TECHNICAL RESEARCH REPORT

Transcription

1 TECHNICAL RESEARCH REPORT Data-Aided ML Parameter Etimator of PSK Burt Modem and Their Sytolic VLSI Implementation by Y. Jiang, W-C. Ting, F.B. Verahrami, R.L. Richmond, and John S. Bara CSHCN T.R (ISR T.R. 99-8)

2 DATA-AIDED ML PARAMETER ESTIMATORS FOR PSK BURST MODEMS AND THEIR SYSTOLIC VLSI IMPLEMENTATIONS Yimin Jiang, Farhad B. Verahrami, Wen-Chun Ting, Robert L. Richmond, John S. Bara+ Hughe Network Sytem, Inc, 77 Exploration Lane Germantown, MD 876, + Intitute for Sytem Reearch, Univerity of Maryland College Park, MD 74, ABSTRACT A high performance Univeral Modem ASIC that upport everal modulation type and burt mode frame format i under development. Powerful and generic data-aided (DA) parameter etimator are neceary to accommodate many mode. In thi paper we preent an approximated maximum likelihood (ML) carrier frequency offet etimator, ML joint carrier phae and timing offet etimator and their ytolic VLSI implementation for PSK burt modem. The performance are cloe to the Cramer-Rao lower bound (CRLB) at low SNR. Compared with theoretical olution the etimator propoed here are much impler and eaier to implement by the current VLSI technology. The CRLB for DA etimation i dicued in ome depth, ome iue on training equence deign i alo addreed in thi work.. INTRODUCTION A high performance ASIC upporting Hughe Network Sytem Univeral Modem product line i under development. Thi ASIC upport a variety of bit rate, modulation (BPSK, QPSK, 8PSK, OQPSK), forward error correction, and frame format. In order to atify the tringent operating condition uch a large carrier frequency offet (up to 3% ymbol rate), low SNR (E b =N around db) and multiple operating mode, powerful and generic etimator are neceary to recover the burt parameter. Maximum likelihood (ML) etimator [5] are optimal etimator. We preent a good approximation of DA ML carrier frequency offet etimator, a joint carrier phae and timing offet etimator and their correponding ytolic VLSI implementation. Several carrier frequency offet etimation method are dicued in [3]. The optimal ML frequency etimator i well known to be given by the location of the peak of a periodogram [8]. However the computation requirement make thi approach prohibitive even with an FFT implementation. Therefore impler approximation method are deired. We preent a DA carrier frequency offet etimator that i baed on autocorrelation and the algorithm derived by Kay []. y(t) e j π ft t = kt g MF ( kt εt ) t = nt z( nt + εt ) Figure : Matched Filter of Optimal Receiver The DA ML joint carrier phae and timing offet etimator i derived in [] (p.96). The preented implementation i hardware intenive. We derived a implified ML joint carrier phae and timing offet etimator, which i uitable for ytolic VLSI implementation. In ection II the etimation algorithm are preented. Section III preent their efficient VLSI implementation. In the lat ection the (for DA cae) are invetigated. The for timing offet etimation cloely depend on training equence, therefore ome iue on training equence deign for timing acquiition i dicued. The performance of the etimator i hown through computer imulation and compared with.. ESTIMATION ALGORITHMS The baeband received ignal i modeled a: y(t) = p E N [(a In g(t nt )+ja Qn g(t nt n= fit)) exp[j(ßft + )]] + n(t) () where g(t) =g T (t)ωc(t)ωf (t), g T (t) i the tranmitter haping function, c(t) i the channel repone, f (t) i the prefilter, n(t) i the additive white Gauian noie (AWGN) with twoided power pectral denity N =, and a n p a In + ja Qn i the data ymbol from complex plane (a n = =(± ± j) for QPSK/OQPSK ignaling). T i the ymbol interval, f i the carrier frequency offet, and fi i the delay factor that i for QPSK and.5 for OQPSK. The matched filter for an optimal receiver can be modeled a [] hown in Figure. y(t) i down converted by carrier frequency offet etimate ^f, and then am-

