FRAME SYNCHRONIZATION FOR PSAM IN AWGN AND RAYLEIGH FADING CHANNELS
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1 FRAME SYCHROIZATIO FOR PSAM I AWG AD RAYLEIGH FADIG CHAELS Haozhang Jia Deartment of Electrical Engineering Univerity of Sakatchewan Sakatoon, Canada, S7 5A9 haj53@mail.uak.ca David E. Dodd Deartment of Electrical Engineering Univerity of Sakatchewan Sakatoon, Canada, S7 5A9 dodd@engr.uak.ca Abtract Pilot Symbol Aited Modulation (PSAM) i a romiing method to comenate for fading in wirele land mobile communication. With PSAM, known ilot ymbol are eriodically inerted into the tranmitted data ymbol tream and the receiver ue thee ymbol to derive the amlitude and hae reference for data ymbol detection One aect of thi rocedure that ha not yet received much attention i frame ynchronization, i.e. the technique ued by the receiver to locate the time oition of the ilot ymbol in the received ymbol equence. Thi aer ue a non-coherent maximum likelihood (ML) frame ynchronization aroach in which only the magnitude of received ignal i ued to obtain the time oition of the ilot ymbol. Comuter imulation reult how good erformance in both AWG and fading channel and excellent tolerance to receiver frequency offet. Moreover, thi method lead to imler analyi and i omewhat imler to imlement. Keyword: Frame ynchronization, PSAM, Maximum Likelihood Etimation.. Introduction One of the mot devatating henomena aociated with mobile communication i channel fading, which can ditort the tranmitted ignal everely and make the recetion very difficult. It degrade the bit error rate and inhibit the ue of ectrally efficient multilevel modulation cheme uch a 6- QAM. Pilot Symbol Aited Modulation (PSAM) can reduce the imact of fading and facilitate the alication of multilevel modulation cheme. PSAM ha been tudied by everal reearcher []-[4]. A illutrated in Fig., known ilot ymbol are eriodically inerted into the data ymbol tream and both data and ilot ymbol are tranmitted over communication channel. At the receiver, ilot ymbol are earated from the data ymbol and then ued to derive the amlitude and hae reference for data ymbol detection. In our literature earch, we find that one aect of thi roce, frame ynchronization, ha been neglected in mot reearch. Frame ynchronization i the technique ued by the receiver to identify the time oition of the ilot ymbol in received ymbol equence. We find the mot alicable work in a aer by Ganman [5]. He reent two frame ynchronization technique: one i a maximum likelihood frame ynchronizer and the other i a equential teting algorithm. Both method rely on coherent detection. Thi aer extend hi work by alying a non-coherent aroach to the maximum likelihood etimation algorithm. The new aroach ue imlifying aroximation baed on relatively high SR a conitent with the recetion of 6-QAM. Comuter imulation ha been ued to tet ynchronizer erformance and everal length and attern of ilot ymbol equence were teted. Every th ymbol wa a ilot ymbol and all other ymbol were randomly elected data ymbol. Thi aer i organized a follow. The PSAM ytem and data model are reented in Section. Section 3 ue the maximum likelihood etimation formulation and develo a ubotimum frame ynchronization technique baed on high SR aroximation. Frame ynchronization criterion for both AWG and Rayleigh fading channel are introduced. Simulation reult are reented and analyzed in Section 4, and concluding remark follow a Section 5.. PSAM Sytem Model In the PSAM ytem model illutrated in Fig., the tranmitter eriodically inert ecific ilot ymbol into the data equence. The combined ilot and data frame illutrated in Fig. contain ilot ymbol inerted at interval of L. The comoite ymbol equence (n) i linearly modulated by a quare root raied coine (yquit) ule, (t) and then tranmitted over a channel characterized by frequency nonelective low fading and additive white Gauian noie. Pilot ymbol have the ame ule hae a the data ymbol. The tranmitted ignal ha a comlex enveloe given by, n= ( t) = A ( n) ( t nt ) ()
