Data-Recursive Algorithms for Blind Channel Identication in. Direct-Sequence Systems. November 30, Abstract

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1 Data-Reursive Algorithms for Blind Channel Identiation in Diret-Sequene Systems Dennis L. Goekel 1, Alfred O. Hero III 2, and Wayne E. Stark 3 November 30, 1995 Abstrat Algorithms for performing blind hannel identiation for a binary phase-shift keyed (BPSK) diret-sequene spread-spetrum (DS/SS) system operating over a fading hannel are presented. These algorithms are derived by identifying the DS/SS system as a disrete oversampled system with intersymbol interferene. In this setting the spreading ode an be viewed as a transmit lter, the knowledge of whih an be used to aid in hannel identiation. An o-line solution to the hannel identiation problem involves the determination of the eigenvetor orresponding to the minimum eigenvalue of a matrix that depends on the orrelation statistis of the samples of the reeived signal. In this paper, the online solution is derived for the ase that the transmit lter and propagation hannel are unknown and jointly identied. Novel low omplexity stohasti gradient algorithms and onjugate gradient algorithms are derived and mean onvergene onditions given. Then it is shown how the knowledge of the spreading ode an be inorporated to aid in identiation. An algorithm is then derived that utilizes trellis searhing for joint data and hannel identiation for oversampled systems. Finally, numerial results in the form of hannel estimation error are presented. Keywords: Blind Channel Identiation, Diret-Sequene Spread-Spetrum Systems, Oversampled ISI Systems Corresponding Author Dennis Goekel Alfred O. Hero III Wayne E. Stark =o Prof. Wayne Stark Univ. of Mihigan - EECS Univ. of Mihigan - EECS Univ. of Mihigan - EECS 1301 Beal 1301 Beal 1301 Beal Ann Arbor, MI Ann Arbor, MI Ann Arbor, MI Tel:(313) Tel:(313) Tel:(313) Fax:(313) Fax:(313) Fax:(313) hero@ees.umih.edu stark@ees.umih.edu goekel@ees.umih.edu 1 This work was supported in part by a National Siene Foundation Graduate Fellowship and a Rakham Pre-Dotoral Fellowship. 2 This work was supported in part by the National Siene Foundation under grant BCS This work was supported in part by the National Siene Foundation under grant NCR and the Army Researh Oe under ontrat DAAH04-95-I-0246.

2 1 Introdution Diret-sequene spread spetrum (DS/SS) systems nd wide appliation in modern digital ommuniations systems, partially due to their ability to suppress multipath interferene. In DS/SS systems, the information-bearing waveform is purposely expanded in frequeny by multiplying the waveform during eah symbol period by a spreading waveform with muh higher bandwidth than the data rate. The spreading waveform is onstruted by dividing the symbol period into many \hips" with eah hip ontaining the produt of a xed hip waveform (i.e. the same for all hips) with either positive or negative one (depending on the hip). In this paper, the fous is on blind identiation for single-user DS/SS systems. The optimal reeiver for a DS/SS system on an additive white Gaussian noise (AWGN) hannel with no intersymbol interferene (ISI) mathed lters the ontinuous time reeived signal with the full spreading waveform and samples the lter output only one per symbol period. However, in pratie the optimal reeiver is implemented with a mathed lter to the hip waveform sampled at the hip rate of the system, whih is muh higher than the symbol rate. This reeiver is simpler in design, only requiring a ontinuous time orrelation over a single hip interval, whih an be implemented as an integrate-and-dump devie, allowing orrelation with the spreading sequene to be done in the digital domain. Sine there are many hips for a single symbol interval, this reeiver eetively samples the reeived signal waveform many times per symbol interval. When a DS/SS system is subjet to known ISI, it is no longer optimal to make deisions on a symbol-by-symbol basis but instead one must do sequene estimation using a Viterbi algorithm. Furthermore, for an ISI system that is oversampled, it is no longer possible to obtain a suient statisti for alulation of the likelihood of a given path of the Viterbi algorithm by multiplying the samples within a symbol period by a transmitter spreading oeient and then summing. The optimal ombining of the samples in a given symbol period depends on 1

3 the hannel impulse response at eah sample period, not just at the symbol period. This is the motivation for performing hannel identiation of DS/SS systems at the hip period. Muh work has appeared on hannel identiation/equalization via oversampling the output of the hannel [2, 3, 4, 5, 6, 7, 8, 9]. The ommon base of this work is that the oversampled proess is ylostationary, thus allowing for blind hannel identiation based on only seond order statistis. Beause estimation of seond order statistis requires fewer samples than that of higher order statistis for a given level of estimation auray, one expets algorithms based on seond order statistis to exhibit faster onvergene. The speial feature of the DS/SS systems onsidered in this paper is that they are naturally oversampled by a fator equal to the proessing gain. The high hip rates in DS/SS ommuniations require the use of eient algorithms for hannel identiation. One option would be to ombine hip periods to lower the oversampling rate and speed up identiation. Besides being suboptimal, as information is being lost in suh an operation, the known spreading ode of the user annot be inorporated to improve onvergene and steady state error performane of the hannel identiation algorithms [10]. As in muh of the previous work, it will be assumed that the hannel has a nite impulse response (FIR) and that the time support of the hannel response is known. Eient adaptive algorithms for highly oversampled systems are onsidered for the mathematially idential algorithms desribed in [2, 4, 11]. The formulation that will be used throughout the paper is similar to the subhannel response mathing (SRM) algorithm [4], a blok-oriented hannel identiation sheme whih minimizes a sum of pairwise dierenes between ltered subhannel outputs. The SRM algorithm requires obtaining the eigenvetor orresponding to the minimum eigenvalue of a matrix of orrelation statistis of the reeived signal. This eigenvetor has dimension equal to the produt of the oversampling fator and the number of symbol periods on whih the hannel response to be identied is nonzero; thus, exat alulation of this eigenvetor requires a number of oating point operations (ops) whih is ubi in the oversampling rate. 2

