C. + c I, b 4) Mnmze C wth respect to P : Select the mnmum allowed ncrement or decrement of power dstrbuton, ΔP. Afterward for each subchannel calculate the terms Δ and + c, P Δ c, P. Fnd subchannels I and J such that Δ Δ + { c } + c I, P = mn Δ, P and { c } c J, P = mn Δ, P 4.58

129 114 where Δ + c I, P <0 and Δ c J, P >0 and I J. Then, f + Δ c I, P > Δ c J, p, transferrng ΔP from subchannel J to I wll result n the largest decrease n C. Thereby, ncrease P I by ΔP and decrease P J by the same amount. Repeat ths procedure wth the new bt allocaton untl Δ + c I, P Δ c J, P. At ths pont, no further transfer wll reduce C. 5) Repeat step 3. In lterature, steps and 3 are consdered as bt loadng process and steps 4 and 5 as power loadng. Both bt loadng and power loadng algorthms are suboptmum subroutnes. Based on applcaton and consderng the complexty of the system each of these suboptmum routnes can be selected nstead of the entre complex optmum algorthm. We appled the mentoned optmzaton algorthm to the channel of Fgure -13. The system s assumed to have 50 MHz bandwdth. For error probablty crteron, we chose p max =10 5. Fgure 4-1 shows the normalzed transmsson rate of the OFDM system usng the optmzaton algorthm for error probablty crteron. Also, the channel capacty s depcted as a reference. Moreover, Fgure 4-1 shows the performance of suboptmum routne, bt loadng, and as t s seen, the bt loadng adaptaton mproves the system but not as much as the total optmum algorthm. These algorthms wth MSSS crteron of ζ max =0.05 are also appled to the same channel and the results are shown n Fgure 4-13.

130 115 Fgure 4-1: Transmsson rate for dfferent adaptve algorthms wth error probablty crteron operatng n channel of Fgure -13 Fgure 4-13: Transmsson rate for dfferent adaptve algorthms wth MSSS probablty crteron operatng n channel of Fgure -13

131 Adaptve OFDM algorthm for mprovng system performance 116 The objectve of ths optmzaton s to maxmze the system performance for a gven total transmsson power P total such that the receved data s transmtted by a certan bt rate N R b = b max = 1. Smlar to the last stuaton the performance crteron s applcaton-dependant and can be ether error probablty or MSSS. We can state the optmzaton problem as bellow: Mnmze C N = 1 = R b c 4.59 wth respect to b and P Є R N N Subject to h 1 = P Ptotal = = 1 h =R b -R bmax = condtons as: Followng the steps mentoned n last secton, we can express the optmzaton C b C b { b *, P * } { b *, P * } + λ + λ = 1 = Smlar to the last case, the optmzaton condtons cannot be explctly solved for any of MSSS and error probablty crtera. Therefore, the teratve steepest descent algorthm needs to be appled. For ths case, the optmzaton procedure follows steps 1, and 4 of ntroduced algorthm n last secton because the data rate s fxed and therefore

132 117 steps 3 and 5, whch adapt the data rate are redundant. Lkewse, suboptmum routnes of bt loadng and power loadng are also applcable at ths stuaton. These optmum and suboptmum algorthms are appled to the OFDM communcaton system ntroduced n secton and the results are llustrated n Fgure The orgnal system uses a QPSK scheme for all the subchannels wth overall data rate of 90Mbts/sec. As t s seen, bt loadng shows better mprovement than power loadng n ths system. The optmum adaptve algorthm mproves the system performance by 6dB at error probablty of Fgure 4-14: The effect of adaptve OFDM loadng to the performance of a communcaton system

133 4.3.7 Impulsve nose cancellaton n OFDM 118 As t s dscussed earler, OFDM system s more robust to mpulsve nose n the envronment than conventonal sngle carrer systems due to the averagng of the symbols at the recever. However, a very long mpulsve nose can cause the entre OFDM symbol to be corrupted. Ths can be devastatng to the overall system performance. Therefore, an mpulsve nose suppresson technque s necessary to tackle ths problem n a hghly mpulsve communcatons envronment such as powerlne channels. Very recently, decson drected mpulsve nose suppresson technques for OFDM systems were developed by researchers for dfferent applcatons [58][59][60]. Some detals of each technque dffer due to ther applcaton stuaton but all of them use a decson drected approach. In ths research, we adopt a varaton of the method used n [60] where our algorthm s more sutable for practcal stuaton of mpulsve nose n powerlne channels Decson drected mpulsve nose cancellaton The block dagram of decson drected mpulsve nose cancellaton OFDM recever s depcted n Fgure The receved sgnal, r(k), s demodulated and then the ~ demodulated data, d(k) s agan modulated by an OFDM modulator resultng ~ s(k). Snce the powerlne channel s slowly varyng compared to the data rate, t s vald to assume that the recever has an accurate estmate of the channel. By convolvng ~ s(k) by the channel estmaton, an approxmaton of the receved sgnal wthout addtve nose s

134 obtaned. Therefore, n such a way we can acheve nose estmate n( k) r( k) h( k)* s ( k) ~ 119 = ~. The most accurate nose estmatons are for the mpulsve nose bursts because these mpulses cause a detecton error at the recever wth a very hgh probablty, especally at low sgnal-to-nose ratos (SNR). Ths error causes a dfference between ~ s(k) and s(k) at the mpulse occurrence tme. Therefore, wth a very hgh probablty ths dfference wll show up at the nose estmaton. Consequently, the estmaton of nose ~ n(k), for that mpulse wll be accurate. Ths wll not be true for the non-mpulsve nose. Therefore, t s sensble to only feedback the estmaton of mpulsve noses as these are the most relable approxmatons. For ths reason, the M largest values of estmated nose are fed back to the orgnal receved sgnal n Fgure These M values are then subtracted synchronously from r(k). The choce of M depends on how many mpulses appear on each feedback process and t s a vtal parameter for the system to perform optmally. Fgure 4-15: Decson drected mpulsve nose cancellaton OFDM recever block dagram