2 where T i the ymbol time, A i an amlitude factor, and (t), i a quare root yquit ule with unit energy uch that: R ( t) = ( τ ) ( τ t ) dτ () and R ( kt ) = δ (. (3) (n) b(n) Deciion Periodic Inertion Delay Channel State Etimation (n) Pule Shaing Data Pilot (t) r ( r(t) Matched Filter r (t) Symbol Samler multilicative fading ditortion factor, which ha a Rayleigh magnitude ditribution. The ower ectrum of fading c( i modeled a in [6], E{ c n c n k }= σ c J (π kf D ) (8) where J i the zeroth-order Beel function, σ c i the variance of the fading comonent, and f D i the rm Doler hift multilied by the ymbol time. The idea behind PSAM ytem i clear. If the fading comonent c(t) can be etimated accurately, then thi channel tate etimation can be ued to counteract the fading effect and make the data deciion more accurate. Frame ynchronization i required in order to imlement uch a roce [4]. A in Fig., frame ynchronization oberve the received ymbol and identifie the timing of the ilot ymbol. Each ilot ymbol give a amle of channel tate and thee amle are then interolated to form a continuou channel tate etimation. Thi etimation i ued to cale and rotate a reference deciion grid and thu otimize the data outut deciion. Reference Grid Frame Synchronizer P DATA P DATA P - L L L DATA Fig. PSAM ytem model For frequency non-elective fading, the delay read of the channel i much le than the ymbol duration, i.e. all of the multi-ath arrive at receiver aroximately at the ame time. Therefore, the channel ha no inherent interymbol interference (ISI) and the multi-ath ditortion can be combined into one multilicative ditortion roce c (t). The received ignal i then given by r ( t) = c( t) ( t) + n( t) (4) where n (t) i zero mean AWG with one-ided ower ectral denity. Paing thi continuou received ignal through a correlator matched to the ule hae ( t) and amled at the ymbol time yield n= r ( kt ) = A ( n) c( τ ) ( τ nt ) ( τ kt ) dτ + n( kt ). (5) For low fading, c (t) i aroximately contant over ymbol durationt, o it may be ulled out of the integral a c (. Uing () and (3), thi i further imlified to r ( kt ) = c( ( n) R (( n T ) + n( kt ) (6) n= and finally we obtain r ( = c( ( + n( (7) where ( i a data ymbol or ilot ymbol, n ( i zero mean comlex AWG with variance, and c( i the Fig. PSAM frame format 3. ML Frame Synchronization Derivation For a PSAM ytem, the frame ynchronization mut etimate the relative oition of the firt ilot ymbol P which correond to the tart of a frame. Conider a full frame obervation, x, having length L = L with ymbol index tarting at, x = [ x, x, x..., x n,..., x L ]. Let µ be the index of the ilot ymbol P within the full frame, where µ i an integer in the range [, L-]. The beginning of the frame (i.e., ilot ymbol P ) aear in any of the L oition in x with equal robability. Therefore, maximum likelihood etimation i to earch for the value of µ that maximize the function f x (x µ) a given by µ ML = argmax f ( x µ ) (9) µ [, L ] x where f x (x µ) aee the imilarity between the known ilot equence P and the received ilot-aced ymbol denoted x tarting at oition µ and exreed a = x k = kl x. () + µ
3 3.. Synchronization in an AWG Channel The non-coherent frame ynchronization cheme i baed only on the magnitude of ymbol and i thu inenitive to the hae and frequency offet in the receiver. In the quare contellation of 6-QAM illutrated in Fig. 3, there are three level of ymbol amlitude. While any tranmitted data ymbol can take any one of the three level, we retrict ilot ymbol to the outermot circle or the innermot circle. Pilot ymbol are therefore eaily introduced into a equence of 6-QAM data ymbol. Q Symbol Maing to Binary Fig. 3 Pilot Symbol maing into 6-QAM Let denote the comlex tranmitted ymbol, r denote the comlex received ymbol, and denote the ower ectral denity of comlex AWG having zero mean. It i well known [7] that, for an AWG channel, the robability denity function (PDF) of the received ymbol i given a r f ( r ) = ex () π Becaue ilot equence i reented here in magnitude, our interet i the robability denity function of r conditioned on. Exreing comlex ymbol r and in olar form and integrating equation over θ r (,π ) yield r r r r f r ( + ) co( θ ) dθ r π π ( ) = ex ex () Uing Beel function of the firt kind and zero order I ( x) = ex( x coθ ) dθ (3) π π and ubtituting thi Beel function into the integral in (), the robability denity function become r ( r + ) r f ( r ) = ex I (4) Changing the variable r into r and into Symbol Maing to Binary I yield ( r + ) r f ( r ) = ex I (5) Under high SR, a it i in [8], the Beel function can be aroximated