Measurement of EVM (Error Vector Magnitude) for 3G Receivers

Transcription

1 REPORT 1 (78) Measuremet of EVM (Error Vector Magitude) for 3G Receivers Master s Thesis by Osvaldo Medoza Iteratioal Master Program of Digital Commuicatios Systems & Techology, Chalmers Uiversity of Techology, Gotheburg, Swede Chalmers Report Number: EX010/2002 The work was performed at Ericsso Microwave Systems AB, Möldal, Swede February 2002 Supervisors: Zalia Ada Vimar Björk Ericsso Microwave Systems AB Examier: Are Svesso Departmet of Sigals ad Systems, Chalmers Uiversity of Techology

2 REPORT 2 (78) PREFACE This report describes a thesis project for the Iteratioal Master Program of Digital Commuicatios Systems ad Techology, at Chalmers Uiversity of Techology, Gotheburg, Swede. The work was doe at the Base Statio Radio Desig Departmet, TB/RB, at Ericsso Microwave Systems AB, Möldal, Swede, from August 2001 to February I would like to thak my family for all the support give durig my studies, my frieds for the good times spet i Swede, ad the people who cotributed ad helped i the developmet of this project, specially to my supervisors ad examier: Zalia Ada Radio Desiger Egieer TB/RB EMW, Möldal Vimar Björk Radio Desiger Egieer TB/RBT EMW, Möldal Are Svesso Professor Departmet of Sigals ad Systems, Chalmers Uiversity of Techology

3 REPORT 3 (78) ABSTRACT The aim of this thesis work is to measure the Error Vector Magitude (EVM) directly at the radio receiver of the 3 rd Geeratio WCDMA base statios, specifically at the gamma bus, ad the output of the aalogue to digital coverters. The EVM is a figure-of-merit for the dow-coversio of the modulated radio sigal ad is a measure of the sigal to oise ad distortio ratio. The error vector is obtaied by subtractig the received sigal from a ideal referece sigal. The set-up cosists of geeratig a RF sigal usig a sigal geerator, feedig it ito the receiver, ad acquirig it from the iterfaces of iterest usig a logic aalyzer. The ideal referece sigal is created by meas of software. The work doe i this project icludes a algorithm proposal to measure the received sigal at the optimal positio, this algorithm is based o the eye diagram, specifically, i the aalysis of the variace of the sigal s magitude. I additio, aother algorithm was developed to compesate for the effects of frequecy offsets itroduced by the RF local oscillator. The frequecy offsets ca be visualized as phase variatios of the IQ costellatio i the time domai, such phase variatios were tracked ad compesated usig a oe-tap Least Mea-Square adaptive filter. Both algorithms were implemeted i software usig Labview, resultig i a EVM RMS of 6.6% at the gamma bus, ad 4.5% at the ADCs. The sigal measured at the gamma bus was affected by quatizatio oise, therefore, it was low-pass filtered to miimize the problem.

4 REPORT 4 (78) TABLE OF CONTENTS Cotets PREFACE...2 ABSTRACT...3 TABLE OF CONTENTS...4 TERMINOLOGY Itroductio EVM Defiitio Difficulties of the measuremet QPSK Modulatio Measuremet Setup Iterfaces at the TRX Methods Normalizatio Iterpolatio Optimal measuremet positio Frequecy-offset compesatio Results, discussio ad future work Results Discussio Future work Coclusios...40 Refereces...41 Appedix A. LabVIEW VIs...43 Appedix B. I/O Summary...74 Appedix C. Labview Mai Pael Descriptio...76 Appedix D. Equipmet...77 Appedix E. Gamma Bus Format...78

5 REPORT 5 (78) TERMINOLOGY 3GPP 3 rd Geeratio Partership Project ADC Aalog to Digital Coverter AGC Automatic Gai Cotrol AWGN Additive White Gaussia Noise BS Base Statio BTS Base Trasceiver Statio CDMA Code Divisio Multiple Access CPICH Commo Pilot Chael DSSS Direct Sequece Spread Spectrum DUT Device Uder Test EVM Error Vector Magitude FDD Frequecy Divisio Duplex GSM Global System for Mobile Commuicatios IF Itermediate Frequecy IMT-2000 Iteratioal Mobile Telephoy 2000 ITU Iteratioal Telecommuicatios Uio LMS Least Mea-Square LSB Least Sigificat Bit MSB Most Sigificat Bit MSC Mobile Switchig Ceter QPSK Quadrature Phase Shift Keyig RF Radio Frequecy RRC Root Raised Cosie Filter SCH Sychroizatio Chael SIR Sigal-to-Iterferece Ratio SMS Short Message Service SNDR Sigal to Noise ad Distortio Ratio SS Spread Spectrum TDD Time Divisio Duplex TRX Trasceiver UMTS Uiversal Mobile Telecommuicatio Services UTRAN UMTS Terrestrial Radio Access Network VI Virtual Istrumet VSA Vector Sigal Aalyzer WBTS WCDMA Base Trasceiver Statio WCDMA Widebad CDMA

6 REPORT 6 (78) 1 Itroductio This report describes a implemetatio to measure the Error Vector Magitude (EVM), at the receiver of the third geeratio (3G) WCDMA base statios, socalled WBTS. The Error Vector Magitude is defied i the base statio coformace testig techical specificatios of the Third Geeratio Partership Program (3GPP). It is defied as a figure-of-merit for the trasmit modulatio, however, the goal of this project is to use the EVM as a figure-of-merit of the 3G base statio s receivers. The origi of the project comes from the eed to evaluate, i a laboratory eviromet, the performace of the receivers of the trasceiver boards (TRX). I this case there are two iterfaces which are of our iterest, the gamma bus ad the aalogue to digital coverters (ADCs). The gamma bus is used for the trasfer of dow-lik ad up-lik user data. The ADCs iterface the aalogue TX/RX ad the FPGAs. EVM provides a isight ito the sigal s quality that other performace measuremets such as the eye diagram or BER measuremets do ot cover. Oe of the advatages is the simplicity of the measuremet set-up, sice there is o eed for a etire commuicatio system. It possesses a direct relatio with the sigal to oise ad distortio ratio (SNDR) ad ca be used to determie the physical error itroduced at differet stages of a commuicatio system, helpig the desiger to troubleshoot specific problems. The measuremet is doe usig the dow-coverted digitally modulated radio sigal. 1.1 EVM Defiitio 3GPP stadards provide the followig defiitio of EVM: The Error Vector Magitude is a measure of the differece betwee the referece waveform ad the measured waveform. This differece is called the error vector. Both waveforms pass through a matched Root Raised Cosie filter with badwidth 3.84 MHz ad roll-off α =0.22. Both waveforms are the further modified by selectig the frequecy, absolute phase, absolute amplitude ad chip clock timig so as to miimize the error vector. The EVM result is defied as the square root of the ratio of the mea error vector power to the mea referece power expressed as a %. The measuremet iterval is oe timeslot as defied by the C- PICH (whe preset) otherwise the measuremet iterval is oe timeslot startig with the begiig of the SCH. The requiremet is valid over the total power dyamic rage. [2] Figure 1-1 shows the error vector ad its compoets.