135 10 To nvestgate the performance of ths recever and observe the effect of M on the system operaton, we smulated an OFDM system over the LV powerlne channel of Fgure -15. The occuped bandwdth of the system s chosen to be 60 MHz. The delay spread of the channel s mcroseconds. To avod ISI and ICI, whle losng less than a 1 db due to guard nterval nserton, we chose an OFDM symbol nterval equal to10 tmes the delay spread, whch s equal to 0 mcroseconds. The subcarrer spacng s now the nverse of 0-=18 mcroseconds, provdng 55 KHz. By consderng 60 MHz bandwdth, at most we can use 1100 subcarrers. We desgned a system wth 104 subcarrers. On each subchannel BPSK modulaton s chosen. For now, we assume a very long nterleaver s deployed n the system and therefore, Mddleton nose model can be used for tme representaton of mpulsve nose. For ths matter, we chose A=0.01 and Γ=0.1. The feedback process for mpulsve nose suppresson happens on each OFDM symbol,.e. 104 bts. Fgure 4-16 shows the error probablty of the system versus M for dfferent SNRs. As t s seen n ths fgure, M=10 s the optmum value for ths system and any number more or less than ths value results n a non optmal performance. It s also mportant to notce that a system wth a hgh value of M s not mprovng the system s performance compared to a system wthout feedback. The nose model used for the system above s a Mddleton nose model and as t s seen from equaton 3.14, ths nose can be generated n tme by a Posson process wth an average arrval rate λ=a. Therefore, the probablty of mpulsve nose for ths nose model s 1-e -A. Wth A=0.01, the probablty of an mpulse s equal to Thereby, the

136 average number of mpulses n an OFDM symbol s equal to =10.4. Ths s the reason why M=10 s the optmum number for the mpulsve cancellaton feedback. 11 Fgure 4-16: Effect of M on the performance of mpulse cancellaton Fgure 4-17 shows the error probablty of the dscussed system wth M=10 on the mentoned channel wth the Mddleton nose model. Also, the performances of a conventonal OFDM system on ths channel wth and wthout mpulsve nose are depcted. The mpulsve suppresson technque has sgnfcantly mproved the system s performance wth mpulsve nose. Ths mprovement at error probablty of 10-4 s close to 10 db. It s mportant to notce that ths system s operatng very smlar to the case of no mpulsve nose, especally at hgh SNRs. Hence, one can clam at these SNRs nose mpulses are completely cancelled.

137 1 Fgure 4-17: Performance of decson drected mpulsve nose cancellaton recever wth M= The teratve mpulsve nose cancellaton Earler, t was argued that the number of feedback mpulses, M, s a vtal parameter for the system to perform optmally and ths number needs to be equal or close to the number of mpulses n each feedback frame. Choosng a fxed M s not advsable for a practcal system, as the number of nose bursts s dfferent for frames at dfferent tmes. Specally, f the tme-based Markov model s used, t wll be seen that over a very long tme, most frames are mpulse free but when a burst nose arrves, those frames ht by the burst are hghly dsrupted. Therefore, an algorthm needs to be employed to estmate the number of mpulses n each frame.

138 13 We propose a threshold algorthm for approxmatng the number of nose mpulses. If the varance of background nose s known to the recever, a threshold value, N T can be set such that the probablty that absolute value of background nose becomes larger than N T s a very small number, P T. If the absolute value of the estmated nose s greater than N T the nose s consdered mpulsve, otherwse t s background nose. Wth ths algorthm, the number of feedback mpulses changes for each frame. Ths algorthm can be repeated teratvely, as well. After the frst teraton, the ~ demodulated data d(k) s more accurate than before the mpulsve nose cancellaton. Therefore, ths more accurate estmate of the sent sgnal could be used agan for the next round of nose estmaton. As the number of feedback mpulses s not fxed, those mpulses that were mssed n the last teraton wth a hgh probablty wll be detected on ths teraton. Smlarly, the ones that were msnterpreted as mpulsve n the former round, wth hgh probablty wll be dscarded ths tme. The number of necessary teratons depends on the mpulsve nose characterstcs. To explore the performance of ths proposed algorthm, we smulated the dscussed system on the mentoned channel wth a Markov-based mpulsve nose model. The frst layer of the nose model has P1 = as ts transton probablty matrx and the second layer s transton probablty matrx s 0.98 P = Ths model has the same PDF as the Mddleton nose model we used before wth A=0.01 and Γ=0.1. We employed the earler system desgned for Mddleton nose model wth M=10, on ths channel condton and the result s llustrated n Fgure The performance curves of a

139 14 conventonal OFDM system n ths channel wth and wthout mpulsve nose are shown n Fgure 4-18, as well. The fxed sze feedback algorthm has mproved the system very slghtly, although the nose model that ths algorthm was desgned for has the same statstcs of the nose n ths envronment. Furthermore, the proposed teratve algorthm wth P T =10-5 was deployed n the same channel condton and ts performances are plotted n Fgure The frst teraton has mproved the system consderably and the second teraton makes the system to operate n an almost mpulsve nose free condton. More teratons mprove the system but they are nsgnfcant n ths system, because the second teraton s performance s very close to the no mpulsve nose stuaton and ths s the lmt of mprovement. Fgure 4-18: Performance of proposed teratve algorthm n a Markov-based nose model envronment

140 4.4 Mult-carrer CDMA (MC-CDMA) 15 The well-known mult-carrer technque, OFDM, s consdered as the modulaton scheme for BPL by most researchers and t was dscussed n last sub-secton. By the applcaton of OFDM, the most dstnct property of power-lne channel, ts frequency selectvty, can be easly coped wth. On the other hand, Code Dvson Multple-Access (CDMA), whch s a form of DSSS, s an attractve scheme due to robustness to nterferences. Ths s very mportant n PBL communcatons snce there are two sources of nterference, the nterference from other wreless devces and the multuser nterference n a home-network. A combnaton of mult-carrer modulaton and CDMA, MC-CDMA has the advantages of both technques MC-CDMA Analyss In MC-CDMA, nstead of applyng spreadng sequences n the tme doman, we can apply them n the frequency doman, mappng a dfferent chp of a spreadng sequence to an ndvdual OFDM subcarrer. Consder a MC-CDMA system wth K users employng bnary phase-shft keyng (BPSK) and bnary spreadng sequences. Input data for each user s frst converted to P parallel stream and each bt s spread over L spreadng chps, as t s shown n Fgure If the bt duraton of the orgnal data s T b, then chp duraton for each transmtted data wll be P Tb L. Therefore, transmtted sgnal of user k s gven as:

