Symbol Error Rate Evaluation for OFDM Systems with MPSK Modulation

Transcription

1 Symbol Error Rate Evaluatio for OFDM Systems with MPS Modulatio Yuhog Wag ad Xiao-Pig Zhag Departmet of Electrical ad Computer Egieerig, Ryerso Uiversity 35 Victoria Street, Toroto, Otario, Caada, M5B 23 Abstract Orthogoal frequecy divisio multiplexig (OFDM) is the most commoly used multiplexig method i multicarrier (MC) systems. I this paper, a ew closedform formula for the symbol error rate () is derived for geeric OFDM systems with M-ary phase shift keyig (MPS) modulatio ad the optimal phase detector for each subchael over a fadig chael. This ew formula ca be used to evaluate both discrete Fourier trasform (DFT)-based OFDM ad discrete wavelet multitoe (DWMT) systems. It ca also be adopted as a criterio for OFDM system desig. Mote-Carlo umerical simulatios are performed to verify the theoretical aalysis, ad it is show that the theoretical results from the preseted ew formula is cosistet with simulatio results. I. INTRODUCTION The orthogoal frequecy divisio multiplexig (OFDM), as the most importat class of multicarrier modulatio (MCM), has bee uder itesive research ad developmet i recet years, ad M-ary phase shift keyig (MPS) has bee adopted as a importat modulatio scheme for OFDM systems [], [2]. A geerally recogized advatage of OFDM is its robustess agaist various types of chael distortios, such as multipath propagatio ad arrowbad iterferece [3], [4]. I OFDM modulatio, subcarriers are mutually orthogoal such that the receiver ca separate these subcarriers eve if there is spectral overlappig amog them. The filter bak formulatio of the modulator ad demodulator of a geeric OFDM system is illustrated i Fig.. I practical OFDM systems, the mutual orthogoality of subchaels is ofte destroyed by o-ideal chaels. Therefore the trasmitted symbols for a give This work is supported i part by Caadia NC research grat RGPIN2393. subchael may be distorted by co-subchael symbols (itersymbol iterferece or ISI) ad symbols of other subchaels (iterchael iterferece or ICI), as well as chael oise. The symbol error rate () is a essetial parameter for OFDM system performace evaluatios. Due to the existece of ISI, ICI ad chael oise i a OFDM system, the expressio of caot be obtaied readily. Recetly some attempts have bee made to aalyze the error performace of discrete Fourier trasform (DFT)-based systems such as i [5], [6]. The DFTbased OFDM system is oly a special case of the OFDM where the DFT filter baks are used as modulators ad demodulators. The error performace of geeric OFDM systems, where geeral orthogoal modulatio/demodulatio filter baks are used, icludig discrete wavelet multitoe (DWMT) ad DFT-based OFDM, has ot bee systematically ivestigated. The cotributio of this paper lies i the systematic ivestigatio of the error performace of a geeric OFDM system, with DFT-based OFDM ad DWMT as two realizatios. Both iterfereces ad additive Gaussia chael oise are cosidered i the aalysis. By studyig the costellatios of received symbols ad iterfereces for a give subchael, ad stochastic modelig the sum of ISI ad ICI, the error performace of a give subchael i a geeric OFDM system is aalyzed i the presece of additive Gaussia oise. Furthermore, by adoptig the covetioal optimal phase detector i each subchael, we preset a ew closed-form expressio for a geeric OFDM system employig MPS modulatio. This ew formula is verified with Mote-Carlo simulatios by employig DFT-based OFDM systems ad covetioal DWMT systems. It is show that the theoretical results from the preseted ew formula is cosistet with simulatio results. The Globecom /4/$2. 24 IEEE