4 For real-time operation, data-reursive algorithms of muh lower omplexity are desirable. The problem of data-reursively obtaining the eigenvetor orresponding to the minimum eigenvalue of a matrix is well studied in the literature on Pisarenko's harmoni retrieval method [12, 13, 14, 15, 16, 17] and diretion-of-arrival estimation [18, 19, 20, 21]. In eah ase, however, the matrix to be analyzed onsisted of the expetation of an outer produt of the reeived vetor with itself, whih allows a very simple rank 1 matrix update formula. Unfortunately, the matrix used in the SRM algorithm is not of this form and these data-reursive algorithms are not appliable. In this work, two data-reursive algorithms for the DS/SS hannel identiation problem are presented. First, stohasti gradient and onjugate gradient algorithms are derived that operate on an error term dened by the Rayleigh quotient to update the minimizing eigenvetor in the SRM algorithm. The omputational load of these algorithms is only linear in the oversampling rate for the ase of unknown transmit lter and hannel identiation. The number of omputations beomes quadrati in the oversampling rate when the knowledge of the transmit lter is inorporated. Furthermore, these algorithms are apable of traking hanges in the hannel unlike the bath or \o-line" algorithms of [2, 4, 11] on whih the algorithms are based. Convergene analyses are presented to aid in seletion of the gain fator in the stohasti gradient algorithm. The onvergene issues that arise due to data bit transitions are also disussed. Finally, a trellis-searhing blind hannel identiation algorithm is desribed by extending the methods of [22] to an oversampled system. In [22], blind hannel identiation is done by extending the optimal known ISI deoder, the Viterbi Algorithm, to blind identiation by allowing eah state to retain multiple paths. Eah path operates with its own hannel estimate, whih is updated at eah time instant. The extension here to the oversampled ase results in a data-reursive algorithm with a omputational load that is only linear in the oversampling rate. 3

5 An outline of the paper is the following. In Setion 2, the system and hannel model are dened, and the ontinuous DS/SS ommuniation system mapped to an oversampled disrete system. The equivalene of the oversampled system to a multihannel system is then demonstrated, whih leads naturally to the subhannel response mathing algorithm. Setion 3 develops the new adaptive algorithm for seond order identiation of oversampled systems, while Setion 4 develops adaptive algorithms for a known transmit lter. Setion 5 ontains the extension of the trellis-searhing algorithm of [22] to oversampled systems. Finally, numerial results for the presented algorithms are shown in Setion 6. 2 Motivation and Problem Formulation 2.1 Mapping of a DS/SS System to a Disrete Oversampled System Assume that binary phase-shift keyed (BPSK) modulation is employed with a data symbol duration of T s. The BPSK signal is spread by multipliation with a spreading sequene having hip duration T T s, thus ahieving a proessing gain of N = T s =T in terms of reeiver SNR. The transmitted data bits, denoted by the sequene (s k ), are assumed to be independent and identially distributed (IID). Sine binary phase-shift keying is being employed, the output of the transmitter an be expressed as (using omplex baseband notation) x(t) = s E b NT s(t)e j where E b is the energy per data bit, is the transmitter phase (whih will be absorbed into the hannel response and suppressed from here forward), and s(t) is the spread spetrum waveform given by s(t) = 1 l=?1 N?1 s l m=0 a m p(t? lt s? mt ) 4

6 where p(t) is the normalized hip waveform, and a m is the value of the spreading ode for the m th hip. If the ontinuous hannel response is denoted by g(t), the reeived waveform is given by r(t) = g(t) x(t) + w(t) where w(t) is a omplex white Gaussian random proess. If g(t) was known, the optimal front end for the reeiver would be a lter mathed to the single symbol response g m P N?1 m=0 a mp(t? mt ). With g(t) unknown, this is not possible and the front end of the reeiver mathes to the ideal hip response, i.e. it simply integrates the reeived signal over a T interval. The sampled output of the reeiver front end orresponding to symbol k and hip n is given by y n (k) = y(kt s + nt ) = Z kts+(n+1)t kt s+nt r(t)dt: It is desired to t the system to an oversampled disrete system so that results from the literature an be applied. The following sequene of simple algebrai manipulations establishes the desired relation: y n (k) = = = s s E b NT Z kts+(n+1)t kt s+nt E b NT 1 l=?1 1 N?1 s l l=?1 m=0 dt a m Z 1 Z 1?1?1 s l h (k?l)n+n + n (k) = d g() d g() 1 l=?1 Z T 0 1 l=?1 N?1 s l m=0 a m p(t?? lt s? mt ) + n (k) dt p(t? + (kn + n? ln? m)t ) + n (k) h n (l)s k?l + n (k) (1) where n (k) is a sequene of IID gaussian random variables with variane N0 2 T, fh n (l)g L?1 l=0 is the n th subhannel response, whih is FIR with L oeients, and h ln+n = s N?1 E b a m g ln+n?m (2) N m=0 5

7 g m = s 1 T Z 1?1 Z T d g() 0 dt p(t + mt? ): The transmitter energy has been purposely worked into the hannel so that the average reeived signal energy for a given hip an be normalized to unity and the variane of the noise adjusted to obtain an overall reeived signal-to-noise ratio (SNR) per symbol period E s =N 0. Relation (1) establishes that for eah symbol period (x k in y n (k)) the hip samples fy n (k)g N?1 n=0 an be viewed as the outputs of N dierent subhannels, h n (l), n = 0; : : :; N? 1, exited by the same symbol sequene fs k g and operating at the symbol rate T s. The hannel identiation algorithms desribed here aomplish identiation of the entire hannel by identiation of eah of the N subhannels. Thus, the DS/SS system fortuitously falls into the setting of hannel identiation via oversampling as presented in [2, 4, 11]. The distintion is that in DS/SS systems the spreading waveform is naturally oversampled and no extra sampling equipment needs to be designed into the reeiver. Observe also that when the spreading ode fa 0 ; : : :; a N?1 g is known, as ours in the majority of appliations, the hannel h ln+n is partially known. This falls into the ategory of hannel identiation problems with known transmit lter; in this ase, (2) indiates that the spreading ode is the transmit lter. 2.2 Subhannel Response Mathing Next the subhannel identiation riterion used here and in [2, 4, 11] is desribed. Consider any two subhannels with impulse responses h n0 (k) and h n1 (k), and outputs y n0 (k) and y n1 (k): y n0 (k) = h n0 (k) s k y n1 (k) = h n1 (k) s k Sine the hannels have the same inputs, if one of the hannels is attahed to the output of the other and vie versa, in the noiseless ase the outputs should be equivalent. In the 6

