Per-Survivor Processing: A General Approach to MLSE in Uncertain Environments

Transcription

1 Per-Survivor Processing: A General Approach to MLSE in Uncertain Environments Riccardo Raheli, Member, IEEE, Andreas Polydoros, Fellow, IEEE, and Ching-Kae Tzou, Member, IEEE Abstract Per-Survivor Processing (PSP) provides a general framework for the approximation of Maximum Likelihood Sequence Estimation (MLSE) algorithms whenever the presence of unknown quantities prevents the precise use of the classical Viterbi algorithm. This principle stems from the idea that data-aided estimation of unknown parameters may be embedded into the structure of the Viterbi algorithm itself. Among the numerous possible applications, we concentrate here on (a) adaptive MLSE, (b) simultaneous Trellis Coded Modulation (TCM) decoding and phase synchronization, (c) adaptive Reduced State Sequence Estimation (RSSE). As a matter of fact, PSP is interpretable as a generalization of decision feedback techniques of RSSE to decoding in the presence of unknown parameters. A number of algorithms for the simultaneous estimation of data sequence and unknown channel parameters are presented and compared with conventional techniques based on the use of tentative decisions. Results for uncoded modulations over InterSymbol Interference (ISI) fading channels and joint TCM decoding and carrier synchronization are presented. In all cases, it is found that PSP algorithms are clearly more robust than conventional techniques both in tracking a time-varying channel and acquiring its characteristics without training. I. INTRODUCTION It is well-known that Maximum Likelihood Sequence Estimation (MLSE) represents the optimum decoding strategy of a data sequence that has undergone coding and transmission over a dispersive and noisy channel, under the assumption that the receiver perfectly knows the parameters characterizing the channel [1] [7]. The cascade of encoder and transmission channel may be described by a finite-state machine with associated state and trellis diagrams. As a consequence, the receiver performs MLSE of the data sequence by means of a Viterbi algorithm that searches for the path with minimum cost in this combined InterSymbol Interference (ISI) and code trellis diagram. Since the number of states increases exponentially with the length of the channel impulse response, high-complexity trellis decoders would be often required for an implementation of the Paper approved by Gottfried Ungerboeck, the Editor for Signal Processing for Storage Systems of the IEEE Communications Society. Manuscript received June 2, 1992; revised July 20, This work was supported by NASA under contract no. NAG 3/1279, the NSF PYI grant no. NCR/ , Ministero dell Università e della Ricerca Scientifica e Tecnologica, and Consiglio Nazionale delle Ricerche. This paper was presented in part at the 1991 IEEE Communication Theory Workshop, Rhodes, Greece, June 30 July, R. Raheli is with the Dipartimento di Ingegneria dell Informazione, Università di Parma, Viale delle Scienze, Parma, Italy. A. Polydoros is with the Communication Sciences Institute, Department of Electrical Engineering Systems, University of Southern California, Los Angeles, CA , U.S.A.. C. K. Tzou is with the Computer and Communication Research Laboratory, Industrial Technology Research Institute, Chutung, Hsinchu, Taiwan 310, Republic of China. optimal algorithm; thus, reduced complexity suboptimal algorithms have been devised based on the well-known Decision Feedback Equalization (DFE) approach [5,8]. This technique entails the truncation of the channel impulse response considered in the combined trellis and cancellation of the remaining (residual) ISI by the use of tentative decisions. An alternative to this classic technique may be obtained by incorporating the decision-feedback mechanism within the Viterbi decoder. This concept seems to have first appeared in [9,10] in an effort to extend the classical structure of MLSE decoders to the case of infinite ISI channels, which obviously cannot support a finite (non-expanding) trellis in a precise way. The basic idea is simple: try to cancel the effects of residual ISI directly within the calculation of each transition metric in the Viterbi algorithm, based on the data sequence associated to the survivor leading to that transition, hence, in a per-survivor fashion. More flexible and general extensions of this concept were subsequently and independently presented by various authors [11] [14]. Further extensions were proposed in [,7,15] in terms of combining channel truncation with set partitioning principles [4]. In accordance with these references, we will refer to these techniques as Decision