2 preseted ew formula provides a viable tool for geeric OFDM system desig ad performace evaluatio. II. SYMBOL ERROR RATE EVALUATION FOR OFDM SYSTEMS WITH MPS MODULATION A. System Modelig The filter bak based modulator ad demodulator, with P subchaels, of a geeric OFDM system are illustrated i Fig.. The modulatio ad demodulatio filters of subchael p, p P, are deoted respectively as f p ad h p, which are of legth L = g, where g is the overlappig factor which is a positive iteger. The upsamplig/dowsamplig factor is doated as. The modulatio filters f p, p P, should fulfill the orthogoality property [3], i.e., for p,p 2 P L l= f p (l)f p 2 (l ) =δ()δ(p p 2 ). () Moreover, the modulatio/demodulatio filters satisfy the perfect recostructio (PR) property defied as [7] L l= f p (l)h p2 ( l) =δ()δ(p p 2 ). (2) The trasmissio chael c() is modeled as a liear time ivariat (LTI) FIR (fiite impulse respose) filteroflegthl c, followed by a statioary zero-mea Gaussia oise source e(), which is idepedet of trasmitted symbols {x p (),p=,,,p }. Note that if a time domai equalizer (TEQ) is employed precedig the filter bak demodulator, c() ca be cosidered as the overall trasfer fuctio resultig from the real commuicatio chael ad the TEQ. For a OFDM system with MPS modulatio scheme, abbreviated as a MPS-OFDM system, a covetioal optimal phase detector is employed for each subchael to calculate the phase of each received symbol ad decode the received symbol with preset phase boudaries. The for a geeric MPS-OFDM system is derived i the followig. B. Symbol Error Rate for Geeric MPS-OFDM Systems I OFDM systems, at each time frame, P parallel symbols {x ( ),x ( ),,x P ( )} are trasmitted, with each symbol modulated to oe of the P subchaels. The trasmitted subchael symbols i subchael p at frame, x p ( ), represet costellatio poits geerated by a modulatio scheme such as MPS. Correspodig to each trasmitted costellatio poit, x p ( ), there is a received symbol y p ( + d), where d is the system delay. If oly the delay ivolved i the covolutio with the modulatio filter ad demodulatio filter, both of legth L = g, is cosidered, the d = g. The followig formulatio gives the expressio of y p ( + d). Formulatio : I a OFDM system show i Fig., suppose the chael impulse respose is c(), =,,, L c, ad the zero-mea additive chael Gaussia oise is e(), the for a give subchael p, the received symbol y p ( + d), correspodig to the symbol x p ( ) trasmitted at time frame, ca be expressed as: y p ( + d) =y p + ξ + ζ, (3) where y p is the cotributio from symbol x p ( ), ξ is the cotributio from chael oise, ad ζ is the iterferece item calculated by summig up all ICI ad ISI items. Note that x p ( ) is the oly trasmitted symbol cotributig to y p, which ca be expressed as y p = α pp ( )x p ( ). (4) The cotributio from the chael oise, ξ, is calculated as ξ = e()h p ( ), (5) = ad the iterferece term ζ ca be calculated as ζ = P = p p,p= +,= α pp ()x p () (6) α pp ()x p (). I (4) ad (6), α pp (), for p, p P, isthe scalar weight of the cotributio from the trasmitted symbol x p () to the received symbol y p ( + d), ad it is calculated as: α pp () = L c j= L c(j) l= f p (l)h p [( ) j l], (7) where L c is the legth of chael impulse respose c(). The derivatio details [8] of the above formulatio are omitted due to the limit of space. Remarks: Whe filter bak based modulator / demodulator ad the commuicatio chael are all liear ad time ivariat, the term y p i (3) is determiistic Globecom /4/$2. 24 IEEE

3 x () f h y () x () f e() h y () x p () f p s() c() + r() h p y p () Fig.. Block diagram for filter bak based modulator ad demodulator i OFDM systems for a give trasmitted symbol x p ( ). Accordig to (4), y p is a weighted versio of x p ( ) ad the weight α pp ( ) is a costat for a give subchael p.therefore i geeral, the costellatio poits of x p ( ) ad y p form a oe-to-oe mappig. For each costellatio poit of x p ( ), there is a correspodig costellatio poit y p, as calculated by (4). All the possible values of y p form the set of costellatio poits at the receivig ed of subchael p. The oise item ξ ad iterferece