a ex x ln x ex( x ) I ( x) (6) π π Thu (5) i finally imlified a ( r r ) f ( r ) ex (7) π Thi conditional robability denity function of magnitude quared received ignal r i imilar to the Gauian denity ditribution in term of ymbol magnitude r. However, thi denity function no longer integrate to due to aroximation factor. To maximize the likelihood function, a in [9], the following notation are defined: P --- ilot ymbol equence P [ P, P,..., P ] = d --- random data ymbol aced L ymbol aart from each other when they aear in a full frame obervation. The uercrit tand for ilot aced. d [ d, d,..., d ] = r --- the ilot-aced obervation, which i the collection of ymbol within a full received frame tarting at the t oition and aced aart from each other by L ymbol. = r [ r, r,..., r ] ote that element in P and d mut be the quare of one of the three ymbol magnitude defined in the 6-QAM contellation (ee Fig. 3). Both r and d are obtained by amling the ymbol tream at the ilot ymbol acing L. Therefore, by uing (7), the frame ynchronization roblem of finding µ, the index of the ilot ymbol P within the full frame, become, = ( ri Pi ) ˆ µ ML argmax ex (8) µ [, L ] π i= Where it i undertood that ilot aced equence r, r,r L begin at the offet µ. The received ymbol index µ + il i modulo L and the ilot aced ymbol wra around within the oberved full frame. By taking the logarithm and neglecting term that are unrelated to P, we obtain the maximum likelihood criterion for the AWG channel, = ˆ µ ML argmax ( ri Pi ) (9a) i=
4 = ˆ µ ML argmax ri Pi ri Pi (9b) i= When viewed in dimenional ace, we elect the received ilot aced equence that ha the minimum Euclidean ditance to the ilot equence. 3.. Synchronization in a Fading Channel We now conider the frame ynchronization roblem in a Rayleigh fading channel. We aume a tranmiion model where all fading occur at the tranmitter; data ymbol ( are firt modulated by fading ignal c(, then the modified ignal c( ( i tranmitted over AWG channel. Thi will give the ame received ignal a if data ignal ha been tranmitted over a fading and noiy channel and i conitent with (7). Prior to ynchronization, the magnitude of the channel fading ignal can be reaonably etimated from ilot aced obervation of the received ignal magnitude. In our analyi, we aume erfect etimation, which doe not ignificantly degrade erformance and greatly imlifie the comutation. Following a imilar rocedure a we did for the AWG channel, we cale the reference ilot equence magnitude in (9) to yield the maximum likelihood criterion for Rayleigh fading channel = ( ) ˆ µ ML argmax ri ci P () i i= In Section 4, Simulink model are ued to tet the erformance of both frame ynchronizer with variou ilot ymbol equence. The model imlement the ML deciion criteria above - an argmax tructure indicate the mot likely oition of ilot ymbol P within a window length of L. 4. Simulation and Performance Analyi In thi tudy, data ymbol and ilot ymbol are both elected from 6-QAM data et. Thi differ from Ganman work [5] where data ymbol are elected from the 6-QAM ymbol et while ilot ymbol are elected from the 8 PSK et. Pilot ymbol are laced in the firt oition of every ubframe, a illutrated in Fig. and the ilot inertion interval i L =. The tatitical erformance of the ynchronizer wa teted with everal ilot equence that have good autocorrelation roertie. Thee equence were Barker code 7,, 3, euman-hoffman 3 and P 5 (ee Table ). For each cae, one full frame obervation i thu 7,, 3 and 5 ymbol reectively. The imulation overview i illutrated in Fig. 4 and each data oint ued, trial of full frame obervation. The frame ynchronizer outut i comared to the outut of a noie-free, error-free, imlified frame ynchronizer and the number of correct ynchronization i recorded for, full frame obervation. The ignal generator continuouly generate eudo-random equence and for each full frame obervation, the frame ynchronizer alway generate an etimate of ilot ymbol P. The tranmitter of the imulation bench and the imlified tranmitter of the reference bench work in a ynchronized mode, o that they inert the ame ilot ymbol in exactly the ame location of the ymbol tream at the ame time, etimate of ilot ymbol P, and never enter the verification mode. The reult roceor comare the outut of the frame ynchronizer to that of the imlified