7 REPORT 7 (78) Figure 1-1 A illustratio of the error vector ad its compoets A time slot has a legth of 2560 chips (666.6 µs), therefore, the EVM calculatio is doe over bursts of 2560 elemets (N=2560). I this project oe chip is equivalet to oe symbol, ad the calculatio of the error vector is doe at each symbol istat, the, the sigal has to be measured at the istats which best represet each symbol. The variable k is used whe the sigal has ot bee measured yet at the optimal positio, ad each symbol is represeted by more tha oe sample. Oce the sigal has bee sampled at the optimal positio, the variable will be used. I other words, the sigal cotais a total of K samples ad a total of N symbols, beig k ad their respective variables. If we have a over-samplig of T (T poits represet each symbol), the K=NT. Sigal Burst before samplig Burst after samplig Sample at istat k Sample at istat Ideal Referece * S() * s Measured waveform Z(k) Z() z k z Modified versio of the * Z ˆ( ) * measured waveform ẑ * Does ot exist Table 1-1 Notatio of each sigal

8 REPORT 8 (78) The measured waveform is deoted as Z(k) ad cosists of a sigal which has bee corrupted by oise ad frequecy offsets. After this sigal has bee measured at the optimal poit, it will be represeted as Z(). The ideal referece sigal is deoted as S(), it is a sigal free of oise whose magitude has bee ormalized to oe. Z! () is referred as the modified versio of the measured sigal where the frequecy, absolute phase, absolute amplitude ad chip clock timig have bee selected so as to miimize the error vector. Each complex elemet of the bursts is represeted as z, s, z!. See Table 1-1 for a summary of the otatio. The istataeous error vector is obtaied by subtractig the ideal referece from the modified versio of the measured waveform. The root mea square EVM is defied by zˆ N s EVM RMS = 2 s N 1.2 Difficulties of the measuremet 2 (1-1) The first step to take is to fid the proper iterface sice multiple tasks are usually itegrated i a sigle compoet ad it is ot possible to acquire the sigal at certai poits of the system. Aother importat issue is the geeratio of a referece waveform. The received sigal is ot kow ad o specific sychroizatio sigal or predetermied sequece is supplied. Thus it becomes aother task to observe ad detect the sigal such that a referece sigal ca be obtaied. The goal is to create the ideal referece waveform by observig the measured oe. The measured sigal cosists of a sequece of N ukow symbols. It would be possible to observe a pilot sigal which could help us i the geeratio of the ideal waveform, but this ivolves the creatio of complex algorithms. Istead, assumig that the SNR is high eough, this ideal waveform is to be created by detectig each received symbol. I additio, two specific problems represet the mai task to solve i this project: fid the optimal measuremet positio ad compesate for frequecy offsets. Both difficulties are described i more detail i the ext sectios Optimal Measuremet positio EVM is calculated at the symbol rate. The sigal to be aalyzed cosists of a burst of N symbols, each symbol cosists of a certai umber of samples (T) equally distributed over the symbol iterval, the umber of samples represetig a symbol is give by the ratio of the samplig frequecy over the symbol frequecy. If the samplig frequecy equals 4 times the symbol frequecy, the the umber of samples per symbol T is 4. The goal is to fid the positio iside the iterval T that miimizes the EVM. After this is doe, the samples must be extracted at that positio, ad used for the EVM calculatio.

9 REPORT 9 (78) Figure 1-2 shows the eye diagram of a QPSK sigal divided ito its I ad Q compoets. The vertical lies idicate the istat with the smallest distortio ad therefore miimizes the cotributio to EVM. Figure 1-2 Eye diagram of a QPSK sigal, divided ito its I ad Q braches. But i our case, the sigal is affected by frequecy offsets, which makes the eye diagram more difficult to aalyze ad also makes it more difficult to fid the optimal samplig istat. I Figure 1-3 it is possible to observe the effects of a frequecy offset of 1 KHz.

10 REPORT 10 (78) Figure 1-3 Eye diagram of a QPSK sigal divided ito its I ad Q braches, the sigal has a frequecy offset of 1KHz Frequecy offsets The RF sigal set is dow-coverted by a ulocked RF local oscillator (LO), which itroduces frequecy offsets. These frequecy offsets affect the sigal makig it impossible to calculate the EVM without a prior compesatio. To see the effects of frequecy offsets i the time domai, we ca make use of oe of the Fourier s trasform properties, the frequecy shiftig property. It states that if F(ω) is shifted by ω 0 its iverse trasform is multiplied by exp(jω 0 ) 1 I { ω -ω )} = f ( x) exp( jω ) F( 0 0 Therefore, compesatig for frequecy offsets i the frequecy domai ca be traslated as compesatig for phase rotatio i the time domai. Phase rotatio ca be visualized if the received sigal is plotted i the IQ costellatio. I Figure 1-4 it is possible to recogize the 4 possible states of the QPSK sigal ad how have they bee rotated aroud the origi due to a frequecy offset.

11 REPORT 11 (78) 1.3 QPSK Modulatio Figure 1-4 QPSK sigal affected by a frequecy offset of 1Khz. The sigal o which the EVM measuremets are to be doe is i Quadrature Phase Shift Keyig (QPSK) format. I Phase-Shift Keyig modulatio, the phase of a costat amplitude carrier sigal moves betwee differet states ad the iformatio is cotaied i the phase of the carrier. For QPSK the phase of the carrier sigal ca take oe of the followig four equally spaced values: π, 4 π 3, 5 4 π, 4 for i=0,1,2,3. π ad 7, ad the sigal is give by 4 2E π si ( t) = cos 2πf c + (2i + 1) 0 t T T 4 (1-2) E is the trasmitted sigal eergy per symbol ad f c is the carrier frequecy. Sice the phase ca assume four possible states, two bits (dibit) ca be ecoded i each symbol (2 2 =4). The biary sequece is take ad divided ito two differet sequeces, oe icludig the eve-umbered bits ad the other icludig the odd-umbered bits, formig the i-phase (I) ad the i-quadrature (Q) compoets respectively. See Table 1-2. Iput Dibit (IQ) Phase π 4 3π 4 5 π π 4 Table 1-2 Sigal space characterizatio of QPSK

12 REPORT 12 (78) The relatio betwee the eergy E carried by a symbol ad the eergy per bit E b. is give by E = 2E b (1-3) The average probability of symbol error for a QPSK sigal i the additive white Gaussia oise (AWGN) chael ca be expressed as E P = b e erfc (1-4) N 0 where N 0 is the oise desity ad erfc(x) is the complemetary error fuctio defied by 2 2 erfc( x) = exp( y ) dy (1-5) π x For more details about QPSK see Ref. [3].

13 REPORT 13 (78) 2 Measuremet Setup The goal is to measure the EVM at the receiver of the base statio i order to evaluate the uplik performace of the TRX, which is our device uder test (DUT). Sice the base statio supports 2 atea diversity, the receiver possess two idetical braches A ad B. The EVM measuremet ca be doe at ay of both braches. The set-up cosists basically i creatig a RF sigal by meas of a sigal geerator, feedig it ito the TRX, obtaiig the digital modulated basebad sigal usig a logic aalyzer, creatig a referece sigal, ad computig the EVM by software. The set-up is show i Figure 2-1. Figure 2-1 Measuremet setup The RF sigal obtaied from the sigal geerator is fed ito the TRX board via the RF iput. The trasmitted waveform is a WCDMA QPSK sigal with equal amplitude i the I ad Q compoets. The sigal has a symbol frequecy of 3.84 M symbols per secod ad has bee passed through a root raised cosie filter with a roll-off factor of The ceter carrier frequecy is 1950 MHz ad the RF power level is 45 dbm. At the TRX receiver, the sigal is first dow-coverted to a itermediate frequecy, the digitized by the aalogue to digital coverters (ADCs) before beig set to the FPGAs (so-called TRX-DIG). The FPGAs perform the dowcoversio to basebad, decimatio, RRC pulsed matched filterig, ad automatic gai cotrol (AGC). The RRC filters at the receiver have 77 taps. To see a detailed list of the equipmet used ad the sigal geerator parameters refer to Appedix D. 2.1 Iterfaces at the TRX Two iterfaces are to be aalyzed for EVM calculatio: gamma bus ad the output of the ADCs. Figure 2-2 shows their locatio iside the WBTS system. The gamma iterface, with a rate of M samples per secod, is used for the trasfer of dow-lik ad up-lik user data. The RX-gamma-bus cosists of 2 high-speed pairs to sed out the data from the TRX board to the TRX-IF. However, i the laboratory eviromet where the measuremets are doe, the data is set to a test board istead to the TRX-IF. The test board fuctio is to provide the sigal to the logic aalyzer.