8 absene of noise, it is obvious that h n1 (k) y n0 (k) = h n0 (k) y n1 (k), sine h n1 (k) h n0 (k) = h n0 (k) h n1 (k). This suggests that to identify the hannels h n0 (k) and h n1 (k), an option is to onsider subhannel estimates ^h n0 (k) and ^h n1 (k) that minimize the mean-squared dierene E 2 (^h n0 ; ^h n1 ) = Ej^h n1 (k) y n0 (k)? ^h n0 (k) y n1 (k)j 2. Clearly E 2 = 0 when the hannels are identied orretly. Furthermore, if the hannels have no ommon zeroes (whih is the wellknown identiability riterion for seond order identiation [3]), E 2 = 0 implies ^h n0 (k) = h n0 (k); ^h n1 (k) = h n1 (k) if the trivial solution ^h n0 (k) = 0; ^h n1 (k) = 0; 8k, is exluded. For the general ase of N > 2, dene the vetor (of length LN) of onatenated subhannels: h = [h 0 (0) h 0 (1) : : :h 0 (L? 1) h 1 (0) h 1 (1) : : :h 1 (L? 1) : : :h N?1 (0) h N?1 (1) : : :h N?1 (L? 1)] T and the vetor (of length LN) of onatenated subhannel observation vetors as: y(k) = [y T 0 (k); yt 1 (k); : : :; yt N?1 (k)]t where y n (k) = [y n (k) y n (k? 1) : : :y n (k? L + 1)] T : Let the notation x[n] denote the n th element of the vetor x. The objetive funtion is then dened as the sum of all pairwise mean-squared dierenes: E(^h) = N?2 N?1 m=0 n=m+1 L?1 E[j i=0 ^h[ml + i]y n (k? i)? L?1 j=0 ^h[nl + j]y m (k? j)j 2 ]; (3) whih when minimized over ^h, the estimate of h, gives a solution h opt. A onstraint must be added to avoid the trivial solution h opt = 0; the onstraint onsidered here is k^hk 2 = 1. 7

9 Note that there is an impliit amplitude ambiguity in the problem; multiplying h by a omplex onstant does not aet the solution h opt. The objetive (3) an be rewritten in an alternate form. Dene the L by L matries: R mn = E[y m (k)y H (k)]. Then n E(^h) = ^h T S^h (4) where ^h is the omplex onjugate (no transpose) of ^h, N?1 S = [I N R nn ]? V; (5) I N is an N by N identity matrix, is the Kroneker produt, and n=0 V = R 00 R 10 : : : R (N?1)0 R 01 R R 0(N?1) : : : R (N?1)(N?1) : The following properties of S are important: 1. S is Hermitian symmetri, i.e. S H = S. 2. From (3) and (4), it an be shown that S is non-negative denite. 3. If the hannels have no ommon zeroes, there will be a unique minimum eigenvalue of S and the onjugate of its eigenvetor will be the desired hannel h [3]. From (5), the minimum eigenvalue in a noiseless or noisy system is equal to 0 or (N?1) 2, respetively, where 2 is the variane of the observation noise per sample. 4. For N > 2, V is not an outer produt of the form E[y k y T k ]. 8

10 3 Adaptive Algorithms for the General Problem For the DS/SS appliation onsidered here, the oversampling rate will be muh higher than the number of periods over whih the impulse response of the hannel has support (i.e. N L). Thus, adaptive algorithms will be sought that have low omplexity in N. Historially, algorithms for iteratively solving for the eigenvetor orresponding to the minimum eigenvalue of a non-negative matrix have fallen into three lasses: stohasti gradient [12, 14, 15, 20], onjugate gradient [13, 23], and Newton's method [16, 17]. It is well known for the harmoni retrieval problem that the latter two algorithms onverge signiantly faster than the stohasti gradient algorithm [13, 17]. However, in the DS/SS problem, Newton's method is impratial beause it requires the omputation of the inverse of the Hessian matrix. Thus, attention here is restrited to the stohasti gradient and onjugate gradient algorithms. 3.1 Stohasti Gradient Algorithm: Denition Although nding the eigenvetor orresponding to the minimum eigenvalue of a matrix orresponds losely to Pisarenko's harmoni retrieval method [12] and diretion of arrival estimation [18], as noted previously S is not expressible as the expetation of an outer produt of the observed vetor unless N = 2; hene, the simple data updates of [12, 20] do not apply. As in early work on Pisarenko's harmoni retrieval [12], a simple stohasti gradient algorithm is onsidered rst. The algorithm updates ^h k?1, the estimate of h opt at iteration k? 1, based on an additional symbol period of data: y 0 (k); y 1 (k); : : :; y N?1 (k). The standard stohasti gradient update equation for ^h k is given by ^h k = ^h k?1? 5 e k (^h k?1 ) (6) where 5 is the gradient operator and e k (^h k?1 ) is the empirial estimate of the Rayleigh quotient of S : 9

11 e k (^h k?1 ) = E k(^h k?1 ) k^h k?1 k 2 = ^h T k?1s k^h k?1 k^h k?1 k 2 ; (7) E k (^h k?1 ) = ^h T k?1s k^h k?1 is the empirial estimate of E(^h k?1 ), and S k is dened by S in (5) exept that the single sample empirial estimates ^R mn = y m (k)y H (k) are used in plae of n R mn = E[y m (k)y H n (k)]. As ontrasted with the algorithms of [12, 16], normalization of ^h k to unity is unneessary sine it is aounted for by the denominator of e k (^h k?1 ). The gradient of e k (^h k?1 ) is simply omputed to obtain the update formula: ^h k = ^h k?1? 5 e k (^h k?1 ) = ^h k?1? (k^h k?1 k 2 Sk^h k?1? (^h T k?1s k^h k?1)^h k?1 ) : (8) k^h k?1 k 4 In (8), the onstant fator of 2 has been absorbed into. Brute fore omputation of this update requires O(L 2 N 2 ) ops beause of the matrix-vetor multipliation S k^h k?1. However, a more areful onsideration of the produt S k^h k?1 reveals that only O(L 2 N) ops are neessary. Noting that 5E k (^h k?1 ) = 2S k^h k?1, element ml + l of S k^h k?1 an be obtained k (^h k?1 k?1 [ml + l] = 2 N?1 n=0;n6=m L?1? j=0 L?1 = 2( i=0 L?1? j=0 L?1 ( i=0 ^h k?1 [ml + i]y n (k? i)y n(k? l) ^h k?1 [nl + j]y m (k? j)y n(k? l)) N?1 ^h k?1 [ml + i] N?1 y m (k? j) n=0 n=0 y n (k? i)y n (k? l) ^h k?1 [nl + j]y n(k? l)): (9) The omputation of the right side of (9) proeeds as follows: the sums over n must be done for eah l and i (or j), thus leading to O(L 2 N) operations; given these sums, S k^h k?1 = 5E k (^h k?1 ) an be found in O(LN) inner produts of vetors of length L. Thus, the update (8) an be done in O(L 2 N) ops. Sine N L for DS/SS systems, this amounts to a large 10