3 pled at rate of =T, typically T = MT, with M an integer. The ampled ignal i filtered by a matched haping filter with repone g( t) and timing offet "T. The output i then decimated down to a rate of =T to obtain a one ample per ymbol ignal z(nt + "T ). The demodulator correct the phae offet and timing offet " of z(nt +"T ) prior to making ymbol deciion and recovering the tranmitted ymbol a^ n. z(nt + "T ) i given by: z(nt + "T ) = k= y(kt )e j(ß ^fkt ) g MF (nt + "T kt ) () Weight w(m ) Weighting Function.. Carrier Frequency Offet Etimation Initially uppoe we have Nz(nT + "T )(n =; ;N ) ymbol without frequency rotation and a = [a ; ;a N ] i known in DA cae. In order to implify the preentation, let u aume perfect timing (frequency etimation performance in the preence of random timing offet i hown through imulation), unit-energy pule (g(t) Ω g( t)), thu z(nt + "T ) can be implified a z(n; f ), which can be expreed a: z(n; f )=a n exp[j(ßfnt + )] + fl n (3) where fl n i additive Gauian noie. Correlation method i adopted to remove data modulation a n, let r n z(n; f )a Λ n = E exp[j(ßfnt + )] + fla Λ n (4) It i eay to how that the autocorrelation of the exponential wave i till an exponential wave at high SNR (imulation how that high SNR condition i not neceary), i.e., R(m) = E N N m n=m r n r Λ n m exp[j(ßfmt )] + noie(m) (5) where m = ; ;L (L < N ). Mengali [4] propoed a frequency etimator baed on modeling noie(m) and the work done by Kay []. From imulation we find that for N large enough noie(m) can be approximated a white Gauian noie. The equence fr(m)g can be treated a a continuou wave (with frequency f) which i paed through a noie removal proce. At high SNR, many good frequency etimation method have been derived. Kay [] preented a frequency etimation method baed on weighted um of phae difference. Hi frequency etimator i ML at high SNR. Let u define the following proce: and m = (m) = arg[r(m)]; m =; ;L (6) ρ (); m = ( (m +) (m))mod(ß); <m<l (7) Index m, L=3 Figure : Weighting Function fw Λ m g We borrow from Kay frequency etimator, that i the weighted um of phae difference. Becaue R(m ) i calculated baed on more data than R(m ) when m < m, after ome arithmetic we derived the following carrier frequency offet etimator: where ^f = ßT L m=! Λ m m (8) wm Λ 3((L = +) (m +) ) ; m =; ((L +) ;L (9) ))(L +) The weighting function i hown in Figure. It i eay to ee that the weight wm Λ decreae a m increae. That i becaue a m get larger and larger, the number of term ued to compute R(m) reduce and thu make m le and le accurate. Compared with Mengali algorithm, our etimator adopt different weighting function, L can be le than N= (e.g. when N =96, L =3can achieve the CRLB at db)... Joint Carrier Phae and Timing Offet Etimator Auming zero frequency offet etimation error, there are K (K = MN) obervation of z(kt + "T ) (k =; ;K ) available for etimating " and, " [ :5; :5). According to the work done in [], the maximization object function of ML joint phae and timing offet etimation in AWGN channel i ( " N #) L(a;"; )=Cexp Re a Λ nz(nt + "T )e j () n= where C i a poitive contant and a =[a ; ;a N ] which i the data pattern and i known to the etimator. Let u define μ(") a: μ(") = N n= a Λ nz(nt + "T ) ()