Feedback Sequence Estimation (DFSE) and Reduced State Sequence Estimation (RSSE) [15], the latter technique being more general in the sense that complexity reduction may be obtained not only through channel truncation but also by partial representation of the residual ISI. Related work also appears in [1] [19]. As a general conclusion, it may be said that the effectiveness of per-survivor DFE techniques is due to a significant reduction in error propagation with respect to conventional techniques, which are based on tentative decisions. The aim of this paper is to show that the idea of canceling residual ISI on the basis of the particular survivor sequence, i.e., in a per-survivor fashion, is not confined to the reduction of decoding complexity, but may be extended to several situations. We view this idea as a particular application of a more general principle which applies whenever the transition metrics in the Viterbi algorithm are affected by some degrees of uncertainty that could be removed or reduced by data-aided estimation techniques. In a typical example, this uncertainty is due to imperfect knowledge of some channel parameters, such as the carrier phase, timing epoch, or the channel impulse response itself. All these cases can be dealt with by a common and unifying approach termed Per-Survivor Processing (PSP). As in the case of reduced complexity decoders, PSP is an effective alternative to the use of tentative decisions in the estimation of the unknown quantities because the negative effects of

2 error propagation are significantly reduced. Additional appealing aspects of the PSP class of algorithms in the case of channel parameter estimation are: (a) the per-survivor estimator associated with the best survivor is derived from data information which can be perceived as high-quality, zero-delay decisions, thus making PSP very suitable for fast time-varying channels; and (b) since many hypothetical data sequences are simultaneously considered in the parameter estimation process, the task of blind acquisition of unknown parameters is substantially facilitated with respect to conventional techniques (here, blind acquisition means acquisition without a training sequence ). This paper expands upon previous work reported in [20] [23]. We note that PSP-type techniques may be applied directly to various generalizations of the VA, such as those described in [13,24], where more than one survivor per state is retained and a list of globally best paths is available at each stage of the decoding process, or those described in [25], in which the VA provides reliability information along with data decisions. PSP techniques are also applicable to reduced complexity MLSE algorithms such as the M-algorithm [2], the (M,L) algorithm [27], or the T-algorithm [28]. In the interest of compactness, we do not discuss these extensions here; nonetheless, this additional flexibility of PSP techniques is worth mentioning. Quite recently, another particular application of PSP has independently appeared in the technical literature in the area of carrier phase synchronization for coded modulations [29] [32]. An application of PSP to joint data and channel estimation [33] was brought to the authors attention during a preliminary review process. The paper is organized as follows: In Section II, we describe the system model and briefly review MLSE. Section III introduces PSP-based MLSE in a general form. Section IV presents a number of possible applications of the PSP concept to joint channel parameter estimation and data decoding. Numerical results are given in Section V and conclusions are drawn in Section VI. II. SYSTEM MODEL AND REVIEW OF MLSE In this section, we briefly review MLSE for coded signal transmission over ISI channels and introduce the notation. A complex code symbol a k, belonging to an M-ary alphabet, is transmitted over a complex linear channel characterized by impulse response h(t) (this filter represents the cascade of the transmitter filter and the physical channel). The complex envelope of the received signal is 1X r(t) = a k h(t? kt ) + n(t) (1) k=?1 wherein the noise process n(t) is the baseband equivalent of white Gaussian noise with power spectrum N 0 =2, independent of the data sequence. Under the assumption that h(t) is perfectly known, the optimum receiver is composed of a filter matched to the pulse h(t) followed by a symbol rate sampler and a Viterbi decoder that searches for the path with minimum metric in the trellis diagram of a finite state machine that models the cascade of encoder and transmission channel [2,3,,7]. Unfortunately, in many practical communication systems this ideal situation is not encountered because of two fundamental