12 savings in omputational eort for the algorithm. By reversing the summations in (9), the update an be performed in O(LN 2 ) ops, whih may be useful in other appliations. Finally, note that there are alternate ways to obtain eient updates; one option is to use the struture of S to write the length L subvetors of S k^h k?1 as the sum of outer produts of subhannel observations. 3.2 Stohasti Gradient Algorithm: Convergene In this setion, the mean onvergene behavior is evaluated. This is used to selet the gain fator of the adaptive algorithm Convergene Analysis Assuming Perfet Estimation of S Global onvergene is diult to demonstrate due to the omplexity of the error surfae. However, it is possible to show that for a noisy system all of the eigenspaes exept the one orresponding to the minimum eigenvetor are unstable; thus, the algorithm will eventually leave these spaes [14]. A less ambitious but desirable goal is to hoose suh that the algorithm will onverge to the orret solution when it is near the minimizing eigenvetor. Here a rst order onvergene analysis is performed, whih is equivalent to assuming S an be estimated without error. In this ase, the error after iteration k of the algorithm is given by the Rayleigh quotient: e(^h k ) = ^h T k S^h k k^h k k 2 = ^h T k S Z ^h k + 0 (10) k^h k k 2 where S Z is the matrix S for a noiseless system (w(t) = 0) and 0 = (N? 1) 2 is the unique smallest eigenvalue of S. It is appropriate to study only the onvergene of the noiseless system as noise only ontributes a onstant noise oor, 0, to the objetive funtion E(^h k ). Sine S Z is Hermitian symmetri, it has an eigendeomposition S Z = P LN?1 i=0 i v i v H i where v i is the i th 11

13 eigenvetor and i the i th eigenvalue of S Z. Then, using the shorthand notation k = e(^h k ), the objetive funtion beomes k = LN?1 i=0 ^h k will onverge to h opt only if i k = i k^h T k v i k2 k^h k k 2 k^h T k v i k 2 LN?1 i = k^h k k 2 i=0 i k: onverges to zero, i = 1; 2; : : :; NL?1. Substituting ^h k = ^h k?1?5e k (^h k?1 ), and retaining only terms that are linear or onstant in k (i.e. assuming k 1 near the solution), one obtains i k = ik^h T k?1v i k 2 k^h k?1 k 2 1! A k^h k?1 k 2 k^h k k 2! i 1? 2 k^h k?1 k i 2 k^h k?1 k 4 whih implies i k i k?1 1? i k^h k?1 k 2! 2 (11) where the last inequality omes from the fat that k^h k k 2 is nondereasing as a funtion of k. This nondereasing property an be derived as follows: note from the denition of 5e k (^h k?1 ) that ^h H k?1(^h k? ^h k?1 ) = ^h H k?1 5 e k (^h k?1 ) = 0, whih implies k^h k k 2 = k^h k? ^h k?1 + ^h k?1 k 2 = k^h k? ^h k?1 k 2 + k^h k?1 k 2 k^h k?1 k 2 : If the initial guess is hosen on the unit irle (i.e. k^h 2 0k = 1), the nondereasing property and (11) an be used to show that onvergene ours in all of the modes if < 2 max. Sine max is diult to obtain, the onservative estimate 2 tr(s Z) 12 is used instead, where

14 tr(s Z ) = L(N? 1) P N?1 i=0 Ejy i(k)j 2? LN(N? 1) 2, whih an be readily estimated if the signal-to-noise ratio is known. Note that the inequality (11) implies that the speed of onvergene depends on k^h k?1 k 2 in general. This suggests dening a normalized algorithm to eliminate the k^h k?1 k 2 in the onvergene ondition. The normalized algorithm is obtained by using a variable gain fator k^h k?1 k 2 at the k th step. Rewriting (6) with this \variable" gain fator yields ^h k = ^h k?1? (k^h k?1 k 2 S k ^h k?1? (^h T k?1 S k^h k?1 )^h k?1 ) k^h k?1 k 2 : (12) and the onvergene analysis follows as before to give i k i k?1 (1? i ) 2 (13) whih implies onvergene if < max 2. Note that this normalized algorithm is similar to the algorithms of [12, 16], but here it has been motivated diretly from the rst order onvergene analysis of the unnormalized stohasti gradient algorithm Rank of S k and Convergene Even in a noiseless system, when the empirial estimate S k is used the algorithm will see a loss in rank relative to the rank NL? 1 of the noiseless ensemble average S Z. To see that the rank of S kz an be somewhat less than NL? 1, onsider the ase where the hannel input bits are all equal to +1 from time k down to time k? 2L + 2 (i.e. all of the hannel input bits that aet S k ). By observing equation (5), one notes that the rst L rows are equal, the seond L rows are equal, et., whih guarantees a rank of no greater than N. In fat, the solution h opt is now only onstrained to satisfy the following set of equations: 13