141 s + P L Eb t) = dk, p ( n) ck, lut c ( t ntc ) cos( wp+ ( l 1) Pt + p+ ( l P ) LT φ 4.64 k ( 1) n= c p=1 l=1 16 where E b and T c are the bt energy and chp duraton respectvely, u Tc(.) s the rectangular waveform of duraton T c. c k, l s the l th chp of the user k, d k, p ( n) s the n th data bt n the p th data stream of user k, w = πf s the th carrer frequency and φ s the phase of the th carrer. In practce the multcarrer modulaton s mplemented by IFFT smlar to OFDM. Consequently, the number of carrers n ths multcarrer system s equal to LP. 1 C k,1 C k, d k (n) Seral to C k,l IFFT s k (t) Parallel P Fgure 4-19: MC-CDMA transmtter block dagram The transmttng and recevng procedure s exactly smlar to the OFDM system. Thereby, the receved MC-CDMA symbol before takng FFT s expressed as + K P L Eb r( t) = H k, p ( l 1) Pdk, p ( n) c, lut ( t ntc ) cos( wp ( l 1) Pt p ( l 1) P ) ( t) LT + k c + + φ + +ν 4.65 n= c k=1 p=1 l=1

142 17 where ν (t) s the effectve nose n the recever after FFT, whch becomes AWGN as descrbed earler. H, s the channel transfer functon for the k th user and th subcarrer. k For downlnk channel we have H = k, H. The block dagram of a MC-CDMA recever for the k th user s shown n Fgure Σ g k,1 C k,1 g k, C k, u k (n) Parallel to g k,l C k,l FFT r(t) seral P Fgure 4-0: MC-CDMA recever block dagram Consderng the decson metrc n the p th stream of the frst user, wthout loss of generalty, we obtan U T s L p = r ( t ) c1, l cos ( w s t + φ s ) g 1, s dt = D p + I + ν 0 l = where s = p + ( l 1) P, g 1, s s the frst user's combnng coeffcents for the s th subcarrer and D p s the desred sgnal whch s obtaned as E T = 4.67 L b c D p H s g1, s L l=1

143 The nose ν has a zero mean and a varance of 18 L z c σ = σ T s 4 g υ 4.68 l =1 where σz s gven by equaton 3.0. I n equaton 4.66 represents the multuser nterference and s expressed by the followng equaton. K L E btc I = d k, p c k, l c1, l H s g s L 4.69 k =1 l =1 There are several schemes for combnng the chps of the same data bt; equal gan combnng (EGC), maxmum rato combnng and Orthogonalty restorng Combnng (ORC). Usng ORC, whch outperforms the other schemes under hgh load condton at hgh SNR values [6], the orthogonalty among dfferent subcarrers s restored. For ORC the combnng coeffcents are 1 gs = 4.70 H s Usng Walsh-Hadamard spreadng codes, whch have zero correlaton, the multuser nterference term n equaton 4.66 equals to zero. Therefore, the BER for the p th stream s obtaned as Pe p = Q ( σ LE b L z H s l =1 ) 4.71 Assumng a unform bt allocaton probablty for dfferent streams, the average BER of the system s obtaned as:

144 P 1 Pe = Pep. 4.7 P p=1 19 Fgure 4-1: The performance of uncoded MC-CDMA system n a fadng channel wth mpulsve nose analytcally and by smulaton For our analyss and smulaton purposes, we desgned an MC-CDMA system for the channel of Fgure -15. The occuped bandwdth of the system s chosen to be 60 MHz. The delay spread of the channel s mcroseconds. To avod ISI and ICI, whle losng less than a 1 db due to guard nterval nserton, we choose an OFDM symbol nterval equal to10 tmes the delay spread, whch s equal to 0 mcroseconds. The subcarrer spacng s now the nverse of 0-=18 mcroseconds, provdng 55 KHz. By consderng 60 MHz bandwdth, at most we can use 1100 subcarrers. We desgned a system wth 104 subcarrers and one known plot channel for estmaton. BPSK s the chosen modulaton scheme for each subcarrer. For the MC-CDMA system, we have

145 130 pcked L = 8 and P = 56 and the spreadng code s Walsh-Hadamard. Fgure 4-1 shows the performance of such a system, analytcally and by computer smulatons. For smulaton results, two nose models were consdered: the tme-markov model of Fgure 3-11 and the statstcal model of equaton Both models are utlzed by the parameters found n chapter 3.

146 Chapter 5 Codng Shannon [63] n 1948 demonstrated that by proper encodng of the nformaton, errors nduced by a nosy channel could be reduced to any desred level wthout sacrfcng the rate of nformaton transmsson, as long as the nformaton rate s less than the capacty of the channel. Snce Shannon s work, much effort has been expended on the problem of devsng effcent encodng and decodng methods for error n a nosy envronment. The channel encoder transforms the nformaton sequence u nto a dscrete encoded sequence v called a codeword. The channel decoder transforms the receved sequence r nto a sequence ũ called the estmated nformaton sequence. The purpose of ths chapter s to brefly ntroduce some major codng technques and nvestgate the effect of codng on our system. 5.1 Convolutonal codng One of the most frequently used codng schemes n dgtal communcatons systems s convolutonal code. It was frst ntroduced by Elas [64] and shortly thereafter, Wozencraft and Reffen [65] proposed sequental decodng as an effcent decodng method for convolutonal codes. Then, n 1967 Vterb [66] proposed a Maxmum

147 13 Lkelhood decodng algorthm that was relatvely easy to mplement for soft decson decodng of convolutonal codes wth small constrant length. A convolutonal code s generated by passng the nformaton sequence through a lnear fnte-state shft regster. In general, the shft regster conssts of K stages and n lnear algebrac functon generators. The parameter K s called the constrant length of code. The nput data to the encoder s shfted the shft regsters k bt at a tme. One can say ths codng scheme has a memory of Kk bts. Consequently, the code rate s defned as k R c =. To be specfc, let us consder a very smple bnary convolutonal encoder n wth K=3, k=1 and n= as shown n Fgure 5-1. Fgure 5-1: A half rate convolutonal code Followng the nstructons gven n [51], the transfer functon of ths codng scheme s gven as: T 5 D s, = 5.1 ( D s ) ( 1 sd ) where D denotes the hammng dstance between the sequence of output bts and all-zero sequence. s s the number of state transtons caused by an nput bt 1.