14 REPORT 14 (78) The format of the parallel data stream cosists of 10 bits i total, 5 bits for the quatized samples (bits 0-4) where bit 4 is the MSB ad bit 0 is the LSB. The quatized samples are time multiplexed with respect to I ad Q. The remaiig bits (5-9) are used for future expasio, bit-parity, gai factor, ad strobe. The format, show i Appedix E, is the same for braches A ad B. The output of each receiver brach is fed ito 10-bit parallel to serial coverters before beig set to the test board. For more details about the gamma bus cosult ref. [11]. Gamma bus ADC s Figure 2-2 WBTS system block diagram The other iterface of iterest is the output of the aalogue to digital coverters (ADCs). The ADCs, placed i the TRX board, iterface the aalogue TX/RX ad the FPGAs, the output is fed ito the iput of each receiver brach. The sigal cosists of a 14 bit sample stream at a rate of M samples per secod. Both braches use 28 pis i total, which are fed ito the sigal aalyzer usig a test coector [8]. The modules for dow-coversio to basebad ad RRC pulse matched filterig, are implemeted i Labview. The RRC filter i the software implemetatio has 161 taps. The measuremet doe at the gamma bus gives us a figure-of-merit of the aalogue RX, ADCs, ad FPGAs performace. O the other had, the measuremet take from the ADCs output gives us a figure-of-merit of the aalogue RX ad ADCs oly, sice the fuctios doe by the FPGAs (dowcoversio ad RRC filterig) are implemeted i software, ad are cosidered to have a ideal behavior. We have assumed that the sigal set by the sigal geerator is ideal ad has a egligible cotributio to the EVM budget. Based o the measured sigal, a theoretical sigal has to be geerated by the software tool iterally. First the measured sigal is sampled at the optimal positio ad the compesated for the phase rotatio geerated by the frequecy offsets. Simultaeously with the phase rotatio compesatio the sigal is detected ad the theoretical waveform is created. The EVM will be calculated by comparig the theoretical waveform ad the modified versio of the measured sigal.

15 REPORT 15 (78) 3 Methods The proposed solutio which fids the optimal measuremet positio ad compesates agaist frequecy offsets, to fially compute the EVM, ca be divided ito 5 differet blocks: Normalizatio Iterpolatio Samplig LMS Adaptive Filterig/Detectio EVM calculatio Figure 3-1 shows a block diagram of the proposed solutio; otice that the first step is to obtai the sigal at the poit of iterest. There are modules which were already implemeted i Labview to obtai the sigal i basebad form. Figure 3-1 Proposed solutio block diagram The followig sectios describe each block i detail.

16 REPORT 16 (78) 3.1 Normalizatio Here, the costellatio diagram of the measured sigal is ormalized, i other words the mea distace betwee the origi ad the samplig poits is set to oe. The operatio extracts the magitude of the IQ samples, computes the magitude mea over the burst ad scales all the samples so the mea at the ed is equal oe. The ormalized sigal is give by Z( k) Z orm( k) = K 1 1 (3-1) zk K k = 0 where N is the umber of elemets i the burst. This block is applied twice durig the program executio: before iterpolatio, ad after samplig. The reaso of ormalizig twice is because the total power i the burst chages after samplig. Therefore the sigal has to be ormalized agai so the mea distace betwee the origi ad the symbols magitude equals oe. The ame of the module that performs this task i Labview is Normalize IQ Magitude.vi 3.2 Iterpolatio Iterpolatio is ecessary to icrease the umber of samples per symbol iterval T. Havig more samples per symbol iterval results i a better precisio whe selectig the optimal samplig poit. If T is ot large eough, the errors ca be itroduced, which leads to a higher EVM. The origial measured sigal take from the gamma bus has T=4. The sigal take from the ADCs has T=16. It is recommeded to iterpolate the data i the gamma bus i order to have at least T=16. However, usig T=64 i both (gamma bus ad ADC s) sigals, will result i a more accurate measuremet. Iterpolatio is doe by first up-samplig the sigal (fillig up with zeroes betwee samples) ad the filterig the sigal i a low pass (LP) filter. The eed of the LP filter comes from the fact that the zero-valued samples itroduced by the up-sampler have to be coverted to iterpolated samples. The explaatio of this effect is based o the time-frequecy duality which is a property of the Fourier trasformatio. The spectrum of the up-sampled sigal is a L-folded compressed versio of the origial sigal spectrum. Multiple images of the compressed spectrum are created after the up-samplig, thus a LP filter is eeded to suppress the extra images. [9] Figure 3-2 Block diagram of a L-iterpolator If the sequece x[] is passed through a factor -L up-sampler, as show i Figure 3-2, havig x u [] as a output, the relatio betwee the Fourier trasform X u (e jw ) ad X(e jw ) is give by

17 REPORT 17 (78) u jw jwl ( e ) X ( e ) X = (3-2) The LP filter must have a cutoff at π/l (relative to the samplig frequecy, f s =2π) ad a gai of L. The effect that the up-sampler ad the LP filter have over the sigal spectrum is show i Figure 3-3. Figure 3-3. (a) spectrum of the sigal before up-samplig. (b) L-folded compressed versio of the origial sigal spectrum whe L=2. (c) iterpolated sigal spectrum, the up-sampled sigal has bee passed through a low pass filter with L=2.

18 REPORT 18 (78) Figure 3-4 Effects of a iterpolator with L=2. (Top) origial sigal. (Ceter) up-sampled sigal. (Bottom) iterpolated sigal. Fially, after applyig the L-factor iterpolator, the sigal posses L times the umber elemets before iterpolatio. See Figure 3-4. I this applicatio a iterpolatio factor of L=2 is applied, the origial sigal is up-sampled by a factor of two ad a half-bad LP filter is used to suppress the L- folded compressed versio of the origial sigal spectrum. The effect of two iterpolators with L=2 applied i cascade is equivalet to the effect of oe iterpolator with L=4. I the same way a iterpolator with L=8 ca be built usig three iterpolators of L=2, ad so o. Thus, a iterpolator with L=2 is to be used ad applied repeatedly i cascade whe a higher factor L eeded. The module created i Labview called Iterpolator Factor 2.vi iterpolates the sigal by a factor L=2, the iput is a array of N samples ad the output cosists of a array of 2N-1 samples. This implemetatio is divided i two parts: upsamplig ad LP filterig. The up-samplig part iterleaves the array cotaiig the origial sequece with aother array of zeroes of the same size. The LP filterig part performed by the Labview module amed Halfbad Filter.vi covolves the resultig up-sampled sequece with a set of coefficiets listed i Table 3-1.

19 REPORT 19 (78) The Labview module Iterpolator Multiple.vi rus repeatedly times a iterpolator of L=2. The iputs are ad the origial sequece with N samples. The outputs are the iterpolated data by a factor of L=2 with N* samples, ad the umber of samples per symbol iterval. 3.3 Optimal measuremet positio This sectio describes a algorithm to fid the optimal measuremet positio. A good way to start is takig a look at the eye diagram. The eye diagram or eye patter is the result of traslatig the waveforms i successive symbol itervals, ito oe iterval o the display. The term eye comes from the similarity of this graphic tool with the huma eye. [3] The iput is the received sigal Z(k). It has a symbol iterval legth of T samples, ad a total legth of N symbols. The total umber of samples i the burst equals K=NT. The output displayed is give by Coefficiet Value h E-3 h E+0 h E-2 h E+0 h E-1 h E-1 h E-1 h E+0 h E-2 h E+0 h E-3 Table 3-1 Half bad filter coefficiets z t, =z k-t-t (3-3) for t=0,1,...,t-1 ad =0,1,...,N The display is periodic ad idetical for each symbol iterval: T t ( +1) T (3-4) where =1,2,...,N.