15 L?1 h i (l)! L?1 l=0 l=1 h opt [jl + l]! = L?1 h j (l)! L?1 l=0 l=1 h opt [il + l]! 8i 6= j whih has a solution spae of dimension L+(N?1)(L?1), implying that the rank of S k is only N? 1. Similar results hold for other sequenes of input bits. This loss of rank of S k points out the requirement for the data bits to hange in suh a way that eah of the modes is persistently exited to avoid onvergene stagnation in an eigendiretion. 3.3 Conjugate Gradient Algorithm In this setion, the appliation of onjugate gradient (CG) algorithms to the minimization of the Rayleigh quotient is reviewed. It is illustrated how data-adaptive versions of these algorithms an be implemented in O(L 2 N) ops for the oversampled blind hannel identiation problem. The two algorithms onsidered here are presented in the notation of [23] - for a more detailed disussion of the appliation of these algorithms to harmoni retrieval, see [13, 23]. The data-adaptive CG algorithms implemented here use the update ~ h k+1 = ^h k + k p k followed by the normalization ^h k+1 = ~ h k+1 k ~ h k+1 k, where k quotient in diretion p k and is given by is hosen to minimize the Rayleigh where k =?B + p B 2? 4CD 2D D = P b (k)p (k)? P a (k)p d (k) B = P b (k)? e k+1 (^h k )P d (k) C = P a (k)? e k+1 (^h k )P (k) P a (k) = ph k S k+1^h k k^h k k 2 P b (k) = ph k S k+1p k k^h k k 2 P (k) = ph ^h k k k^h k k 2 P d (k) = ph k p k k^h k k 2 : The new searh diretion p k is seleted by p k =? 5 e k+1 (^h k ) + b k p k?1. In Chen et. al. [13], b k is hosen suh that p H k S p k?1 = 0, whih yields 14

16 b k = 5H e k+1 (^h k )S kp k?1 p H k?1 S kp k?1 : On the other hand, Yang et. al. [23] hoose b k suh that p HHp = 0 where H is the Hessian k k?1 of the Rayleigh quotient. This yields b k =?[5H e k+1 (^h k )S kp k?1 + k 5 e k+1 (^h k )k 2^hH k p k?1 ] p H k?1 S k p k?1? e k+1(^h k )kp k?1 k 2 : The algorithms above dier from the algorithms presented in [13, 23]; similarly to the stohasti gradient algorithms presented in Setion 3.1, they use only the instantaneous data estimates of S k to step one per symbol period. The omplexity of the data-adaptive CG algorithms is primarily due to the matrix-vetor multipliations S kp k?1 and S k^h k?1 whih an be done in O(L 2 N) operations as per (9). Note the nie feature that the onjugate gradient algorithms do not require speiation of an additional gain fator. 4 Adaptive Algorithm for a Known Spreading Code When NL is fairly large, the stohasti algorithm may take a long time to onverge beause of the high dimensionality (NL) of the searh spae. However, in [10], it is shown how knowledge of the transmit lter an redue this dimensionality. In the ase where the spreading ode is the known transmit lter as in (2), the searh spae dimension is redued to NL? N + 1. Beause the DS/SS appliation onsidered here assumes large N and fairly small L, this saving of (N? 1) dimensions promises to greatly redue the dimensionality of the problem with an assoiated improvement in performane. When the spreading sequene fa k g is xed and known, the subhannel responses depend only on the unknown hannel response g = [g 0 g 1 : : :g NL?N ]. As a funtion of ^g, the estimate of g, the objetive funtion (3) beomes: 15

17 E(^g) =? N?2 N?1 m=0 n=m+1 L?1 N?1 j=0 u=0 E[j L?1 N?1 i=0 u=0 a u^g jn+n?u y m (k? j)j 2 ] a u^g in+m?u y n (k? i) with the onstraint k^gk 2 = 1. In [10], it was shown that E(^g) = ^g T QSQ H^g where Q, an (L?1)N+1 by LN matrix of spreading oeients, is dened as Q = [Q 0 Q 1 : : :Q N?1 ], where Q n = [q n (0) q n (1) : : :q n (L? 1)] with length LN? N + 1 vetors q n (k) = [a kn+n a kn+n?1 : : : a kn+n?ln+n ] T where a l is dened to be zero for l 62 [0; N? 1]. The stohasti gradient algorithm an be applied as before to yield the normalized update formula ^g k = ^g k?1? (k^g k?1 k2 Q S kq T ^g k?1? (^g T k?1 QS kq H^g k?1 )^g k?1 ) k^g k?1 k 2 (14) but the omplexity of this algorithm is O(L 3 N 3 ) if implemented with a brute fore matrix multipliation, and still O(L 2 N 3 ) if one uses the property that eah q n (k) ontains at most N nonzero entries. The key is to reognize that Q S k QT ^g k?1 = 5E k (^g k?1 ) where the gradient in this ase is with respet to the elements of ^g k?1. Applying the hain rule for gradients [24, pg. 134], element j of Q S kq T ^g k?1 is given by 16

18 @(^g T k?1 QS kq H^g k?1 k?1 [j] = = N?1 L?1 m=0 l=0 N?1 L?1 m=0 T k?1 QS kq H^g k?1 k?1 [ml + k?1 [ml + k?1 k?1 S k^h k?1 [ml + l] a ln+m?j (15) The NL partial derivatives required on the right side of (15) are obtained from (9). Also, for a given j, only N of the terms on the right side of (15) are non-zero. Thus, the proedure at eah step is: 1. Form the gradient with respet to ^h k?1 using (9) in minfo(l 2 N); O(LN 2 )g operations. 2. Apply (15) to get the gradient with respet to ^g k?1 in O(LN 2 ) operations. 3. Use the result of step 2 in the numerator of (14) to update ^g k?1. The O(LN 2 ) operations required for this algorithm is larger than for the stohasti gradient algorithm when the knowledge of the transmit lter is not used. Finally, note that the onjugate gradient algorithms an take advantage of the transmit lter knowledge in an analogous way to the stohasti gradient algorithm. 5 Adaptive Algorithm Using Trellis Searhing In this setion, a joint data and hannel estimation algorithm due to Seshadri [22] is extended to oversampled systems. 5.1 Joint Data/Channel Data Estimation It is well known that the maximum likelihood sequene estimator for a disrete hannel with known impulse response of duration L symbol periods an be implemented with a Viterbi Algorithm with 2 L?1 states, one state for eah possible set of the previous L? 1 data bits [25, pg. 610]. The Eulidean distane metri (equivalent to likelihood for the AWGN ase) for a path in the trellis originating at state i at time k is updated as follows: the states between 17