148 133 Followng the Vterb [66] decodng algorthm, t s possble to fnd an upper bound for error probablty of ths codng scheme used n an OFDM system. The Parwse Error Probablty (PEP) represents the probablty of choosng the coded sequence X ˆ = ( xˆ, xˆ,..., ˆ ) when ndeed another code sequence X = ( x x,..., ) was transmtted, 1 x L where L s the frame length. Under the assumpton of perfect channel state nformaton (CSI), the condtonal PEP wth respect to the channel coeffcents h = ( h h,..., ) s gven as: where ( X, X ˆ ) ε s the energy dfference between two codewords. Wth the assumpton of BPSK we have, P ( ) ( ) ˆ ˆ ε X, X X, X hk = Q N ( ) Es,Xˆ hk = Q Ωk N0 0 1, x L 1, h N 5. P X 5.3 where E s s the total transmtted energy and Ω k s the set of bt ntervals locatons where X and Xˆ dffer n the k th subchannel and where Ω s the cardnalty of Ω, whch also corresponds to the length of the error event. Defnng the sgnal-to-nose rato as τ and usng the upper bound on Gaussan-Q functon,.e. Q( z ) 0.5exp( z / ) =E s obtan N 0 1 (, Xˆ ) exp ( τ Ω ) P X h k k 5.4, we

149 134 Under the assumpton of symbol-by-symbol nterleavng whch guarantees ndependency among bts n the k th subchannel, 5.4 yelds: PEP s the basc tool for the dervaton of unon bounds on the error rate performance of a coded communcaton system. A unon bound on the average BER on the k th subchannel can be found as n [51] where ( X) P s the probablty that the sequence X s transmtted, ( X, X ˆ ) q s the number of nformaton bt errors n choosng another coded sequence Xˆ nstead of X. For unform error probablty codes, a symmetry property exsts, elmnatng the need for averagng over all possble transmtted sequences, whch leads to In the case that PEP s gven n a product form, the transfer functon technque [51] provdes an effcent method for the computaton of 5.7,.e. where ( D s) T k, s the transfer functon assocated wth the error state dagram of the code under consderaton, s s an ndcator varable takng nto account the number of bts n error and D k s gven as the base of the k th subchannel PEP expresson derved for the channel model under consderaton. P 1 (, Xˆ ) exp( τ ) Ω [ ] k X h k 5.5 ( X) q( X, Xˆ ) P( X, Xˆ ) P k P 5.6 b h k X X Xˆ ( X, Xˆ ) P( X, Xˆ ) P k q 5.7 b h k X Xˆ Pb k T ( Dk, s) 5.8 s s=1

150 135 Fgure 5-: Effect of convlutonal codng on the system performance and ts upper bound. As an example, we consder a convolutonally coded OFDM system to demonstrate the BER performance mprovement. Ths system s the same system desgned formerly for the channel of Fgure -13. The system occupes 50 MHz bandwdth and uses 104 subcarrers and BPSK s chosen as modulaton scheme on each subcarrer. By now, the mpulsve nose s not consdered; thereby background whte nose s the only nose n the system and adjacent nose sgnals are ndependent. Ths lets us use 5.8 to fnd the unon bound of codng scheme performance. The convolutonal code under nvestgaton s the one descrbed by 5.1. Therefore, from 5.8 the total error probablty of OFDM system s bounded by: P b 1 N N k ( 1 D k ) k = 1 D 5 5.9

151 136 where N s the number of subcarrers. Fgure 5- shows the performance curves of such system wth and wthout codng, as well as the unon bound of equaton 5.9 for ths communcatons lnk wth codng. It s seen that the codng mproves the system performance substantally and unon bound s qute tght and accurate for ths codng desgn OFDM nterleavng technques to tackle the effect of burst nose on convolutonal codes performance As t was dscussed before, convolutonal code s a codng technque wth Kk bts memory. Ths memory s then used for the decodng process. Therefore, f a burst error occurs longer than ths memory, the decodng wll be faulty and error spreadng happens [66]. In any communcatons system burst error can happen ether by burst nose n tme or narrowband nterference (NBI) n frequency. In BPL both these defcences exst. To solve ths problem, nterleavers are ntroduced n communcatons engneerng. Interleaver shuffles the nput data stream so that the adjacent sgnals n orgnal data stream spread over the sequence. At the recever the receved data stream s denterleaved. Thereby, f the sent sgnal s ht by a burst nose and nterleavng sze s much longer than the burst, wth a hgh probablty the receved data stream after denterleavng wll not contan any burst error. OFDM communcatons systems have an nherent property that automatcally separates errors caused by NBI, due to the spreadng of the nput data stream over several subcarrers. For smplcty, we assume an OFDM system wth 8 subcarrers, smlar to the

152 137 one n Fgure 5-3. If ths system does not apply any nterleavng, the receved data stream s read to recever column by column. If now ths system s ht by an NBI at subcarrer 4, the errors caused by ths nterference wll be separated by 8 bts. If the same system s ht by a burst mpulsve nose at OFDM symbol 11, ths burst wll be transferred to the recever, causng a decodng error. Therefore, an nterleavng technque s needed to overcome ths problem. Fgure 5-3: Sequence of 16 OFDM symbols wth 8 subcarrers each, affected by mpulsve nose (symbol 11) and a narrowband nterferer (subcarrer 4). Two types of nterleavers are commonly used: convolutonal and block nterleavers [67]. Convolutonal nterleavers are more complex, but ther delay s only half that of correspondng block nterleavers. Block nterleavers are smple: The bts are flled nto a table column by column and read out row by row; for the de-nterleaver the operaton s reversed. Assume now that a block nterleaver drectly algned to a transmsson block of 16 OFDM symbols wth 8 subcarrers each s used, as ndcated n Fgure 5-3. If the OFDM