20 REPORT 20 (78) At the ed we observe the patters displayed i Figure 1-2. The eye opeig is the distace from the zero level to the closest trace at the optimal samplig istat (where the eye opes widest). The wider the opeig the better the quality of the sigal. The closig of the eye alog the time axis is idicative of distortio ad jitter. [3] [10]. The proposed algorithm is based o the aalysis of the variace of the IQ sigal magitude at each samplig istat 0 t T 1 whe lookig at the eye diagram. The lowest variace will be obtaied at that istat where most of the crossigs coicide, i.e. the best samplig istat. The algorithm cosists i the followig steps: 1. Get burst of K samples: Z(k) k=0,1,2,...,k-1 2. Divide Z(k) i T vectors: v 0 =[z k-0, z k-t, z k-2t,...,z k-(n-1)t ] T v 1 =[z k-1, z k-1-t, z k-1-2t,..., z k-1-(n-1)t ] T v 2 =[z k-2, z k-2-t, z k-2-2t,..., z k-2-(n-1)t ] T : v T-1 =[z k-(t-1), z k-(t-1)-t, z k-(t-1)-2t,..., z k-(t-1)-(n-1)t ] T i geeral v t, =z k-t-t for t=0,1,2,...,t-1 ad =0,1,2,...,N-1 3. Obtai the magitude of each elemet of the vectors v t : m 0 =[ v 0,0 v 0,1 v 0,2... v 0,N-1 ] T = [ z k-0, z k-t, z k-2t,..., z k-(n-1)t ] T m 1 =[ v 1,0 v 1,1 v 1,2... v 1,N-1 ] T = [ z k-1, z k-1-t, z k-1-2t,..., z k-1-(n-1)t ] T m 2 =[ v 2,0 v 2,1 v 2,2... v 2,N-1 ] T =[ z k-2, z k-2-t, z k-2-2t,..., z k-2-(n-1)t ] T : m T-1 =[ v T-1,0 v T-1,1 v T-1,2... v T-1,N-1 ] T =[ z k-(t-1), z k-(t-1)-t, z k-(t-1)-2t,..., z k-(t-1)-(n-1)t ] T 4. Calculate the variace of each magitude vector m t { 1 } { } 2 σ 1 = var m 2 σ 2 = var m 2 " 2 σ 1 var m { } = T T 1 Each of the variaces σ 2 t calculated correspods to a samplig istat t from 0 to T-1. Therefore it ca be cosidered as a fuctio of time σ 2 (t) 5. The optimal samplig positio is the oe that leads to the lowest variace. I other words, the argumet of the miimum of the variace fuctio, give by

21 REPORT 21 (78) { mi{ ( ) } t opt = arg σ 2 t (3-5) 6. Oce the optimal samplig positio has bee determied it is possible to extract the sample represetig each symbol i the followig fashio for =0,1,2,...,N-1. Z ) = Z( k T t ) (3-6) ( opt Below i Figure 3-5, the magitude of I+jQ of the burst, has bee plotted i the same way as a eye diagram, observe how most of the values coicide at samplig positio 60. Figure 3-5 Eye diagram of I+jQ. N=2560. T=64. At Figure 3-6 is possible to observe the variace at each istat correspodig to the same sigal show i Figure 3-5. Note how the variace reaches its miimum at t opt =60.

22 REPORT 22 (78) Figure 3-6 Variace observed at each istat iside the symbol iterval. N=2560. T=64. This algorithm performs well regardless of the phase rotatio, sice it is based o the magitude of the samples. The larger the umber of samples per symbol the more accurate solutio. Iterpolatio represets a trade-off sice as we use more samples, more data has to be processed ad the algorithm becomes slower. However, i this case, it is recommeded to have a larger umber of samples per symbol iterval which results i more exact measuremets. Sice this method works based o statistical iformatio of the sigal, it is suggested to use more tha 50 symbols per burst processed to esure a good performace. Nevertheless, for this applicatio the umber of symbols to be processed will be equal or greater tha Figure 3-7 IQ costellatio of the QPSK sigal before ad after samplig. The proposed algorithm described is implemeted i the Labview module called Sample.vi, this block is composed by two parts: the first oe which fids the optimal measuremet positio is performed by the module called Fid Samplig Poit.vi, ad the secod oe which extracts the samples at the idicated measuremet positio is performed by the module Extract Samples.vi. The iputs for the first part are the data burst ad the umber of samples per symbol iterval, the output is the optimal measuremet positio. For the secod part the iputs are the data burst, the umber of samples per symbol iterval ad the optimal measuremet positio. The output is the array of samples take at the optimal measuremet positio.

23 REPORT 23 (78) The Labview module Sample.vi has as iputs the data burst ad the umber of samples per symbol iterval. Ad as outputs the array of samples take at the optimal measurig positio, ad the optimal measurig positio itself. Eve though we have maaged to sample the sigal, the effects of the frequecy offsets are still preset, ad the EVM caot be computed yet, otherwise its value would go to ifiity. I the followig sectio a method to compesate for these frequecy offsets is described. 3.4 Frequecy-offset compesatio Compesatig for a frequecy offset ivolves a trackig process for correctig the phase rotatio chages ad extractig the iformatio relevat for EVM calculatio which is phase oise ad magitude errors. The problem becomes more difficult sice phase rotatio has to be de-embedded from the phase oise. However, we take the advatage of the fact that the phase respose as a fuctio of frequecy is cosidered as a determiistic parameter ad phase oise as a radom parameter. Figure 3-8 illustrates the problem for 1000, 2000, 3000, ad 4000 symbols, as more ad more data is plotted the rotatio of the IQ costellatio becomes more remarkable.

24 REPORT 24 (78) Figure 3-8 Illustratio of phase rotatio Here we ca create a model for the measured sigal. Each received symbol has a magitude which has bee affected by AWGN, ad its phase possess 3 compoets: the ideal phase of the trasmitted symbol exp(jθ ), the phase error added due to AWGN exp(jψ ), ad the phase compoet due to frequecy offsets exp(jφ ). Therefore the model for each symbol z is give by z = A exp( j( θ + ψ + φ )) (3-7) Our objective is to de-embed each of the phase parameters. I this case, we assume that the compoet due to a frequecy offset has a liear behavior defied by φ = v b (3-8) +

25 REPORT 25 (78) where v is the parameter which defies the phase rotatio (umber of radias per symbol), ad b is the iitial phase offset. We will make use of a Least Mea- Square adaptive filter to extract the phase compoet due to a frequecy offset exp(jφ ), ad durig this process it is possible that part of the phase oise exp(jψ ), is extracted together. However the assumptio that φ behaves liearly will help us to differetiate it from ψ which is a zero-mea Gaussia process. The Least Mea-Square (LMS) algorithm belogs to the family of stochastic gradiet-based algorithms ad its mai feature is the simplicity, which leads to a low complexity implemetatio. The LMS describes a feedback cotrol system ad cosists of a combiatio of two basic processes [4]: 1. A adaptive process, which ivolves the automatic adjustmet of a set of tap weights. 2. A filterig process, which ivolves (a) formig the ier product of a set of tap iputs ad the correspodig set of tap weights emergig from the adaptive process to produce ad estimate of a desired respose, ad (b) geeratig a estimatio error by comparig the estimate with the actual value of the desired respose. The estimatio error is i tur used to actuate the adaptive process, thereby closig the feedback loop. However, the sigal at this momet is ot cotiuous ad each received symbol z has a arbitrary ideal phase (θ =π/4, 3π/4, 5π/4, 7π/4). Subsequet samples are cosidered u-correlated betwee each other with respect to the ideal phase, therefore, the umber of filter coefficiets is oe for this applicatio (w ), ad the algorithm performs aalyzig sample per sample, obtaiig the error at each time, ad usig it to correct the ext icomig symbol so as to miimize the cost fuctio. The error obtaied at each istat is a fuctio of ψ ad φ ad plays ad importat role to de-embed both variables, as we will see further o. Notice that the estimatio error computed here is ot the same as the error vector defied i sectio 1. The estimatio error e is used to de-embed the phase rotatio that affects the received sigal. The cost fuctio is defied by 2 ξ = E{ } (3-9) e e is the estimatio error at istat, give by e = sˆ s (3-10) where s is the value of the desired respose ad ŝ is the estimate. The we eed to apply a correctio update to the filter coefficiet w, the update must lead to a miimizatio of the cost fuctio ad has the followig form w +1 = µ ξ (3-11) w