frame ynchronizer, and count the number of P ymbol that are correctly detected during, full frame obervation. Tranmitter Signal Generator Simlified Tranmitter Fig. 4 Simulation tet model In comuter imulation, AWG i modeled by uing the Simulink integrated block. SR i defined by the ratio of the average ower of inut ignal to noie ower. Fading channel i modeled by a multi-ath Rayleigh fading channel block Doler fading rate wa et to % of the ymbol rate, which i conitent with that of [5]. Table Sequence Fading & AWG Channel Reference Bench Ideal Channel Receiver Simulation Bench Simlified Receiver Polar Binary Sequence BK7 [-,-,-,,,-,] BK [-,-,-,,,,-,,,-,] BK3 [-,-,-,-,-,,,-,-,,-,,-] Frame Synchronizer Reult Proceor Simlified Frame Synchronizer -H3 [,,,,,,-, -,,, -,, -] P5 [-,,,,-,-,-,-,,-,,-,-,,] Another characteritic of thi frame ynchronization method need to be highlighted. Although it i uually aumed that carrier ynchronization i achieved before frame ynchronization, there i often ome mall frequency offet reidue f m. Moreover, fading channel introduce Doler hift, f D, which alo caue frequency offet in the received ignal. Thu it i intructive for u to tet the frame ynchronizer tolerance to thee mall frequency offet. Therefore, the deign arameter for an AWG i SR and frequency offet f m, while the deign arameter for Rayleigh fading channel are frequency offet f m and Doler hift f D. The channel arameter f m, f D and SR can ignificantly affect erformance,
5 however, imulation how our ynchronizer to be robut to modet frequency offet. A in [5], our ML ynchronizer i comared through imulation to the tandard correlator and to the non-coherent ynchronizer of Liu & Tan [8], which are, reectively, * c = argmax k xkl + [, L ] µ µ k= ˆ µ () * ˆ µ lt = argmax k xkl + µ f ( x k + µ µ [, L ] k = ) () where f ( x k ) i a data correction term which we have choen to be r in our imulation. Fig. 5 comare imulated erformance in AWG and how that our ynchronizer erform much better than () and (). In each cae, BK wa ued a ilot equence. Fig. 6 how that on a fading channel with f D =. of the ymbol rate, our ynchronizer erform well, while the other fail. where T f = L = * L i one frame eriod, d robability of true ilot ymbol detection and i the i the robability of fale alarm, i.e. detection of a non-ilot ymbol. Mean time to ynchronization calculation wa baed on full frame ML obervation and a verification tage that declare ynchronization after two identical frame location etimate in ucceion. The AWG ynchronizer ha good erformance over a wide range of SR. Fig. 7(a) how the robability of fale acquiition of AWG without frequency offet. The imulation reult how that with the increae of the frame obervation length, the robability of true ilot ymbol detection increae, which i conitent with the theoretical analyi. To tet the robutne of the ynchronizer to frequency offet, we et frequency offet f m =. a in [5]. The robability of fale acquiition and the comuted mean time to acquiition are hown in Fig. 7 (b) and Fig. 7(c) reectively. Simulation how that although our frame ynchronization method aume high SR, it alo work well in moderate SR. f Cor L & T U:ML Fig. 5 Acquiition Performance in AWG Channel with no receiver frequency offet and ilot equence BK We now examine in more detail the robability of fale acquiition a a funtion of ilot ymbol bit attern and channel SR. Since SR i generally not known, a range of value are exlored. In addition, we calculate mean time to acquiition v. SR by the following exreion of Tacq that wa adated from [] T acq = T f + d d + T f f f + ( f ) d (3) Cor L & T U:ML Fig. 6 Acquiition Performance in Fading Channel with f D = % of the ymbol rate and ilot equence BK The erformance in a Rayleigh fading channel i wore than for an AWG channel. Fig. 8(a) how the robability of fale acquiition v. SR with arameter f D =. and f m =.. The erformance i eecially oor at low SR becaue of the high SR aroximation ued in ignal roceing. Fig. 8(b) illutrate the robability of fale acquiition v. SR for a Rayleigh fading channel with frequency offet f D =. and f m =.. Fig. 8(c) how the mean time to acquiition for the ame cae with frequency offet. Frequency offet ha little effect on the erformance of the ynchronizer and thi i attributable to the non-coherent ignal roceing.