153 138 symbol 11 s ht by a nose mpulse, the de-nterleaver wll read out the symbols row by row, such that the errors of symbol 11 are seperated evenly, resultng n dspersed bt errors wth 15 correct bts n between. Such an error pattern could be corrected easly wth convolutonal codes. If, however, an NBI hts subcarrer 4, the output of the denterleaver wll contan an error burst of length 16. The decoder can no longer correct such an error event. It s therefore necessary to fnd a better nterleaver. For smplcty, n order to desgn an approprate nterleaver for OFDM system, we consder a system wth eght subcarrers. The bts shall be nterleaved wthn a block of 16 symbols,.e. there are 18 bts n a block. The nput of the nterleaver s denoted as x and ts output as y, both of them are vectors wth 18 elements. Element y s the bt transmtted n OFDM symbol dv 8 at subcarrer mod 8. The nterleaver shall be capable of separatng burst errors caused by NBIs as well as burst errors caused by mpulsve nose such that the bt errors at the nput of the Vterb decoder are dspersed. To measure the qualty of an nterleaver, the followng metrcs are defned: 1. d 1 : Mnmum dstance between any two bt errors at the output of the denterleaver caused by an NBI.. d : Mnmum dstance between any two bt errors at the output of the denterleaver caused by an mpulsve nose burst (affectng a sngle OFDM symbol) 3. d 3 : Mnmum dstance between any two bt errors at the output of the denterleaver caused by two adjacent mpulsve nose bursts (affectng two consecutve OFDM symbols)

154 d 4 : Mnmum dstance between any two bt errors at the output of the denterleaver caused by two NBIs at adjacent subcarrers. d 1 and d are the mnmum dstances between any two-bt errors caused by a sngle error event n the frequency and the tme domans, respectvely, whereas d 3 and d 4 defne the correspondng dstances when two adjacent error events occur. We wll optmze the nterleaver as follows: Frst we attempt to maxmze d 1, as the mpact of an NBI on performance s crtcal. Then, we wll maxmze d. If possble, also the mpact of two error burst events, as descrbed by d 3 and d 4, wll be mnmzed. As shown n Fgure 5-3, an NBI affects every eghth bt,.e. the maxmum dstance that can be acheved n ths case s d 1max =8. A sngle mpulsve nose burst destroys up to eght of the total of 18 bts. Therefore, n ths case d max =16 cannot be exceeded. Two adjacent nose burst destroy up to 16 out of the 18 bts, such that d 3max =8. A smlar argument yelds d 4max =4. As dscussed earler, OFDM technque automatcally separates errors caused by an NBI. Hence, OFDM wthout an nterleaver leads to d 1 =8, whch s a favorable property. However, an mpulsve nose burst affects all bts of an OFDM symbol, such that n ths case d =1, whch s undesrable. Usng the smple block nterleaver mentoned prevously would ndeed maxmally separate the errors caused by a nose burst (d =16), but at the same tme fal to separate NBI errors (d 1 =1), thereby destroyng the nherent property of OFDM. An deal block nterleaver should separate both types of error bursts and combne the features of both approaches. Ramseer n [68] has ntroduced a very effectve nterleavng technque, whch we wll dscuss here.

155 140 In the desgn process of the optmum nterleaver t s frst attempted to keep the nherent advantage of OFDM,.e., d 1 =8. There are 64 possble nterleavers wth ths property. By constructon t was found that such nterleavers can be descrbed as follows: Interleaver y = x j=(k 1.) mod 18, = j 5.10 De-nterleaver x = y j=(k.) mod 18, = j (k 1.k ) mod 18=1 5.1 where k 1 and k are odd ntegers (1,3,5,...,17). x j s the j th bt of the vector x to be nterleaved, and y s the bt transmtted n OFDM symbol dv 8 at subcarrer mod 8. Note that 5.1 assures that the combnaton of nterleaver and de-nterleaver actually returns the orgnal sequence of the bts. There are 64 combnatons of (k 1,k ) that satsfy 5.1. Each combnaton s smulated and the mnmum dstances d, d 3 and d 4 are computed n [68]. The detaled results are shown Table 5-1. All these nterleavers have the mnmum dstance for a sngle NBI d 1 =8. Havng found nterleavers wth maxmum d 1, we can now pck the nterleaver wth best mnmum dstance for a sngle NBI, d. Accordng to Table 5-1, ths s d =15 and t s acheved by ether k 1 =113, k =17 or k 1 =15, k =111. For both of these systems d 3 =7,.e., two adjacent nose burst are separated almost perfectly (recall that d 3max =8). However, none of these pars can separate two adjacent NBIs, because d 4 =1. The matrx of nterleaved bts correspondng to the system wth k 1 =113, k =17 s shown n Table 5-. Assume that the orgnal bts (as output by the convolutonal encoder) are numbered from 0 to 17. Wthout an nterleaver, bts 0..7 would be

156 141 transmtted n the frst OFDM symbol, bts n the second, and so on. However, wth an nterleaver, the bts transmtted n each OFDM symbol correspond to the rows of Table 5-. Bt 0 s therefore transmtted n subcarrer 1 of the frst symbol, bt 1 n subcarrer of the 3rd symbol, etc. Table 5-1: Mnmum dstances for an 8 16 nterleaver k1 k d d3 d4 k1 k d d3 d4 k1 k d d3 d4 k1 k d d3 d If the transmtted block s dsturbed by an NBI at the frst subcarrer, the bts n the frst column of Table are 5- affected. However, after the de-nterleaver, these bt errors wll be dstrbuted (bts 0, 8, 16, etc.), such that the Vterb decoder does not see any error burst. The dstance between any two affected bts s d 1 =8. An mpulsve nose burst dsturbs an entre symbol, e.g. the bts transmtted n the frst row of Table 5-. However, after the de-nterleaver, ths error burst s spread out to bts 0, 3, 38,... 98, 113. Obvously, the mnmum dstance between to affected bts s d =15.