26 REPORT 26 (78) The explaatio of this update comes from the fact that the gradiet of the cost fuctio poits towards the directio it icreases. Thus, the update cosists of a small correctio toward the directio it decreases, ad that is the reaso why we use the egative sig. The parameter µ affects the rate at which the weigh coefficiet moves toward the optimal solutio. The gradiet correspods to the derivative of E{ e 2 } with respect to w *, give by 2 ξ = E{ e } = E{ e = E{ e e The gradiet of the estimatio error is give by * 2 } } * * z (3-12) e = (3-13) thus, the gradiet of the cost fuctio becomes * ξ = E{ e z } (3-14) The simplicity of the LMS algorithm comes whe the expressio E{e z * } is replaced by a oe poit sample mea e z *. The the update equatio assumes the followig form give by w + = w µ e z (3-15) * 1 this equatio is the key i the adaptatio process of the LMS algorithm ad we have to recall that it was derived for the case whe the umber of taps i the filter is oe. The filter coefficiet at the istat, w, is updated for each ew sample as defied i Eq. (3-15). The estimated sigal sample ŝ is defied as sˆ = w z (3-16) * for N. Each estimated symbol ŝ has to be compared with a referece symbol s to compute the error, but at that time it is ukow which symbol has bee set. The ŝ is detected to create the ideal referece. Detectio theory deals with the desig ad evaluatio of a decisio-makig processor that observes the received sigal ad guesses which particular symbol was trasmitted accordig to some set of rules [3].

27 REPORT 27 (78) I QPSK the sigal costellatio cosists of 4 equally spaced symbols: m 0, m 1, m 2, m 3 : m 0 π = cos + 4 π j si 4 m m m π = cos π = cos = cos 7 π + 4 π j si 3 4 π j si 5 4 π j si 7 4 therefore, the observatio space is partitioed i 4 regios B 0, B 1, B 2, B 3. Each regio is divided by the decisio boudaries, which i this case are the I ad Q! axis. The rule for detectio cosists i settig m = mi if the distace betwee the received symbol z k ad m i is the miimum for i=0,1,2,3. This method is called maximum-likelihood detectio ad basically selects the message poit closest to the received sigal poit. [3] Each geerated symbol resultig from detectio has the form give by s = Aexp( jθ ) (3-17) where the ideal amplitude A equals always 1, ad θ ca assume oe of the four possible ideal agles π/4, 3π/4, 5π/4, 7π/4. Oce we cout with a referece symbol ad a estimated symbol it is possible to obtai the estimatio error at the istat. The estimatio error is used to close the feedback loop of the adaptive process, as show i Figure 3-9. Figure 3-9 LMS adaptatio process.

28 REPORT 28 (78) The respose of the algorithm ca be tued through the step size µ. The step size µ modifies the speed of covergece, if this value is small we get a more accurate respose whe t but the speed of covergece is slow. If µ is large the the respose whe t is less accurate ad speed of covergece is fast. Nevertheless, for stability, µ is upper ad lower bouded by 2 0 < µ < (3-18) λ max where λ max is the maximum eigevalue of the autocorrelatio matrix R z. It is possible to use the fact that λ max is upper bouded by the trace of R z p λ max λi = tr( R z ) (3-19) i= 0 Eq. (3-19) is desiged for the geeral case where the umber of taps i the LMS filter equals p+1, beig p the order of the filter (0 i our case). If we cosider Z() as a wide-sese statioary process, the the trace ca be replaced by 2 tr( R ) (umber of taps) E{ Z( ) } (3-20) z = therefore, E{ Z() 2 } ca be estimated from a average as N E ˆ{ Z( ) } = z (3-21) N = 0 ad the boudaries for the step size ow become 2 0 < µ < N (3-22) (umber of taps) z N = 0 However, our applicatio requires a umber of taps equal to oe, the Eq. (3-22) ca be simplified to 2 0 < µ < N (3-23) z N = 0 The upper boud is geerally cosidered too large to esure stability. It might coverge i the mea but probably with a large variace, which most of the times is uwated. Commo values rage aroud oe order less tha the upper boud. Thus, for this applicatio we set µ equal the upper boud ad itroduce a ew factor called β whose default value is 0.1 ad ca also be adjusted by the user. Hece, we defie a ew step size parameter µ defied by µ is defied as µ ' = β µ (3-24)

29 REPORT 29 (78) µ 2 = N 1 1 z N = 0 fially β ca be varied betwee the rage give by 2 (3-25) 0 < β <1 (3-26) w 0 is iitialized as z 0 is the first icomig symbol. w = 1 0 (3-27) The LMS algorithm ca be summarized i the followig steps [5]: 1. Parameters: Number of taps= 1 β=step size adjustmet (0.1 default) 2. Iitializatio: w = Computatio: For =0,1,2,...,N-1 a) ŝ = w * z b) s =detect( ŝ ) c) e = ŝ - s * d) w +1 = w + µ e z After the etire sequeces S() ad Ŝ () have bee computed it is still ot possible to calculate the EVM. Observe Figure 3-10 at the right Figure 3-10 (Left) detected sigal. (Right) illustratio of the trasiet respose.

30 REPORT 30 (78) the first values of Ŝ () formig a wake towards the referece poits belog to the trasiet respose. They should ot accout for the EVM sice the algorithm has ot bee adapted yet ad might lead to a erroeous measuremet. Oe way to avoid this problem is to calculate the time that takes for the algorithm to coverge ad skip the symbols icluded i that period of time. Before goig further lets first state the misadjustmet. The misadjustmet is defied as the ratio of the excess mea-square error to the miimum mea square error, ad is thus a measure of how closely the adaptive process tracks the true Wieer solutio, [6]. Usually a misadjustmet of 10% is cosidered satisfactory. The misadjustmet is give by Excess MSE M = (3-28) ξ mi tr( R ) µ z 1 µ (umber of taps) N N 1 = 0 2 z The misadjustmet M ad the time costat τ MSE are related through the followig approximatio (umber of taps) M (3-29) 4τ MSE this approximatio assumes that all the eigevalues are equal, however, it has show to be a good guess eve whe the eigevalues are ot equal [6]. The algorithm takes four time costats to coverge, ad the total trasiet time ca be expressed as Number of taps t s (3-30) Missadjustmet for our case 1 t s (3-31) Missadjustmet Figure 3-11 shows the squared error after a arbitrary sigal has bee passed through the oe-tap LMS filter, with a step size equal to oe teth of the upper boud ad a misadjustmet of 10%. This gives the followig settlig time: 1 t s = = 10 (3-32) 0.1

31 REPORT 31 (78) Figure Trasiet respose of the squared error. We have prove i practice that the expressio derived for the settlig time i Eq. (3-30) is a good estimate. If we take a look to the phase of the filter coefficiet w after the algorithm has coverged, we observe that we have a rough estimate of φ which we ca deote as φˆ. Therefore, the filter coefficiet w is defied by w = W exp( jφˆ ) (3-33) where W is the magitude of w. To be able to de-embed φ it is ecessary to obtai the best liear fit of φˆ alog the burst, give by for =0,1,2,...,N-1. φ = liear fit { φˆ } (3-34) We ca defie a ew coefficiet c whose phase equals exp(-jφ ). The magitude of c is give by c = 1 (3-35) The multiplyig c by z for all N, results i a estimated sigal Z! () whose magitude of each symbol equals the magitude of its respective symbol i Z(). The resultig ẑ is give by zˆ = z c (3-36) Fially ψ is the differece betwee the phase agle of ẑ ad θ. The phase oise ψ must have a mea equal to zero for all N, ad to be sure of it the mea is calculated ad removed if preset. Ay residual phase offset take away i the followig way N 1 1 Zˆ( ) = Zˆ( ) exp( j ψ ) (3-37) N =0

32 REPORT 32 (78) Z! () is referred as the modified versio of the measured sigal where the frequecy, absolute phase, absolute amplitude ad chip clock timig have bee selected so as to miimize the error vector. The compesatio doe agaist frequecy offsets ivolved oly the phase. The magitude of the origial sigal remaied uaffected. Now it is possible to compute the EVM as defied i Eq. (1-1).