item ζ are both radom processes. The followig theorem ca be proved [8] to provide a closed-form formula for MPS modulatio by calculatig the probability desity fuctio (PDF) of y p ( + d) ad settig the detectio boudaries for optimal phase detector of each subchael. Theorem : Whe the trasmitted symbols, x p (), p P, are coded by the MPS scheme with symbol eergy ɛ x, ad the optimal phase detector for each subchael is adopted at the receiver ed, the of the received symbol y p ( + d) at subchael p, which correspods to the trasmitted symbol x p ( ), ca be determied as: P e,p = π (M )π M exp [ γ si2 (π/m) si 2 (φ) ] dφ, (8) where α pp γ = ( ) 2 ɛ x, α pp () 2 + α pp () 2 ɛ x + σ 2 p,p p, (9) D. Expressio for Fadig Chaels is the sigal to oise ad iterferece ratio (SNIR), σ 2 is the variace of zero mea additive Gaussia oise e(), ad α pp () is defied i (7). The overall for the α OFDM system, P e, ca be obtaied by averagig P pp p γ = ( ) 2 ρ over all subchaels, P e = P P p = P e,p. () C. Expressio for Additive White Gaussia Noise (AWGN) Chaels For a AWGN chael or a chael which have bee perfectly equalized, c() =δ(). The expressio for a geeric OFDM system ca the be simplified. From (2) ad (7), { p = p ad = α pp () = () otherwise, i.e., the oly symbol that has ozero cotributio to y p ( + d) is x p ( ).Siceσy 2 = σ2,thesnirγ i (9) becomes γ = ρ = ɛ x, (2) which is the sigal-to-oise ratio (SNR). From (), the overall, P e,is σ 2 P e = P e,p. (3) By substitutig () ito (8), we get the for a OFDM system which is idetical to the formula for sigle carrier systems over a AWGN chael. With the icrease of SNR, will decrease quickly. We ca also see from (3) that for a AWGN chael, the umber of subchael, P, has o effect o average system error performace. Fig. 2 shows the results for both sigle carrier modulatio ad the covetioal DWMT modulatio, with differet umbers of subchaels, over a AWGN chael. The modulatio scheme adopted is biary phase shift keyig (BPS). For fadig chaels, we rewrite γ i (9) i a alterative expressio as p,p p. α pp () 2 + α pp () 2 ρ +, (4) Globecom /4/$2. 24 IEEE

4 2 P=64 P=32 P= DWMT: N=32 DWMT: N=8 Sigle carrier Fig. 2. Error performace of sigle carrier modulatio ad the covetioal DWMT (g =2)modulatio with BPS. Fig. 3. Error performace of covetioal DWMT systems (g =2) over chael c() =[,.5e jπ/6,.3e jπ/3,.2e jπ/2,.]. It ca be see that whe SNR ρ is much larger tha (db), P e,p is domiated by the weights α pp (), ad ca be approximated as γ p,p p α pp ( ) 2 α pp () 2 +, α pp () 2. (5) I this case, the performace will ot chage much with the icrease of SNR ρ. Fig.3showsthe error performace of a DWMT system, with overlappig factor g =2, over a sample fadig chael. For DFT-based OFDM systems, from (7), the iclusio of zero-padded cyclic prefix does ot chage the value of umerator i (5), but the value of deomiator will decrease because of the icrease of the upsamplig/dowsamplig factor. Therefore, OFDM systems with cyclic prefix will demostrate superior error performace tha OFDM systems without cyclic prefix. Expressios of (8) ad () provide a systematic approach to calculate the of a geeric OFDM system, which fulfills the orthogoality ad PR coditios i () ad (2). Moreover, the formula ca be used as objective fuctios i the desig of modulatio ad demodulatio filter baks for a OFDM system to achieve better error performace. III. SIMULATIONS To verify our theoretical aalysis of error performace of a geeric OFDM system preseted i Theorem, Mote-Carlo umerical simulatios are performed with three types of OFDM systems, amely, DFTbased OFDM with zero-padded cyclic prefix, the DFTbased OFDM without cyclic prefix ad the covetioal DWMT systems [3] with overlappig factor g = 2, of which the modulatio filters are geerated with prototype filter [ ( π w(l) = si l + )], (6) 2P 2 for l =,,, 2P. Trasmitted data stream x p (),p =,,,P, are geerated usig