6 a) Fale Acquiition in AWG Channel with no freq. offet a) Fale Acquiition with Fading Channel and no freq.offet ML-H3 ML-P ML-H3 ML-P b) Fale Acquiition in AWG Channel freq. offet =.f y b) Fale Acquiition with Fading Channel and freq. offet =.f y Probability of Fale Acquition ML-H3 ML-P ML-H3 ML-P c) Acquiition Time with AWG Channel and freq. offet =.f y c) Acquiition Time with Fading Channel and freq. offet =.f y Mean Time to Acquiition 5 5 ML-H3 ML-P5 Mean Time to Acquiition 5 5 ML-H3 ML-P Fig. 7 Synchronization Performance in AWG Channel a) with no receiver frequency offet, b) frequency offet = % of ymbol rate and c) ynchronization time with frequency offet Fig. 8 Synchronization Performance in Fading Channel a) with no receiver frequency offet, b) frequency offet = % of ymbol rate and c) ynchronization time with frequency offet
7 5. Concluion In thi aer, a non-coherent maximum likelihood frame ynchronization technique for PSAM wa develoed and teted with AWG and Rayleigh fading land mobile channel. When comared to a reviou tudy uing coherent detection [5], our non-coherent ytem how omewhat better erformance in both AWG and fading channel and ignificantly better erformance in reence of receiver frequency offet. In addition, our method lead to imler analyi and i omewhat imler to imlement. Acknowledgement Student funding for thi work wa rovided by a SERC Dicovery grant. Comuting facilitie were rovided by the Univerity of Sakatchewan and the author are grateful for the aitance of Trevor Zintel and David Karaloff in etting u comuting oftware. Reference [] J. H. Lodge and M. L. Moher, TCMP-a modulation and coding trategy for Rician fading channel, IEEE I. Select. Area Commun., vol.7, , Dec.989. [] A. Aghamohammadi and H. Meyr, A new method for hae ynchronization and automatic gain control for linearly modulated ignal in frequency flat fading, IEEE Tranaction on Communication, vol. 38,.5-9, 99. [3] J.K.Caver, An analyi of ilot ymbol aited modulation for rayleigh fading channel, IEEE Tranaction on Vehicular Technology, vol.4, , ovember 99. [4] S.Samei and T.Sunaga, Rayleigh fading comenation for QAM in land mobile radio communication, IEEE Tranaction on Vehicular Technology, vol.4,.37-47, May.993. [5] A. Ganman, M.P. Fitz and J.V. Krogmeier, Otimum and Subotimum Frame Synchronization for Pilot-Symbol-Aited- Modulation,IEEE Tranaction on Communication, vol.45, o., , October 997. [6] W. C. Jake, Microwave Mobile Communication, IEEE Pre, 974. [7] John G. Proaki, Communication ytem engineering, Prentice Hall, 994. [8] G. L. Lui and H. H.Tan, Frame Synchronization for Gauian Channel, IEEE Tran. Commun., vol. COM-35, , Aug [9] D.E. Dodd, K. Takaya and Q. Zhang, Frame Synchronization for Pilot Symbol Aited Modulation ; Proceeding of IEEE Canadian Conference on Electrical and Comuter Engineering,. 5-58, May 9-, 999, Edmonton, Canada. [] B. Peron, D.E. Dodd, and R.J. Bolton, A Segmented Matched Filter for CDMA Code Synchronization in Sytem with Doler Frequency Offet Proceeding IEEE Globecom, San Antonio, Texa, ov.
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