19 whih the path travels determine an estimate of the input bit sequene ^b k = [^b k ^bk?1 : : :^b k?l+1 ]. This estimate is used to evaluate the path metri by L?1 i k(^b k ) = i k?1? (y(k)? ^bk [l]h(l)) 2 where i k?1 is the aumulated metri of state i at time k? 1 (or, alternatively, the metri of the path that was retained at state i at time k? 1), y(k) is the data sample at time k, and h(l) is the hannel impulse response at delay l. Two paths enter eah of the states at time k. At eah state, the path with the highest metri is kept and the other disarded. The algorithm onsidered in [22] extends the above idea to hannels where the ISI is unknown. The algorithm is nearly idential to the algorithm desribed above exept that multiple paths (the number denoted M) are kept for eah state at eah time. In this ase, eah path ontains and updates its own estimate of the hannel. Metris are now assoiated with a path and are alulated for a given path based on its hannel estimate instead of the atual hannel. In equation form, the metri for the j th path from state i with bit estimate ^b k is given by l=0 L?1 ij k (^b k ) = ij? (y(k)? k?1 ^bk [l]^h ij (l)) 2 where ij k?1 is the metri of the j th path of state i at time k?1 and ^h ij (l) is the hannel estimate of the j th path from state i for delay l. Note that 2M paths enter eah state at time k; the M paths with the best metris are retained and the others disarded. For the M surviving paths in eah state, the hannel estimate is updated using an LMS algorithm. 5.2 Extension to Oversampled Systems As disussed in Setion 2.2, the oversampled hannel an be viewed as N subhannels operating at rate T s, whih all need to be equalized. Sine the method of [22] is able to equalize a hannel that is not oversampled, eah subhannel ould be separately equalized with this algorithm 18 l=0

20 and the results ombined (although there would be a sign ambiguity for eah subhannel). Of ourse, this would be suboptimal sine it is known that the input to eah of the subhannels is the same. Instead, a single Viterbi Algorithm is used with 2 L?1 states and M paths kept per state, eah with its estimate of all of the subhannels. The metri for the j th path from state i orresponding to input ^b k is given by ij k (^b k ) = ij k?1? N?1 m=0 (y m (k)? L?1 l=0 ^bk [l]^h ij m (l))2 where ij k?1 is the metri of the j th path of state i at time k? 1, and ^h ij m (l) is the hannel estimate of the j th path from state i for subhannel m at delay l. A method of updating the hannel estimate for a given path must now be onsidered. Although the LMS algorithm ould be used just as it was in [22], a dierent method is proposed here for the following reason. Stohasti gradient algorithms (like LMS) are largely utilized to redue omplexity as has been done in the previous setions here. In the hannel estimation portion of the trellis-searhing algorithm, however, the minimal mean square error for ensemble averages ours for the orret path at h opt n (l) = R n (l) = E[y n (k)^b k [l]]. whih is easily approximated. Thus, instead of using a gradient searh algorithm, for eah path this ensemble average an be approximated by the sample mean ^h ij n (l) = ^R n (l) = 1 T?1 y n (k)^b T k [l]: For a hannel that does not vary with time, this appears to be an attrative alternative to the LMS algorithm. If the onsidered hannel does vary with time, a \forgetting fator" an be added to the distribution so that it an adapt to new statistis. Note the added benet that there is no need for the hoie of a gain fator. k=0 Finally, the trellis-searhing algorithm has one large advantage over the SRM-based algorithms. Not only an it be easily modied to take advantage of a known spreading ode, but 19

21 it an also perform blind hannel identiation when the (known) spreading ode varies aross symbol periods as in many pratial systems. Sine the fous here is on omparisons with the SRM-based algorithms, this will not be addressed further. 6 Numerial Results 6.1 Implementation Issues It is important to point out the omplexity and limitations of the algorithms before the numerial results are presented. Both the \standard" stohasti gradient algorithm and the \normalized" stohasti gradient algorithm along with both onjugate gradient algorithm algorithms based on subhannel response mathing algorithm require only O(L 2 N) operations at eah iteration. As mentioned earlier, the algorithms have the drawbak that they are only orret to within a omplex onstant; in other words, they have both an amplitude and phase ambiguity. As disussed in [26], this is not a severe limitation. Furthermore, the algorithms only identify the hannel and do not deode the transmitted bits. One possible method of doing so is to run a Viterbi Algorithm onurrent with the hannel estimator. At eah symbol period, path metris in the trellis are evaluated using the urrent hannel estimate. Note that this adds O(2 L N) operations for a total omplexity that is maxfo(2 L N); O(L 2 N)g. The trellis-searhing algorithm requires O(L2 L NM) operations per symbol, signiantly more than the stohasti gradient algorithm, but also deodes the bits as it runs with no need for an additional sequene estimator. Note that the algorithm suers from a sign ambiguity. It annot distinguish between the true bit stream aross the true hannel and the negative of the true bit stream aross the negative of the true hannel beause they both give the same outputs. Thus, some sort of dierential enoding at the transmitter is required [25, pg. 266]. 20

22 6.2 Simulation Results The algorithms presented in the previous setions are ompared in this setion through simulation. As in [11], the normalized mean squared error at iteration k is dened by NMSE k = ^h k k ^h k k? h khk The goal is to determine both how the algorithms ompare to eah other and also how these results ompare to results reported on similar problems suh as those from Pisarenko's harmoni retrieval. Eah of the algorithms ertainly an be tweaked to obtain better performane. In partiular, the trellis-searhing algorithms have higher omplexity than the SRM-based algorithms. Sine the SRM-based algorithms alulate e k (^h k?1 ) at eah step, it ould be used to implement a multiple starting point iteration to lose both the performane and omplexity gap [27]. Slight extensions, however, will be ignored and only the base algorithms onsidered. In the rst set of numerial results, two hannels are onsidered with zeroes as tabulated below. For both hannels, N = 4; L = 5, whih yields = 0:14 from the onvergene analysis in Setion : Subhannel Good Bad j j j j j j j j j j j j j j j j Note in Figure 1 how the zeroes in the \Bad" hannel are loser to the unit irle than the zeroes for the \Good" hannel. Also note that on the \Bad" hannel, subhannels one and two 21