33 REPORT 33 (78) 4 Results, discussio ad future work 4.1 Results Followig the setup described i sectio 2 ad the algorithm implemetatio described i sectio 3, two differet results were obtaied: EVM RMS at the gamma bus, ad EVM RMS at the aalogue to digital coverters. EVM RMS measured at the gamma bus was 6.6%. Figure 4-1 shows Z ˆ( ) obtaied from the gamma bus, the crossigs of the straight grid lies represet the four ideal states. ˆ Figure 4-1. Z ( ) measured at the gamma bus. EVM RMS measured at the ADCs was 4.5%. Figure 4-2 shows Z ˆ( ) take from the ADCs. Note how i this case the received symbols are closer to the ideal refereces tha i the sigal take at the gamma bus.

34 REPORT 34 (78) ˆ Figure 4-2. Z ( ) measured at the ADCs. Figure 4-3 shows a sigal obtaied from the gamma bus, before ad after beig measured at the optimal positio. ˆ Figure 4-3. Z ( ) measured at the gamma bus. (Left) before samplig. (Right) after samplig.

35 REPORT 35 (78) The origial umber of samples per symbol iterval at the sigal show i Figure 4-3 was 4. After beig iterpolated with a factor of L=16, the resultig oversamplig was T=64. For the sigal take at the ADCs the origial umber of samples per symbol iterval was 16, ad after iterpolatio with L=4 the resultig over-samplig is T=64. Below i Figure 4-4, φˆ, φ, ad -φ, have bee plotted. They correspod to the sigal aalyzed i Figure 4-3. The vertical axis is i radias ad the horizotal axis represets. Observe that the iitial phase offset b is equal to 0.15 rads ( 8.6 degrees), ad v has a value of 1.95*10-5 rads/symbol. Figure 4-4. Illustratio of Hece, φ for this specific burst ca be modeled as φˆ, φ, ad -φ. φ 5 = rads Figure 4-5 shows the EVM versus time, correspodig to the same sigal aalyzed i Figure 4-3. Figure 4-5. EVM vs time. After applyig the same liear compesatio over the origial received sigal Z(k), it is possible to plot the eye diagram of I ad Q braches as show i Figure 4-6.

36 REPORT 36 (78) Figure 4-6. Eye diagram of Z(k) after removal of frequecy offsets. Observe how the widest opeig of the eye occurs at t opt =60. Coicidig with the results show i Figure 3-5 ad Figure 3-6. The Labview virtual istrumets (VI s) which resulted from the implemetatio of the algorithms are show i Appedix A. The results preseted here were obtaied usig such VI s, ad every iput ad output is described i Appedix B. 4.2 Discussio The differece betwee the EVM RMS obtaied at the gamma bus ad the oe obtaied at the ADCs is attributed to the differet umber of taps used i the RRC filters of the TRX ad the software implemetatio. A larger umber of taps leads to a better sigal quality ad lower EVM. Besides, the RRC matched filters applied to the sigal observed at the ADCs are implemeted i software ad have a ideal behavior. A problem was foud i the sigal measured at the gamma bus. The values of the origial 4 samples represetig each symbol fall iside certai regios ad create gaps betwee each regio. This effect is attributed to the quatizatio oise caused by roud-off errors, sice 5 bits are used to represet each sample. Figure 4-7 shows the I+jQ magitude of a sigal obtaied from the gamma bus. The sigal has ot bee iterpolated. The optimal measuremet positio is t opt =2; observe the gaps caused by quatizatio oise. Oce the samples were extracted at t opt ad plotted at the IQ costellatio, they follow a patter of circular lies separated by small gaps. See Figure 4-8. I order to solve this problem, the origial sigal was low-pass filtered prior to ay processig. The low pass filter removes the quatizatio oise which is spread all over the spectrum. Sice the badwidth of the RRC filters equals oe quarter of the samplig frequecy (0.25f s ), the cut-off frequecy of the low-pass filter must be higher tha this quatity. For istace at 0.4f s. If the LP cut-off frequecy gets closer to the RRF filters badwidth the the quality of the sigal is degraded ad the EVM icreased.

37 REPORT 37 (78) Figure 4-7. Illustratio of quatizatio oise i the eye diagram of the sigal s magitude. Figure 4-8. Illustratio of quatizatio oise i the IQ costellatio.

38 REPORT 38 (78) Usig LP filterig the effects of quatizatio oise ca be lowered. See Figure 4-9 which shows the same sigal as i Figure 4-8 but ow LP filtered. Figure 4-9. Illustratio of the quatizatio oise removal results. The problem becomes imperceptible at the sigal measured at the ADCs where 14 bits are used to represet each sample. At the ADCs the quatizatio oise is egligible. So far the results have demostrated how the EVM measuremets help the desiger i the troubleshootig of sigal impairmets. First by showig the differece i the sigal quality betwee two iterfaces observed. Secod, by makig visual the effects of quatizatio oise. Although 3GPP stadards do ot specify EVM testig for the receiver side, the tool has provided a figure-of-merit of it, ad its use ca still be exteded to other areas. We followed the defiitio, which refers to the trasmit modulatio, wheever it had applicability to the receiver side. But the reaso that we used LP filterig, evertheless it is ot specified i the stadard, is to reduce the effects of parameters which could lead to a erroeous measuremet. Ufortuately at this momet we do ot cout with aother tool which could give us a compariso poit of our results. However, similar outcomes are expected if ay other device is used. Some iterestig papers relate similar work doe to measure the EVM. I Ref. [12] a method to compute EVM i GSM (8-PSK) systems is described. The device uder test is a oliear amplifier. A steepest descet method is used to de-embed frequecy offsets, amplitude chages, carrier leakage, ad arbitrary phase ad output power of the trasmitter.

39 REPORT 39 (78) Aother paper which may be of iterest to the reader is Ref. [13]. This paper gives a full descriptio of EVM as well as hits to use the measuremet for troubleshootig i digital RF commuicatios systems. Other research papers that may also be of iterest ad have iflueced my work are the listed i Ref. [14] to [18]. Such papers describe specific EVM applicatios over differet scearios. 4.3 Future work If this project is to be cotiued, it is ecessary to fid aother tool which could give us a poit of a compariso. Either a commercial implemetatio or aother algorithm so as to validate our results. If aother algorithm is to be created, it is recommeded to make use of a pilot sigal. This may ivolve some complex work i decodig but ca provide aid i the compesatio of frequecy offsets. It is also ecessary to keep track of the evolvig stadards, sice o method is specified to compesate agaist frequecy offsets. This leads to a ambiguity i the iterpretatio of how the sigal must be modified prior to EVM measuremets. This leads to slight differeces i the outcomes depedig o the algorithms used i differet measuremet equipmet. If EVM has bee itroduced to the stadards, the the method to obtai it should also be stadardized.