157 14 Table 5-: Matrx of 8 16 nterleaver for k 1 =113, k = Fgure 5-4 shows the performance results of the coded MC-CDMA system wth mpulsve nose n the channel medum of the system analyzed n Fgure 4-1 for dfferent nterleaver szes. These results are also compared to the upper bound of equaton 5.9 n Fgure 5-4. The smulaton results clearly show that ncreasng the sze of the nterleaver mproves the performance. Specfcally, the nterleaver wth sze yelds more than 4dB gan n SNR at BER of 10-5 compared to a system wthout code and nterleaver. However, further ncreasng the nterleaver depth from 18 to 104 gves less than 1dB gan. Fgure 5-4 also shows the calculated upper bound n 5.9. As t can be seen, the upper bound s very close to the curve of nterleaver sze of for hgh SNR values, snce t s calculated under the assumpton of deal (nfnte) nterleaver sze. Ths fact shows that ncreasng the nterleaver sze more than wll not mprove the performance of the system.

158 143 Fgure 5-4: Coded MC-CDMA smulaton results wth dfferent nterleaver szes and analytcal upper bound assumng nfnte nterleaver 5. Concatenated codng Concatenated codng, devsed by Forney n 1966 [69], s another powerful technque for constructng codes resstant to burst errors. Concatenated codes use a nonbnary code as an outer code and a bnary code as an nner code. The nner code s generally short and s decoded wth a soft-decson decodng algorthm and the outer code s generally larger and decoded wth an algebrac decodng method. Normally, the nner code s responsble for random errors happened by background nose and the outer code

159 144 s cleanng up the burst nose. The nner code s usually a convolutonal code. In most conventonal applcatons Reed-Solomon (RS) codes [70] are used as the outer codes. RS codes guarantee that successful recept of any k dstnct packets enables reconstructon of the source data. If the codng system transforms the k nput packet to n=k+l packet, stretch factor for ths code s defned as n s =. When l redundant packets k and k-l source data packets arrve, there s a system of l equatons correspondng to the l redundant packets receved. Substtutng all values correspondng to the k receved packets nto these equatons takes ( k l + 1) la exclusve-ors of source packets, where A s the length of a symbol. The remanng subsystem has l equatons and l unknowns correspondng to the source data packets not receved. Wth RS codes, ths system has a specal form that allows one to solve for the unknowns n tme proportonal to l va a matrx nverson and matrx multplcaton. RS codes are theoretcally the most perfect set of codes for outer codng (or erasure channel codng) but they suffer from some practcal lmtatons, whch are stated brefly n the followng paragraph. For RS codes, the sze of the fnte feld symbol alphabet s an upper bound on n, and ths sze lmts the stretch factor. In most practcal mplementatons, the alphabet sze s 56 (each symbol s one byte), whch lmts n to values of 56 or less. It s possble to use a larger alphabet sze for RS codes, e.g., 65,536 (each symbol s two bytes), but n ths case, the practcal stretch factor s severely lmted to small values due to processng consderatons. On the encodng sde, the operatons needed to generate n encodng packets requres kla exclusve-ors of source packets, where A s the length of a symbol

160 145 (16 n ths example). Thus, for example, f k=10,000, and n=0,000 (a moderate stretch factor of two), then t takes 800,000,000 exclusve-ors of source packets to produce 0,000 encodng packets from 10,000 source packets or around 80,000 exclusve-ors of source packets per source packet, whch s prohbtvely expensve. Thus, even a moderate stretch factor of two s not practcally possble for moderate values of k. For all but very small values of k, the practcal lmtaton on l s a few hundred at most, as the la processng overhead to produce the encodng per source packet s lnear n. The decodng tme s typcally comparable to the encodng tme for RS codes. The large encodng and decodng tme for RS codes arses from the dense system of lnear equatons used. An alternatve to RS codes for erasure channels s Raptor codes [71] or a varaton of them, Tornado codes [7]. These codes are two layered codes of Low Densty Party Check Codes (LDPC) [73] as the frst layer and LT codes or Fountan codes as second layer [74]. Both of these codes n each layer use a sparse party check code and therefore the order of operatons for encodng and decodng s small. The only drawback of usng Raptor code relatve to RS codes s that the decoder requres slghtly more than k of transmtted packets to reconstruct the source data. Any subset of sze k(1+ε) s suffcent to recover the orgnal k symbols wth hgh probablty. Shokrollah n [71] proves that each output symbol s generated by usng O(log(1/ε)) operatons, and the orgnal symbols are recovered from the collected ones wth O(k log(1/ε)) operatons. Thereby, the number of operaton for decodng and encodng of the symbols are ndependent of the sze of the packets, A. These codes can encode very large data blocks (compared to RS

161 146 where each block s a GF( m ) symbol that prohbts large encodng blocks for complexty of encodng and decodng, both.) whch s very essental n our case, snce we are consderng broadband communcatons wth hgh rates. Table 5-3 shows comparson between decodng tme for RS code and Tornado code for erasure channels [7]. Table 5-3: Comparson between decodng tme for RS codes and Tornado codes. For smulaton purposes, we adopted communcatons scheme and condtons used for Fgure 5-. As mentoned earler, the Raptor code conssts of a precoder and Fountan code. The precoder that we have used n our smulatons s a hgh rate (1088, 14) LDPC code and the Fountan encoder s an LT-code. LT-codes share the same error floor problems as LDPC codes over erasure channels. However, t has been shown that usng the LDPC as a precoder for LT-code solves the error floor problem. The degree dstrbuton of our LT-code s as follows: Ω ( x) = x x x x x x x x x x

162 where the coeffcent of every term shows the probablty of choosng the exponent of x as the number of nput nodes ncorporated n generated output symbol. 147 Fgure 5-5 shows the smulaton results of the frame error rate (FER) versus SNR performance of our system for two rates of = compared to Convolutonal codes of rate R = 1/ and R = 1/ 3 R and = R whch are also. As t can be observed, the results for the concatenated code wth rate of 0.36 acheve an FER of 10-6 at an SNR value of 8.75 db. In fact, ths code acheves waterfall reducton of BER as a functon of SNR. The observed error floor n the case of R = code s manly due to the low percentage of the uncovered nodes, whch comes from the random nature of the code. However, the rate of concatenated code n both cases s hgher, comparng to the case of usng a R = 1/ 3 Convolutonal code alone Concat. Rate=0.375 Concat. Rate=0.36 Conv. Rate=0.5 Conv. Rate= FER SNR Fgure 5-5: FER performance versus SNR comparson of codes wth dfferent rates