23 1 1 Good Bad Figure 1: Zeroes of the \Good" and \Bad" Channels have a set of zeroes that fall nearly on top of eah other, making it plausible that identiation is more diult. The \standard" stohasti gradient algorithms and \normalized" stohasti gradient algorithms were run on hannels \Good" and \Bad" at signal-to-noise ratios of 20 db and 10 db. The best gain fators from among = 0:06; = 0:14; = 0:30 (the gain fator the onvergene analysis suggests and one on either side) were seleted and the results are shown in Figure 2 through 5 along with the other SRM-based algorithms. The results are averaged over 200 sample trials. Only the data-adaptive version of the onjugate gradient algorithm of Chen et. al. [13] is shown as the data-adaptive version of the onjugate gradient algorithm of Yang et. al. [23] was found to have nearly idential performane. Comparisons with the trellis-searhing algorithms are made in Figures 6 and 7. 22

24 Key to the Figure abbreviations: 1. \SG": The standard stohasti gradient algorithm. 2. \Norm": The normalized stohasti gradient algorithm. 3. \SRM": The o-line O(L 3 N 3 ) algorithm of [4] where S k is obtained at time k by averaging all of the data through time k to estimate ensemble averages. 4. \Trellis LMS": The adaptive trellis-searhing algorithm where hannel updating is done with an LMS algorithm with (empirially hosen) gain fator = 0: \Trellis Mean": The adaptive trellis-searhing algorithm where hannel updating is done via sample mean. 6. \Conj Grad": The data-adaptive onjugate gradient hannel identiation algorithm. A few things are immediately apparent from Figures 2 through 5. First and foremost is that all of the SRM-based data-adaptive algorithms (SG, Norm, CG) perform muh worse than the \o-line" SRM algorithm. The onjugate gradient algorithm shows faster initial onvergene for both \Good" and \Bad" hannels at both signal-to-noise ratios, but for SNR=10 db, its performane levels o very quikly and is surpassed slightly by the stohasti gradient algorithms. Note in partiular how the normalized stohasti gradient algorithm with properly hosen gain fator omes very lose in performane to the onjugate gradient algorithm. In omparison with the trellis-searhing algorithms, one notes that for M = 2 the trellis-searhing algorithms perform quite poorly. For M = 8, a omplexity muh greater than the SRMbased algorithms, the trellis-searhing algorithms perform markedly better, espeially for the \Bad" hannel as they suer no performane impairment based on the plaement of the zeroes of the hannels. Initial onvergene is espeially impressive. Although the trellis-searhing algorithm that uses sample means for updating the hannel estimates has the advantage of not 23

25 requiring the hoie of a gain fator as in the trellis-searhing algorithm with the LMS-based hannel updating, the performane advantage that was expeted is not in evidene. In the ase displayed in Figure 6, it atually performed slightly worse. Sine the LMS hannel updating strategy will also trak better for hannels that vary with time, it is onluded that it is superior to the sample mean hannel updating strategy. There are some eets not evident from the plots - whereas the average values displayed are representative of all of the sample runs for the SRM-based stohasti gradient algorithms, the performane urves for the other methods show wider variation. For the onjugate gradient algorithm, the error on a given sample run tends to boune around a lot in the steady state with the smooth displayed urves being a byprodut of averaging over many sample runs. The only moderately good performane of the trellis-searhing algorithm with either hannel updating strategy is aused by outstanding performane on ertain sample paths and a omplete lak of onvergene in others. This was an unexpeted phenomenon as it was not noted in [22]. Finally, the benet of knowledge of the transmit lter is onsidered. In this ase, a hannel with N = 8; L = 3 is formed by onvolving a length 17 randomly generated omplex hannel with exponentially dereasing average power (drop from peak to e?1 at sample 8) with a length 8 spreading ode. Results for the normalized stohasti gradient algorithm and normalized known transmit lter algorithms are shown in Figure 8. Note that the known hannel transmit lter algorithm shows better initial onvergene and steady-state performane but at a higher omputational ost per step as disussed in Setion 4. 7 Summary and Conlusions In this paper, it has been shown how a diret-sequene spread spetrum system sampled at the hip rate an be mapped to a disrete oversampled system with the spreading ode as the transmit lter. This motivates the searh for eient data-adaptive blind hannel identiation algorithms when the oversampling rate is high. Stohasti gradient and onjugate gradient 24

26 algorithms have been presented that are only linear in the oversampling rate. However, if one wants to take advantage of the knowledge of the transmit lter to redue the dimensionality of the searh spae, the algorithms beome quadrati in the oversampling rate. Finally, for omparison a trellis-searhing algorithm for performing non-oversampled equalization has been extended to the oversampled ase. Numerial results suggest that there is quite a bit of performane loss between the o-line SRM algorithm and the data-adaptive implementations of it. Furthermore, it appears that the gain of the onjugate gradient algorithms over the stohasti gradient algorithms is not as prominent as was shown in the Pisarenko's harmoni retrieval; however, the onjugate gradient algorithm showed markedly faster initial onvergene in general. The huge gains shown in [10] for the known transmit lter are also not evident for the ase that was onsidered in this paper. In general, the trellis-searhing algorithms perform muh better than the SRM-based algorithms while still only being linear in the oversampling rate. However, while still only linear in the oversampling rate, the omplexity is muh higher if a large number of paths are kept per state, whih appears to be a prerequisite for aeptable performane. If the SRM-based algorithms are allowed to lose this omplexity gap, they an take into aount the urrent error estimate at eah iteration and use a multiple starting point SRM-based algorithm to narrow the performane gap onsiderably. 25