40 REPORT 40 (78) 5 Coclusios We have preseted the work doe to measure the EVM directly at the output of the third geeratio WCDMA base statio receivers. Algorithms for fidig the optimal measuremet positio ad frequecy-offset compesatio were preseted. EVM provides a figure-of-merit of the dow-coverted basebad sigal. Although the stadards refer to it as a way to measure the quality of the trasmit modulatio, we have prove that it ca also represet a figure-of-merit for the receivers. EVM is defied as the magitude of the vector resultig from the differece betwee a ideal referece sigal ad a measured sigal. Such differece is calculated at symbol rate. The sigal aalyzed was a WCDMA QPSK with equal amplitude i the I ad Q compoets. It was created by a sigal geerator, fed it ito the RF iput of the receiver ad acquired at the iterfaces of iterest usig a logic aalyzer. The ideal referece sigal was created by meas of software. Oe of the mai tasks solved i this project was the implemetatio of a algorithm which fids the optimal measuremet positio. The solutio preseted for this problem is based o the eye diagram, specifically, i the aalysis of the variace of the sigal s magitude. Oce the samples were extracted at the optimal samplig positio we still faced the problem of frequecy offsets itroduced by the RF local oscillator. Frequecy offsets ca be see as a phase rotatio of the IQ costellatio. The proposed solutio which compesates for this effect cosists of a oe-tap LMS adaptive filter. It tracks the movemets of the IQ costellatio ad provides a rough estimate of the phase rotatio. Such rough estimate was used to obtai a best liear fit, cosidered as the compesatio to be made to the sigal. The reaso of the liear fit is the assumptio that the phase rotatio is liear. Withi the adaptive algorithm loop, a detector was implemeted, its fuctio is to geerate a referece ideal sigal which is used to compute the EVM. Both algorithms were implemeted i Labview ad two iterfaces were aalyzed, the gamma bus ad the output of the aalogue to digital coverters. The EVM RMS measured at the gamma bus was 6.6%. The EVM RMS measured at the ADCs was 4.5%. The differece is attributed to the umber of taps used i the RRC filters at the receiver, which is lower tha the umber of taps used i the software implemetatio of the RRC filters applied to the ADCs sigal. A problem of quatizatio oise was foud at the gamma bus, affectig our EVM measuremet. The solutio to compesate agaist the quatizatio oise effects was a LP-filter with a cutoff frequecy of 0.4f s. Although this problem was out of the scope of project, it was ecessary to correct i order to avoid errors i the measuremets.

41 REPORT 41 (78) Refereces [1] Holma, H. Ad Toskala Atti, WCDMA for UMTS: Radio Access For Third Geeratio Mobile Commuicatios, Joh Wiley & Sos, Eglad, [2] 3GPP Techical Specificatio 25141, Base statio coformace testig (FDD). [3] Hayki, S., Digital Commuicatios, Joh Wiley & Sos, New York, [4] Hayki, S., Adaptive Filter Theory, 2 d ed., Pretice Hall, Eglewood Cliffs, N.J., 1991 [5] Hayes, M., Statistical Digital Sigal Processig ad Modelig, Joh Wiley & Sos, New York, [6] Widrow, B., Stears,S., Adaptive sigal processig, Pretice Hall, N.J., [7] Rappaport T., Wireless Commuicatio: Priciples & Practice, Pretice Hall, N.J., [8] CEH /1 Ue, Desig Specificatio for the Digital Part of the TRX, TRX-DIG. EMW Iteral Documet. [9] Mitra, S., Digital Sigal Processig: A Computer-Based Approach, McGraw Hill, [10] Jeruchim, M., Babala, P., Shamuga, S., Simulatio of Commuicatio Systems, Modelig Methodology ad Techiques.2 d ed., Pleum, [11] 8/ CSX Ue, Commo Parts IWD Gamma IF TRX/RX. EMW Iteral Documet. [12] Mashhour A., Borjak A., A Method for Computig Error Vector Magitude i GSM EDGE Systems Symulatio Results. IEEE Commuicatios Letters, Vol. 5, 3, March [13] Agilet Product Note , Usig Vector Modulatio Aalysis i the Itegratio, Troubleshootig, ad Desig of Digital RF Commuicatios Systems. [14] Hassu R., Flaherty M., Matreci R., Taylor M., Effective evaluatio of lik quality usig Error Vector Magitude techique, i Proc. IEEE Wireless Commuicatios Coferece, 1997, pp [15] Heutmaker M., The Error Vector ad Power Amplifier Distortio, i Proc. IEEE Wireless Commuicatios Coferece, 1997, pp

42 REPORT 42 (78) [16] Nakagawa T., Araki K., Effect of Phase Noise o RF Commuicatio Sigals, i Proc. IEEE VTC, 2000, pp [17] Spriger A., Frauscher T., Adler B., Pimigsdorfer R., Weigel R., Impact of Noliear Ampliers o the UMTS System, IEEE 6 th It. Symp. O Spread- Sprectum Tech. & Appli., 2000, pp [18] Dow S., Yag J., Ye K., Matreci R., Spotted-Elk E., Pettis S., Trih L., Vector Sigal Characterizatio of 38GHz Power Amplifier with 100 Mbps QPSK Modulatio, i Proc. IEEE MTT-S Digest, 2000, pp

43 REPORT 43 (78) Appedix A. LabVIEW VIs QPSK Samplig & EVM.vi -Reads Data from: Gamma bus & ADC's at the TRX -Measures the QPSK data at the optimal positio. -Compesates agaist the efects of frequecy offsets. -Computes the Error Vector Magitude over bursts of 2560 symbols Coector Pae Frot Pael

44 REPORT 44 (78)

45 REPORT 45 (78) Cotrols ad Idicators Visible symbols Eye Diagram Phase Correctio Testmux Yes/No Brach A/B Boolea Offset smiq Dig mod o/off W3GPP o/off Geerator RF RF Level Frequecy Beta Iterpolatio Factor Iterface Aalyzer LP filterig to remove quatizatio oise Cutoff Freq. Best Samplig Poit Sampled sigal Z() Received sigal before samplig Z(k) EVM Peak % EVM RMS% Estimated Sigal Z^() without trasiet EVM vs Time % Eye diagram Q brach Eye diagram I brach Step Size u' Output Samples per Symbol Phase rotatio, C(). (rads) Block Diagram

46 REPORT 46 (78)

47 REPORT 47 (78) List of SubVIs LMSew.vi /proj/gr_labview/measure/osvaldo_temp/evm/lmsew.vi Remove Mea & Normalize Magitude.vi /proj/gr_labview/measure/osvaldo_temp/evm/remove Mea & Normalize Magitude.vi Sample.vi /proj/gr_labview/measure/osvaldo_temp/evm/sample.vi Calculate EVM.vi /proj/gr_labview/measure/osvaldo_temp/evm/calculate EVM.vi Complex Eye Diagram.vi /proj/gr_labview/measure/osvaldo_temp/evm/complex Eye Diagram.vi

48 REPORT 48 (78) Upsample phase correctio.vi /proj/gr_labview/measure/osvaldo_temp/evm/upsample phase correctio.vi Get_Logic_Aalyzer_data.vi /proj/gr_labview/istrumet/hp/logic_aalyzer/hp16700.llb/get_logic_aalyzer_data.vi Covert_HP16700_data_testboard1.vi /proj/gr_labview/measure/wbts/trx/rx_test/wbts_testboard1/bbtt_trxctrl.llb/cover t_hp16700_data_testboard1.vi ge_offset.vi /proj/gr_labview/istrumet/geeral/ge_offset.vi Toggle_Data_Testboard.vi /proj/gr_labview/measure/wbts/trx/rx_test/wbts_testboard1/bbtt_trxctrl.llb/toggle _Data_Testboard.vi Covert_HP16700_data_testco.vi /proj/gr_labview/measure/wbts/trx/rx_test/rx measuremets from test coector/rx_measure_usig_hp16702.llb/covert_hp16700_data_testco.vi Complex Remove mea.vi /proj/gr_labview/measure/wbts/trx/rx_test/wbts_rxsub.llb/complex Remove mea.vi Normalize IQ magitude.vi /proj/gr_labview/measure/osvaldo_temp/evm/normalize IQ magitude.vi Complex Up ad Dow Coverter.vi /proj/gr_labview/measure/wictoria_rx/wic_rxsub.llb/complex Up ad Dow Coverter.vi Halfbad Filter.vi /proj/gr_labview/measure/wictoria_rx/wic_rxsub.llb/halfbad Filter.vi RRC filter.vi /proj/gr_labview/measure/wictoria_rx/wic_rxsub.llb/rrc filter.vi Iterpolator Multiple.vi /proj/gr_labview/measure/osvaldo_temp/evm/iterpolator Multiple.vi Phase_Rotatio_Compesatio.vi /proj/gr_labview/measure/osvaldo_temp/evm/phase_rotatio_compesatio.vi LP Filter.vi /proj/gr_labview/measure/osvaldo_temp/evm/lp Filter.vi History "QPSK Samplig & EVM.vi History" Curret Revisio: 342 rev. 1 Tue, Ju 28, :57:21 AM Apostolos K SubVIs of this VI were modified. VI was coverted from LabVIEW versio to LabVIEW versio 3.1a16.