4 that i the cro correlation between training equence a and received ample fz(nt + "T )g. The ML joint phae and timing etimator i given by []: µ ( t) x x ideal ample x ^" = arg max jμ(")j () " ^ = arg[μ(^")] (3) T T t According to the Equivalence Theorem [], and auming that c(t) and f (t) are all-pa filter, z(nt + "T ) i equivalent to the following: z(nt + "T )= where p N E r(t) =g T (t) Ω g T ( t) = in(ßt=t ) ßt=T a k r(nt + "T kt )e j + N n (4) k= co(ffßt=t ) 4ff t =T The above expreion alo aume that raied coine haping i adopted with ff denoting the rolloff factor. N n i the ampled verion of n(t), Gauian noie, after being filtered by g MF (t). Arriving at a olution to Eq. () i a difficult tak and the reulting hardware tructure preented in [] i quite complicated. It i well known that a quadratic form can be ued to approximate the central egment of a convex function around it peak. The location of the peak of a econd order polynomial i eay to compute from it coefficient. The expreion for jμ(")j can be approximated by a quadratic equation a hown below. If "!, the inter-ymbol-interference (ISI) and noie N n can be ignored and we can implify jμ(")j a p N jμ(")j ß E ja n j p r("t )=N E r("t ) (5) n= where ja n j =(n =; ;N ). Furthermore by letting t = "T and uing Taylor erie approximation for ine and coine function and after ome implification, we arrive at p jμ(t)j ßN E ß t 6T (6) From Eq. (6) we can ue a econd order polynomial to approximate the relationhip between ampling time and the magnitude of correlation jμ(t)j given that thee ampling point are cloe enough to the ideal ampling point (i.e. t i cloe enough to ). Uing a general form of the econd order polynomial jμ(t)j = bt + bt + b (7) ugget that a joint phae and timing etimator can be derived baed on three adjacent ample of jμ(t)j. Thee ample are the cloet one to the ideal ampling point a hown in Figure 3. In order to meet the condition that t i cloe enough to, two meaure are adopted: one i that the ampling rate M (ample per ymbol) i large enough (imulation how that M = 4 Figure 3: Three Sampling Point Model can achieve good performance);econd i locating the larget available magnitude x through peak earch. Let u define the ampling time of x a nominal on time axi. Therefore the ampling time of x and x are T and T, repectively. A LaGrange interpolating polynomial can be adopted baed on the value of x k (k =; ; ): jμ(t)j = k= Y x k i=;i6=k = bt + bt + b t t i t k t i (8) uing the fact that t = T ; t =; t = T, we can get b = x T x x + (9) b = x T x () b = x () The ML timing offet etimator () i the ^" which maximize jμ(")j. It i eay to compute the ampling time of the peak of jμ(t)j from a econd order polynomial, i.e. t peak = b (x b = x)t x 4x +x therefore, the ML etimate of " i ^" = t peak T = x x M (x 4x +x) () (3) The phae etimator i hown in Eq. (3). Interpolation technique can be applied to correct the timing offet before phae etimation. Thi however introduce an additional delay in the demodulation proce. Simulation how that uing the time for the non ideal ample of x i ufficient for meeting the CRLB (ampling time of x i t). Thi lead to ^ =arg[μ(t)] (4) In order to locate the larget available value x eaily, a highly correlated data pattern a i elected. [6] dicue thi problem in depth. Here unique word (UW) and alternating (one zero) data pattern are invetigated.

5 L f = wm ( m) πt m = 3[(L + ) (m + ) ] wm =, m =,..., L [(L + ) ](L + ) + (m) Mod pi - w m + + πft y(t) t = kt g MF exp[ j(πfkt )] ( kt εt ) ε = Multi-Sample Correlator x x 4 ( x 4 x + x ) θ= arctan[ µ ( t )] Peak Locator x, x x, Timing Offet Et. Phae Offet Et. ε θ Full Quadrature atan Look Up Table (LUT) θ ( m ), m =,..., L Parallel to Serial Figure 5: Joint Carrier Phae and Timing Offet Etimator x 4 x3 x x r i R () R () R ( L) R(L) Complex multipler are hared y baed on high peed clock a a a N yin x in a i y out y out = yin + xinai yin yout y out = y in y 4 y3 y y xin R xout xout = xin, R = xin xin + R, y = R, complex conjugate Figure 6: Multi-Sample Correlator x in Figure 4: Sytolic VLSI Structure of Carrier Frequency Offet Etimator 3. VLSI IMPLEMENTATIONS For the frequency etimator, the calculation of R(m) (Eq. (5)) i a hardware intenive tak that require (N L )L= complex multiplication and (N L 3)L= addition. In order to make full ue of each input data and exploit concurrency, we propoe the ytolic VLSI