27 Referenes [1] S. Haykin, Ed., Blind Deonvolution, Prentie Hall, Englewood Clis, [2] H. Liu, G. u, and L. Tong, \A Deterministi Approah to Blind Equalization," Conferene Reord of the Twenty-Seventh Asilomar Conferene on Signals, Systems, and Computers (1993), pp [3] L. Baala and S. Roy, \Time-Domain Blind Channel Identiation Algorithms," Proeedings of the 1994 Conferene on Information Siene and Systems, Vol. 2, pp [4] S. Shell and D. Smith, \Improved Performane of Blind Equalization Using Prior Knowledge of Transmitter Filter," Conferene Reord of the 1994 IEEE Military Communiations Conferene, Vol. 1, pp [5] L. Tong, G. u, and T. Kailath, \Blind Identiation and Equalization Based on Seond-Order Statistis: A Time Domain Approah," IEEE Transations on Information Theory, Vol. 40, pp , Marh [6] E. Moulines, P. Duhamel, J. Cardoso, and S. Mayrargue, \Subspae Methods for the Blind Identiation of Multihannel FIR Filters," IEEE Transations on Signal Proessing, Vol. 43, pp , February [7] K. Meraim, P. Duhamel, D. Gesbert, P. Loubaton, S. Mayrargue, E. Moulines, and D. Slok, \Predition Error Methods for Time-Domain Blind Identiation of Multihannel FIR Filters," Proeedings of the 1995 International Conferene on Aoustis, Speeh, and Signal Proessing, Vol. 3, pp [8] G. Giannakis and S. Halford, \Performane Analysis of Blind Equalizers Based on Cylostationary Statistis," Proeedings of the 1994 Conferene on Information Siene and Systems, Vol. 2, pp [9] D. Slok and C. Papadias, \Further Results on Blind Identiation and Equalization of Multiple FIR Channels," Proeedings of the 1995 International Conferene on Aoustis, Speeh, and Signal Proessing, Vol. 3, pp [10] S. Shell, D. Smith, and S. Roy, \Blind Channel Identiation using Subhannel Response Mathing," Proeedings of the 1994 Conferene on Information Siene and Systems, Vol. 2, pp [11] L. Baala and S. Roy, \A New Blind Time-Domain Channel Identiation Method Based on Cylostationarity," IEEE Signal Proessing Letters, Vol. 1, pp , June [12] P. Thompson, \An Adaptive Spetral Analysis Tehnique for Unbiased Frequeny Estimation in the Presene of White Noise," Conferene Reord of the Thirteenth Asilomar Conferene on Ciruits, Systems, and Computers (1979), pp [13] H. Chen, T. Sarkar, S. Dianat, J. Brule, \Adaptive Spetral Estimation by the Conjugate Gradient Method," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 34, pp , April

28 [14] M. Larimore and R. Calvert, \Convergene Studies of Thompson's Unbiased Adaptive Spetral Estimator," Conferene Reord of the Fourteenth Asilomar Conferene on Ciruits, Systems, and Computers (1980), pp [15] M. Larimore, \Adaptive Convergene of Spetral Estimation Based on Pisarenko Harmoni Retrieval," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 31, pp , August [16] V. Reddy, B. Egardt, and T. Kailath, \Least Squares Type Algorithm for Adaptive Implementation of Pisarenko's Harmoni Retrieval Method," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 30, pp , June [17] G. Mathew, S. Dasgupta, and V. Reddy, \Improved Newton-Type Algorithm for Adaptive Implementation of Pisarenko's Harmoni Retrieval Method and Its Convergene Analysis," IEEE Transations on Signal Proessing, Vol. 42, pp , February [18] R. Shmidt, \Multiple Emitter Loation and Signal Parameter Estimation," IEEE Transations on Antennas and Propagation, Vol. 34, pp , Marh [19] M. Kaveh and A. Barabell, \The Statistial Performane of the MUSIC and the Minimum-Norm Algorithms in Resolving Plane Waves in Noise," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 34, pp , April [20] D. Torrieri and K. Bakhru, \The Reursive Suppression Algorithm for Adaptive Superresolution," IEEE Transations on Antennas and Propagation, Vol. 40, pp , August [21] J. Yang and M. Kaveh, \Adaptive Eigensubspae Algorithms for Diretion or Frequeny Estimation and Traking," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 36, pp , February [22] N. Seshadri, \Joint Data and Channel Estimation Using Blind Trellis Searh Tehniques," IEEE Transations on Communiations, Vol. 42, pp , February/Marh/April [23]. Yang, T. Sarkar, and E. Arvas, \A Survery of Conjugate Gradient Algorithms for Solution of Extreme Eigen-Problems of a Symmetri Matrix," IEEE Transations on Aoustis, Speeh, and Signal Proessing, Vol. 37, pp , Otober [24] J. Marsden and A. Tromba, Vetor Calulus, W. H. Freeman, New York, [25] J. Proakis, Digital Communiations, MGraw-Hill, New York, [26] S. Verdu, B. Anderson, and R. Kennedy, \Blind Equalization without Gain Identi- ation," IEEE Transations on Information Theory, Vol. 39, pp , January [27] D. Goekel, A. Hero III, and W. Stark, \Blind Channel Identiation for Diret- Sequene Spread-Spetrum Systems," Conferene Reord of the 1995 IEEE Military Communiations Conferene, pp

29 SG(u=0.30) 10 1 SG(u=0.30) Norm(u=0.30) NMSE Norm(u=0.30) Conj Grad Norm(u=0.14) NMSE 10 2 Conj Grad 10 4 SRM 10 3 SRM Iteration Figure 2: Comparison of the SRM-based algorithms for the \Good" hannel, SNR=20 db Iteration Figure 4: Comparison of the SRM-based algorithms for the \Bad" hannel, SNR=20 db SG(u=0.14) Conj Grad NMSE NRMSE Conj Grad SG(u=0.14) Norm(u=0.14) Norm(u=0.14) NMSE 10 1 SG(u=0.30) Norm(u=0.14) Conj Grad 10 3 SRM SRM 10 2 SRM Iteration Figure 3: Comparison of the SRM-based algorithms for the \Good" hannel, SNR=10 db Iteration Figure 5: Comparison of the SRM-based algorithms for the \Bad" hannel, SNR=10 db 28

30 10 1 NRMSE Conj Grad Trellis Mean ( M=2) SG (u=0.14) Trellis Mean ( M=8) Trellis LMS ( u=0.02, M=8) 10 3 SRM Iteration Figure 6: Comparison of SRM-based stohasti gradient algorithms with the trellis searhing algorithm, \Good" hannel, SNR=10 db NMSE Unknown (u=0.06) Unknown (u=0.14) 10 2 Known (u=0.14) Known (u=0.06) NMSE Trellis LMS (M=2) Conj Grad Trellis LMS (M=8) Iteration Figure 8: Comparison of the normalized stohasti gradient algorithm with that when the transmit lter is known, N = 8; L = 3, SNR = 10 db 10 3 SRM Iteration Figure 7: Comparison of SRM-based stohasti gradient algorithms with the trellis searhing algorithm, \Bad" hannel, SNR=20 db 29