163 148 Fgure 5-6 shows the performance of the system versus nverse of the code rate value. As can be realzed, no error floor s observed wthn the range of the smulaton wth a decrease n the rate FER SNR=8.5 db SNR=9 db SNR=9.5 db SNR=10.5 db Rate - 1 Fgure 5-6: FER performance of the system versus nverse of the code rate for dfferent SNR values

164 Chapter 6 Indoor Whte LED communcatons Indoor wreless connectvty s always appealng to consumers because of ts ease of use. One of the conventonal wreless access systems s W-F. These systems and smlar other wreless schemes suffer from so many shortages, such as nterference, not provdng qualty-of- servce (QoS), adequate coverage, etc. A better canddate for wreless home networkng s optcal wreless. Use of conventonal lasers for optcal ndoor communcatons has not been feasble as yet, because of the hgh cost of laser sources. Instead of lasers, LEDs can be used as communcatons transmtters connected to electrc grd, recevng hgh data rate sgnals va BPL. Recently, whte LEDs emerged n the market and are consdered as future lamps. Apparently, n the near future, the ncandescent and fluorescent lamps wll be replaced by the low cost, effcent and mnature whte LEDs. Researchers pledge that by 01, these devces wll reach 7W and 1000lm. Ths s brghter than a 60-w bulb and yet draws a current provded by 4 D-sze batteres [75]. A Japanese research team suggested the use of the same whte LEDs not only for lghtng the homes, but also as lght sources for wreless n-house communcatons [76]. Usng ths new and developng technology along wth MV/LV powerlne communcatons can create a revoluton n the area of consumer networkng due to ts effcency and affordablty.

165 150 Whte LEDs are consdered as strong canddates for the future of lghtng technology. The reason s that LEDs offer very favorable characterstcs such as hgh brghtness, very low power consumpton and hgh lfetme expectancy. Therefore, t s predcted that n near future whte LEDs wll replace the conventonal ncandescent and fluorescent lamps. Moreover, LEDs can be used as a wreless communcatons transmtter. Ths s not possble for any other knds of lamps n broadband transmsson. Ths functonalty of LEDs as a transmtter s based on a fast response tme and modulaton of vsble lght for wreless communcatons. Fgure 6-1 shows a very general realzaton of vsble lght communcaton system usng whte LEDs. Ths system s a wreless optcal ndoor system that uses vsble lght as communcatons carrer. The concept of ndoor optcal communcatons has been an actve area of research snce early 1980 s. Most of the research n ths area s done based on Infrared (IR) as the communcaton carrer and results from these efforts are nearly all applcable to any parts of lght frequency spectrum. There are several advantages usng whte LEDs for communcatons over W-F and IR for ndoor communcatons: Installaton s easer than most of wreless systems. Whte LED communcatons do not need any band lcensng because t does not cause or suffer from any electromagnetc nterference. Whereas, there are always concerns n usng W-F or any other RF communcatons systems regardng nterference from or to other wreless communcaton systems.

166 151 Dfferent users n dfferent rooms and buldngs do not nterfere wth one another because LED sgnal rays do not go through walls. On the other hand, n W-F, t s possble that dfferent transmtted access pont sgnals nterfere and cause a degraded performance. Shadowng effect s so much less compared to IR case because LED lght fxtures are dstrbuted throughout the room. LEDs are less expensve than laser sources used n IR. Recever obtans at least one strong Lne of Sght (LOS) sgnal as the transmtters are on the celng. Ths s not the case n most IR transmsson stuatons. Fgure 6-1: Vsble lght communcatons usng whte LED 6.1 Cellular whte LED communcatons plannng For any optcal transmtter half-power (HP) angel and for any optcal recever, feld-of-vew (FOV) s defned. Half-power angle, s the hghest angle that the transmtter

167 15 can llumnate and FOV s the hghest angle that the recever can receve sgnal rays from. The mathematcal defntons of these parameters are gven by equatons 6.1 and r1 FOV = tan ( ) 6.1 H 1 r HP = tan ( ) 6. H where r 1, r and H are shown n Fgure 6-. H Fgure 6-: An optcal communcatons scenaro In order to llumnate a room wth whte LED, we need to use several LEDs. For modelng purposes, we used the scheme shown n Fgure 6-3. In ths model room, nne LED lamps (whch consst of several LEDs) are employed wth meters spacng between the lamp rows on the celng and 1 meter dstance to the walls. It s shown n [76] to have a unform llumnaton anywhere n the room the HP angle has to be greater than 70 degrees.

168 153 Fgure 6-3: A model room usng whte LED communcaton system confguraton The llumnaton coverage area of the center lamp s shown n Fgure 6-3 by a dashed crcle. Recever should be desgned n a way that ts FOV s hgh enough to at least receve a LOS sgnal ray from one transmtter. In ths way, there would be no blnd spot n the room. The nearly blnd spots n the room are near the corners. Accordng to Fgure 6-3, f the recever s coverage area radus s greater than meters, recevers at the corners wll at least receve one LOS sgnal from the closest transmtter. If we assume the room heght s 3 meters, ths coverage area wll correspond to an FOV equal to tan 1 ( ) 5degrees. Therefore, the desgn for the recever needs a FOV equal or 3 greater than 5 degrees. Jvkova and Kavehrad n [77] have desgned an effcent recever confguraton for a FOV of 5 degrees. Accordng to ths research, a greater FOV needs more complexty and a greater recever area. In the same paper, Jvkova and Kavehrad have nvestgated dfferent scenaros of wreless optcal communcatons for coverng an

169 154 ndoor space. In ths paper, they argue that an FOV of approxmately equal to 30 degrees can optmze the lnk budget of the system both n cellular schemes, lke whte LED, and dffusng multspot regmes. 6. Indoor optcal channel modelng Smlar to other communcatons channels, modelng of wreless optcal channels has been a challenge. The very frst mpulse response model for these channels was presented n [78] by Bapst and Gfeller. Equaton 6.3 s the result of ths research. { h(t)= t 3 τ 0 sn( FOV) τ < t < 0 elsewhere 0 τ 0 cos( FOV 6.3 where τ 0 s the delay of the shortest path from transmtter to the recever. From 6.3, t s notced that obtanng a narrower FOV results n a smaller delay spread n the channel. Ths delay spread s caused by reflectons of the optcal sgnals off of the walls. For a room wth dmensons of our model room and wth a recever wth 5 degrees FOV, the delay spread cannot exceed nanoseconds. Furthermore, equaton 6.3 confrms that reflected sgnals have so much less power at the recever compared to LOS path sgnals. Ths causes the mpulse response to have a very sgnfcant LOS sgnal and some resdual attenuated sgnals. Due to ths fact some researchers n some papers, lke n [79] have assumed that n whte LED communcatons, the channel only conssts of one straght LOS path.