implementation a hown in Figure 4. If higher peed clock i available, the complex multiplier can be hared on time diviion bai. fr(m)g will be available on the clock cycle following the one latching the Nth data ymbol into the etimator. Frequency offet can then be calculated via Eq. (8). One advantage of thi tructure i that it i calable. If we want to increae L to get a better performance, more element can be added at the right hand ide hown in Figure 4. The hardware block diagram for the joint phae and timing etimator i hown in Figure 5. The multi-ample correlator generate output at a higher rate than one ample per ymbol. A ytolic VLSI implementation of the correlator i hown in Figure 6, where x ij denote the ith ymbol (i =; ;N ), jth ample (j =; ; 3) of the output from the matched haping filter. In QPSK cae, a n = ± ± j, only adder are neceary therefore the computational complexity i relatively mall epecially when uing the correlator a oft-deciion UW detector. Through peak earch module, we can locate x, x and x. An Arctan Lookup table (LUT) i ued when etimating the phae offet. 4. PERFORMANCE BOUNDS AND SIMULATION RESULTS The variance of an unbiaed etimation i lower bounded by the CRLB. The for DA frequency etimation i given by [8] a follow: E[(fT ^ft) ] 6 ρ ff 4ß E N (N ) (5) N The for joint carrier phae and timing offet i given by [6]. If a i choen properly (e.g. a i BPSK type data or real) the in joint etimation are the ame a thoe in the ingle parameter etimation with the knowledge of the other parameter. The for phae etimation i given by ρ ff E[( ^ ) E ] N (6) N that i independent of training equence if Nyquit hape i adopted. Moeneclaey propoed the CRLB for DA timing etimation with the aumption that training equence a i zero mean and i.i.d. in [7]. The bound for the cae where the ampling rate =T B (B i the bandwidth of r(t)) and N large enough i given by E[(fi ^fi ) ] T ρ E N N 4ß f R(f )dfff (7) with R(f ) the Fourier tranform of r(t). Jiang ha propoed the following expreion for with arbitrary training equence in [6]: 4 K= 39 ßk k = R ja[k]j 5 (8) 8 < : E N NT k= K= N NT where A[k] i the kth element of P N-point dicrete Fourier tranform (DFT) of a, i.e. A[k] = n= a ne j(ßnk=n). Accord- N ing to Eq. (8), ha different value for different data pattern. Eq. (8) give u inight on training equence deign immediately if etimation variance i choen a performance criteria. A equence i optimal if it minimize the for ;

6 of Peudo Random Data Pattern α=. α=.5 α=.5.5. Timing Offet Detector Characteritic with Random Phae (No Noie).5, Standard Deviation Timing Offet Etimate Signal to Noie Ratio E /N (db) b o Timing offet ε, (the ymbol period i ), no noie, uknown phae Figure 7: The for Timing Etimation with UW Pattern Normalized RMS Frequency Offet Etimation Error 3 Performance Compariion of Carrier Frequency Etimation and Perfect Timing Random Timing Eb/No (db), Df=.3RS, Random Timing, Random phae, N=96, L=3 Figure 8: RMS Carrier Frequency Offet Etimation Error v. Figure 9: Timing Offet Etimate ^" v. Timing Offet " the k that maximize (ßk=K) R(k=NT) with it maximum R max, the the following hold RHS of Eq:(9)» K= k= K= = R max E a R max ja[k]j (3) The optimal equence can be derived immediately, which ha the following DFT ( p Ea if k = ±k m ; ja[k]j = (3) otherwie It i a tone with frequency k m =N. The only concern here i that the optimal equence perhap i not binary. Two data pattern have been invetigated in our ASIC deign: alternating one-zero pattern (i.e. a i = ( ) ip =( + j)), and a unique word pattern. A 48-ymbol UW wa elected. According to Eq. (8) for the alternating one-zero data pattern timing etimation under ome contraint. The following preentation how a imple example on training equence deign under energy contraint, i.e., aj op = arg max with the condition that K= k= K= N= k= N= ßk R k N NT ja[k]j (9) ja[k]j = E a (3) Becaue (ßk=K) R(k=NT) i non-negative (R(f ) i aumed to be non-negative, e.g., raied-coine hape, otherwie jr(f )j i adopted in Eq. (8)), ja[k]j i alo non-negative, let k m be j = ρß E N Nff (33) and thu the performance i independent of rolloff factor ff given that ff >. For the UW pattern, the timing etimation i cloely related to the rolloff factor. It follow from Eq. (8) that the larger the rolloff factor, the maller. Figure 7 how Eq. (8) plotted a a function of SNR for three different value of rolloff factor. The parameter for the computer imulation were QPSK ignaling, N =96and L =3in an AWGN channel for frequency etimation, N =48and M =4in the AWGN channel for joint phae and timing etimation. Figure 8 how normalized root mean quared (RMS) frequency etimation error with f = :3=T, which i compared with the for frequency etimation. From imulation we can ee that the etimation RMS error i very cloe to the even at db, the

7 Timing Offet Etimation Simulation: rolloff factor.5 Phae Etimation Performance of QPSK CRLB CRLB UW STD STD UW STD UW Root Mean Squared Timing Offet Etimation Error Root Mean Squared Phae Error in Radian Signal to Noie Ratio E /N (db), N=48,α=.5 b o EbNo(dB), N=48, QPSK, random phae, random timing, α=.5 Figure : Timing Offet Etimation Performance (one zero pattern v. UW pattern, ff =:5) Figure : Phae Offet Etimation Performance (UW pattern, ff =:5) performance degradation caued by timing error i very mall. Figure 9 how the aw tooth characteritic of Eq. (3) under no noie condition with random phae. From imulation we can ee that Eq. (3) i an unbiaed etimate of ". Peak earch (i.e. locating x) reolve the m=4 (m = ±; ±) ambiguity. For phae and timing etimation, different rolloff factor for the raied coine haping function were alo teted. Simulation how that the RMS timing etimation error meet the CRLBDA of timing etimation for all ff and data pattern. Simulation alo upport that for the one-zero pattern the RMS timing error i independent of ff, while for the UW pattern it decreae a ff increae. Thi i in agreement with the evaluation of the CRLBDA. Figure how the timing offet etimation performance with ff = :5, where one-zero pattern and UW pattern of QPSK are illutrated. Figure how the phae etimation performance. The RMS phae etimation error meet the CRLBDA for phae etimation. 5. CONCLUSIONS The RMS etimation error of our algorithm that have moderate complexitie meet the CRLBDA at low SNR, therefore they are efficient. The technique propoed here can be ued in high performance PSK burt modem working under large carrier frequency offet and low SNR condition. A training equence deign method for timing acquiition i alo illutrated in thi paper. 6. REFERENCES [] S. Kay, A Fat and Accurate Single Frequency Etimator, IEEE Tran. Acout., Speech, Signal Proceing, vol. 37, no., pp , December 989. [3] Y. Jiang, W. Ting, F. Verahrami, R. Richmond, J. Bara, A Carrier Frequency Etimation Method of MPSK Signal And It Sytolic VLSI Implementation, Proc. CISS 99, Jan., 999. [4] U. Mengali, M. Morelli, Data-Aided Frequency Etimation for Burt Digital Tranmiion, IEEE Tran. Comm., vol. 45, no., pp.3-5, Jan [5] H. L. Van Tree, Detection, Etimation and Modulation Theory, Part I, New York: Wiley, 968. [6] Y. Jiang, J. S. Bara On the true Cramer-Rao lower bound for the DA joint etimation of carrier phae and timing offet, ubmitted to IEEE ICC, Aug [7] C. Georahiade, M. Moeneclaey, Sequence Etimation and Synchronization from Nonynchronized Sample, IEEE Tran. Inform. Theory, vol. IT-37, pp , Nov. 99. [8] D. Rife, R. Boortyn, Single-Tone Parameter Etimation from Dicrete-Time Obervation, IEEE Tran. Info. Theory, vol. IT-, no. 5, pp , Sept [] H. Meyr, M. Moeneclaey, S. Fechtel, Digital Communication Receiver, Synchronization, Channel Etimation, and Signal Proceing, New York: Wiley, 998.