49 REPORT 49 (78) LP Filter.vi Coector Pae Frot Pael Cotrols ad Idicators Iput Data low cutoff freq: fl Filtered Data Block Diagram List of SubVIs Complex Covolutio.vi /proj/gr_labview/measure/wictoria_rx/wic_rxsub.llb/complex Covolutio.vi FIR Widowed Coefficiets.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/3filter.llb/fir Widowed Coefficiets.vi History "LP Filter.vi History" Curret Revisio: 14

50 REPORT 50 (78) Normalize IQ magitude.vi Normalizes the icommig data. The output possess absolut magitude. Coector Pae Frot Pael Cotrols ad Idicators Data Iput Normalized Data Block Diagram List of SubVIs Mea.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/5stat.llb/mea.vi History "Normalize IQ magitude.vi History" Curret Revisio: 2

51 REPORT 51 (78) Iterpolator Multiple.vi Iterpolates the Iput Data by a factor of L=2^. ca have values equal t 1,2,3,4 (L=2,4,8,16 respectively) =5 turs off the iterpolator Coector Pae Frot Pael Cotrols ad Idicators Iput Data Samples per Symbol (iput) Iterpolated Data Samples per Symbol (output) r

52 REPORT 52 (78) Block Diagram List of SubVIs Iterpolator Factor 2.vi /proj/gr_labview/measure/osvaldo_temp/evm/iterpolator Factor 2.vi History "Iterpolator Multiple.vi History" Curret Revisio: 12

53 REPORT 53 (78) Iterpolator Factor 2.vi Iterpolates the iput data by a factor of two, first by up-samplig (fillig up with zeroes), ad the by filterig the data with a half-bad LP filter. The output posses 2N-1 samples, where N is the umber of samples of the origial sequece. Coector Pae Frot Pael Cotrols ad Idicators Iput Data Iterpolated Data Origial Sigal Up-Sampled Sigal Iterpolated Sigal Block Diagram

54 REPORT 54 (78) List of SubVIs Halfbad Filter.vi /proj/gr_labview/measure/wictoria_rx/wic_rxsub.llb/halfbad Filter.vi History "Iterpolator Factor 2.vi History" Curret Revisio: 9

55 REPORT 55 (78) Sample.vi Fids the optimal measuremet positio ad extracts the samples. The iputs are the umber of samples per symbol, ad the IQ data. The output are the sampled data ad the istat at which it was sampled. Coector Pae Frot Pael

56 REPORT 56 (78) Cotrols ad Idicators Samples per symbol Number of samples per symbol Iput Data Visible symbols Optimal Samplig Positio

57 REPORT 57 (78) Data Out Origial Received Data Eye diagram Q brach Eye diagram I brach Sampled Data Block Diagram List of SubVIs Fid Samplig Poit.vi /proj/gr_labview/measure/osvaldo_temp/evm/fid Samplig Poit.vi Extract Samples.vi /proj/gr_labview/measure/osvaldo_temp/evm/extract Samples.vi Complex Eye Diagram.vi /proj/gr_labview/measure/osvaldo_temp/evm/complex Eye Diagram.vi History "Sample.vi History" Curret Revisio: 27

58 REPORT 58 (78) Fid Samplig Poit.vi Returs the optimal samplig poit for a set of data. Coector Pae Frot Pael Cotrols ad Idicators Iput Data Data i

59 REPORT 59 (78) Samples per symbol Number of samples per symbol Optimal Samplig Poit (topt) I+jQ Eye Diagram Variace Block Diagram List of SubVIs Variace.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/5stat.llb/variace.vi History "Fid Samplig Poit.vi History" Curret Revisio: 23

60 REPORT 60 (78) Extract Samples.vi Extracts the samples at the idicated positio. Coector Pae Frot Pael Cotrols ad Idicators Samples per symbol Samplig Poit Data Iput Sampled Data Block Diagram List of SubVIs History "Extract Samples.vi History" Curret Revisio: 11

61 REPORT 61 (78) LMSew.vi Applies a Least Mea-Square trasversal filter to the received QPSK sigal Z(k). This VI detects the received symbols ad compesates for variatios of phase. The step size is determied by the Beta (0<Beta<1) The default umber of taps is 1 The outputs modiified versio of the measured sigal sigal Z^(), ad the detected sigal S(), both with ad without the trasiet. The effects of frequecy offsets have bee removed i Z^(). Coector Pae Frot Pael

62 REPORT 62 (78)

63 REPORT 63 (78) Cotrols ad Idicators Received Sigal Number of taps Beta Liear Fit of phase compesatio W Elemet Data Phase Error Detected Sigal S() Detected Symbol W Magitude W Phase (rads) Squared Error Step Size u' Trasiet time Ideal Referece S() without trasiet Detected Symbol Modified versio of the measured waveform Z^() without trasiet X * Y Misadjustmet EVM vs Time G() Phase Error mea C() & G() C() G() without trasiet Block Diagram

64 REPORT 64 (78)

65 REPORT 65 (78) List of SubVIs Complex Dot Product.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/7lialg.llb/complex Dot Product.vi Variace.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/5stat.llb/variace.vi Liear Fit.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/6fits.llb/liear Fit.vi Ulimited phase.vi /proj/gr_labview/measure/osvaldo_temp/evm/ulimited phase.vi Detectio.vi /proj/gr_labview/measure/osvaldo_temp/evm/detectio.vi History "LMSew.vi History" Curret Revisio: 70

66 REPORT 66 (78) Detectio.vi Maximum likelyhood detectio For every received symbol selects the ideal symbol with the closest distace Coector Pae Frot Pael Cotrols ad Idicators Received Symbol Detected Symbol Block Diagram List of SubVIs History "Detectio.vi History" Curret Revisio: 15

67 REPORT 67 (78) Phase_Rotatio_Compesatio.vi Extracts the phase of the origial data array, compares it a ideal referece ad removes the mea. Coector Pae Frot Pael Cotrols ad Idicators Detected Data Array Origial Data Array Corrected Output Array Agle Correctio (rads) Block Diagram List of SubVIs Mea.vi /opt/labview/5.1/sol2/fds/lv51/vi.lib/aalysis/5stat.llb/mea.vi History "Phase_Rotatio_Compesatio.vi History" Curret Revisio: 8

68 REPORT 68 (78) Ulimited phase.vi Trasforms the phase limited to a rage of -3.14<phase<3.14 rads to a ulimited rage - if<phase<if Coector Pae Frot Pael Cotrols ad Idicators Iput Array cycle Output Array Block Diagram List of SubVIs History "Ulimited phase.vi History" Curret Revisio: 5

69 REPORT 69 (78) Complex Eye Diagram.vi Complex Eye Diagram : Prits eye diagrams for I ad Q for icomig RX data. The data must be corrected for phase, amplitude ad time shifts. Coector Pae Frot Pael Cotrols ad Idicators stop Oversample rate Corrected RXData

70 REPORT 70 (78) Visible symbols Rev I brach Q brach I+jQ Phase Block Diagram List of SubVIs History "Complex Eye Diagram.vi History" Curret Revisio: 95

71 REPORT 71 (78) Calculate EVM.vi Calculates the RMS Error Vector Magitude give the received sigal Z(k) ad a referece ideal sigal S(k). EVM = sqrt(sum( Z(k) ^2)/sum( S(k) ^2))x100 Coector Pae Frot Pael Cotrols ad Idicators Received Sigal Referece Sigal EVM RMS% EVM Peak EVM vs Time Peak idex Block Diagram