170 155 The more rgorous channel modelng process for ndoor optcal channels was suggested by Alqudah and Kavehrad n [80]. In ths paper, authors consder up to three sgnal reflectons from the transmtter to the recever. Ther models s used for nfrared applcatons, whch they have ther sgnal source on the floor amng a beam to the celng. By some modfcatons, ther method can be adapted for our system confguraton. The complexty n obtanng channel mpulse response s brought about by the multple paths sgnals take, n travelng from a transmtter to a recever. Ths multpath results from reflecton off of walls, celng, furnture, etc. Room surfaces act as Lambertan reflectors that reflect an ncdent sgnal n all drectons. Assumng room surface exposed to a transmtter s made of N surface elements, each reflecton produces N-1 new reflectons, as llustrated n Fgure 6-4. In ths case the room has assumed to have three elements and the transmtter has LOS path to the recever and two paths to ponts a and b. The mpulse response of any reflecton s found by consderng all the elements wthn the recever FOV. In ths example, all the elements are wthn the recever FOV. Complexty n calculatng mpulse response s caused by reflectons. Each reflecton results n N-1 new reflectons. When determnng channel mpulse response, contrbuton of each element on a surface wthn recever FOV should be consdered. Snce surfaces do not offer perfect reflecton and sgnal strength s nversely proportonal to the dstance traveled, a fnte number of reflectons are consdered n obtanng an mpulse response.

171 156 Fgure 6-4: Illustraton of sgnal propagaton. The room surface s composed of 3 elements: a, b, and c. Impulse responses are obtaned by dvdng the reflectng surface nto a fnte number of reflectng elements N [81]. If N s large, accurate samples of contnuous mpulse response are obtaned. The number of elements N for a rectangular room, of dmensons equal to (W, L, H), s gven by: where N ( n n + n n + n n ) = 6.4 x z x y y z W n x = L n y = H n z = d 6.5 The constant d represents the dstance between centers of neghborng elements, whch s taken to be the same for all surfaces. Every surface element contrbutes drectly to receved sgnal f that element s wthn recever FOV, or ndrectly through reflectons off of other surfaces, as llustrated n Fgure 6-5.

172 157 (a) (b) (c) Fgure 6-5: Illustraton of (a) drect path, (b) sngle reflecton, and (c) two reflectons. Reflectons are counted from dffusng spots to a recever. The radaton pattern of surface elements s assumed a frst order Lambertan. They reflect ncdent lght wth equal ntensty n all drectons. Therefore, the ntensty at an angle θ from the surface norm s proportonal to cos(θ ). The lne-of-sght response h 0 (t), when the source T s wthn the FOV of the recevng element R can be expressed as [81]: h 0 TR ( t ) cos( ϕ TR ) cos( θ TR ) A = R δ ( t π R TR R c TR ) 6.6 where cos( ϕ TR ) s equal to dot product of two unt vectors. The frst s perpendcular to T (Transmtter locaton pont at ts surface) and the second orgnates from R (Recever locaton pont at ts surface) and extends toward T. The angle θ TR s the angle between a vector perpendcular to R and a vector that lays on the straght lne that connects T and R. A R s the recevng element area, R TR s the dstance between T and R, and c s the speed of lght. The response after a sngle reflecton off an element s obtaned by treatng as a recever, and then as a source as shown n Fgure 6-5(b). The mpulse response s gven by: h 1 TR cos( ϕ T ) cos( θ T ) A ρ cos( ϕ R ) cos( θ R ) A R ( t ) = δ ( t π R T π R R R T + c R R ) 6.7

173 158 where A s the area of the reflectng element, and ρ s ts reflectvty. The response resultng from two reflectons off elements frst and then off element j s found by extendng 6.7 to nclude mpulse response between j and recever, as llustrated n Fgure 6-5(c), and s expressed as: h, j, R ( t ) = cos( ϕ ρ j T π R cos( ϕ ) cos( θ T j R π R T ) A ) cos( θ jr jr ρ ) A cos( R ϕ j π R R δ ( t ) cos( θ j T + j R c j ) A j + R jr ) 6.8 Equaton 6.8 makes t clear why receved power through reflectons becomes nsgnfcant as hgher reflecton orders are consdered. Ths s the case snce n th order mpulse response s equal to n-1 th order multpled by a quantty that s much smaller than 1. In ths study, up to thrd reflecton s consdered n calculatng the mpulse response. In ths model, the transfer functon between a transmtter and a recever s dvded nto 4 components. The frst represents the transfer functon between sources and surface elements. The second block contans the transfer functon between surface elements. The thrd has the transfer functon from surface elements to a recever. The last component accounts for drect response between a source and a recever. These relatons are represented n Fgure 6-6.

174 159 Fgure 6-6: New representaton of drect, frst, and second reflecton for a mutple source and a sngle recever. The frst component n ths model represents transfer functon between sources on the celng and surface elements. It s referred to as Source Profle and s modeled by a multple-nput multple-output system wth N outputs. The transfer functon between each source and each of surface elements s expressed by an entry n matrx F. The second component consoldates dependence on ndoor geometry, dmensons, and reflecton coeffcents. Ths component contans the transfer functons between any two reflectng elements. In matrx format, and consderng up to n reflectons, t s expressed as: Φ n = I NxN + φ + φ I 3 + φ NXN n + Lφ 1, n, n = where I NxN s the N N dentty matrx, and φ s gven by