the QPS scheme. The composite data sequece before eterig the chael, s(), is obtaied as illustrated i Fig.. AWGN oise with a particular SNR is added to obtai distorted data sequece r(). Ther() is passed through the demodulatio filter of each subchael, ad the result sigals are dowsampled by factor. The optimal phase detector is employed i each subchael. The detected symbols are compared with the origial data ad errors are couted to obtai for a particular SNR. Figs. 4 ad 5 show the results of these OFDM systems, with P = 32 subchaels, over two sample multipath fadig chaels with impulse respose c () = [,.5e jπ/6 ] ad c 2 () = [,.5e jπ/6,.3e jπ/3,.2e jπ/2,.2,.] for QPS symbols. The simulatio results demostrate cosistece with the correspodig theoretical aalysis. Globecom /4/$2. 24 IEEE

5 As expected, DFT-based OFDM system with cyclic prefix has superior over the other two OFDM systems, with the cost of trasmissio efficiecy due to zero paddig. The DWMT has the poorest sice both chaels have o arrow bad iterferece ad hece the ISI has more sigificat effect o tha ICI (which DWMT is desiged to miimize). Moreover, sice the filters i DWMT are of loger legth, the dispersive chael could distort more symbols ad produce more ISI. IV. CONCLUSIONS I the past, the error performace of a geeric OFDM system, i which the modulatio filters form a set of geeric orthoormal basis, has ot bee systematically ivestigated. I this paper, a ew closed-form formula is preseted for a geeric OFDM system with MPS modulatio ad optimal phase detectio i each subchael, based o the costellatio poits of received symbols ad statistical aalysis of ISI ad ICI. This formula ca be used to evaluate both DFTbased OFDM ad DWMT systems. Simulatios are performed with DFT-based OFDM systems with ad without zero-padded cyclic prefix, ad the DWMT system with overlappig factor g = 2. The results show the cosistecy betwee theoretical aalysis ad umerical simulatios. The preseted ew formula provides a importat tool to aalyze the performace of a geeric OFDM system ad it provides a realistic way to formulate a objective fuctio to desig a OFDM system [8]. REFERENCES [] V. Migoe ad A. Morello, CD3-OFDM: a ovel demodulatio scheme for fixed ad mobile receivers, IEEE Tras. o Commuicatios, vol. 44, o. 9, pp. 44 5, Sept [2] C. Liu, The effect of oliearity o a QPS-OFDM-QAM sigal, IEEE Tras. o Cosumer Electroics, vol. 43, o. 3, pp , Aug [3] S. D. Sadberg ad M. A. Tzaes, Overlapped discrete multitoe modulatio for high speed copper wire commuicatios, IEEE Joural o Selected Areas i Commu., vol. 3, o. 9, pp , Dec [4] A. N. Akasu ad X. Li, A comparative performace evaluatioofdmt(ofdm)addwmt(dsbmt)baseddsl commuicatios systems for sigle ad multitoe iterferece, i proc. ICASSP, vol. 6, pp , May 998. [5] S. Lei ad V. Lau, Performace aalysis of adaptive iterleavig for OFDM systems, IEEE Tras. o. Vehicular Tech., vol. 5, o. 3, pp , May 22. [6] T. Tjhug, X. Wag, ad C.S. Ng, Error performace evaluatio of the MDPS-DMT systems i AWGN ad impulse oise, IEEE Tras. o Cosumer Electroics, vol. 46, o., pp. 3 36, Feb. 2. [7] P. P. Vaidyaatha, Multirate Systems ad Filter Baks, Eglewood Cliffs, NJ: Pretice Hall, 993. [8] Y. Wag ad X.-P. Zhag, Symbol error rate evaluatio ad filter bak desig for OFDM systems with MPS modulatio, submitted for joural publicatio, DFT based w. CP: aalytical DFT based w. CP: simulatio DFT based w/o. CP: aalytical DFT based w/o. CP: simulatio DWMT: aalytical g=2 DWMT: simulatio g= Fig. 4. Compariso of the aalytical ad simulated performace of OFDM systems with QPS codig over chael c () (P =32). 2 DFT based w. CP: aalytical DFT based w. CP: simulatio DFT based w/o. CP: aalytical DFT based w/o. CP: simulatio DWMT: aalytical g=2 DWMT: simulatio g= Fig. 5. Compariso of the aalytical ad simulated performace of OFDM systems with QPS codig over chael c 2() (P =32). Globecom /4/$2. 24 IEEE