reasons: (a) Suboptimal receiver structures based on a simplified state description of the channel must be adopted to combat excessive state complexity [,7,13,15]; (b) A set of channel parameters, or the impulse response h(t) itself, are unknown and must be estimated jointly with data decoding. The standard approach to suboptimal decoding in the presence of uncertainty due to these reasons is the use of data-aided parameter estimators. In order to review suboptimal MLSE receivers, let us first introduce a combined code and ISI reduced trellis diagram with a state k defined as [3,,7] k =? k ; a k?1 ; a k?2 ; :::; a k?k in which k is the encoder state at epoch k, K is the reduced channel memory (K L, with L being the true memory parameter), and the K-tuple (a k?1 ; a k?2 ; :::; a k?k ) represents the reduced channel state. This description does not include set partitioning principles also employed in RSSE in order to keep the notation simple. A generalization is straightforward. Let the transitions be denoted by k! k+1 and also let k denote a time dependent parameter vector representing possible unknown channel parameters and/or residual ISI due to state reduction. Define the transition (or branch ) metrics at epoch k as ( k! k+1 ) = F k! k+1 ; r(t); k (3) where F [] denotes the functional dependence of ( k! k+1 ) on the time continuous received signal r(t) and the parameter vector. Perfect knowledge of the parameter vector k would enable the realization of ideal MLSE, for the assumed state complexity. In practical systems, k is not known and must be estimated in order to be inserted in the computation of (3). A common approach is based on data-aided parameter estimation techniques, in which the aiding data sequence is obtained in a decisiondirected mode from tentative low-delay decisions at the Viterbi decoder output [3,4,,34,35]. Let this tentative decision on the code sequence at epoch k be denoted by â k?d?1, where d represents the reduced decoding delay. Based on this sequence of tentative decisions fâ i g?1 k?d?1 and the received signal, a dataaided parameter estimator provides the Viterbi decoder with an estimate of the unknown parameters as ˆ k = G h r(t); fâ i g k?d?1?1 with G[] denoting the functional dependence of the estimate ˆ k on the received signal and the sequence of tentative decisions. Note that a delay of d symbols is inherent in the estimate ˆ k with respect to the true parameter vector k. The standard approach to suboptimal MLSE in the presence of unknown parameters consists of using (4) in the computation of the branch metrics (3) according to i ( k! k+1 ) = F h k! k+1 ; r(t); ˆ k : (5) Denoting with Γ( k ) the survivor metrics, the usual updating step of the Viterbi algorithm then follows: For all successor i (2) (4)

3 u k n(t) a - k r(t) ENCODER j h(t) j.. - WMF - -? y( - - k ) SAMPLER - e?j(t) VITERBI DECODER - û k?d â k?d PHASE ESTIM. WMF ESTIM. TIMING EPOCH ESTIM. DISCRETE CHANNEL ESTIM. Fig. 1. Conventional MLSE with unknown channel parameters. states k+1, the accumulated metrics Γ( k+1 ) are determined by performing a minimization over the current states k Γ( k+1 ) = min Γ( k ) + ( k! k+1 ) : () k Finally, the survivors terminating in the current states are extended by incorporating the transitions that comply with (). III. PER-SURVIVOR PROCESSING As an alternative to the above classical approach to suboptimal MLSE in the presence of uncertainties, a per-survivor estimation of the unknown parameters can be implemented. In this technique, the code sequence associated to each survivor is used as the data-aiding sequence for the per-survivor estimation of the unknown parameters. A formal description can be given by defining the code sequence associated to the survivor of state k as fâ i ( k )gi=?1 k?1. Per-survivor estimates of the unknown vector k based on the data-aided estimator G[] and the code sequences associated to each surviving path can be defined as ˆ( k ) according to ˆ( k ) = G h i r(t); fâ i ( k )gi=?1 k?1 : (7) These per-survivor estimates are then used in the computation of the branch metrics (3) according to h i ( k! k+1 ) = F k! k+1 ; r(t); ˆ( k ) : (8) Decoding proceeds as in the standard Viterbi algorithm. As an example, in reduced state decoding (with perfect knowledge of the channel) the residual ISI component is an unknown scalar parameter. Obviously, perfect side information regarding this residual ISI term would allow us to perform optimum reduced state MLSE; however, in practical realizations this ISI term must be estimated. In early approaches to state complexity reduction [8], this ISI term was estimated by the feedback section of a Decision Feedback Equalizer (DFE) driven by a sequence of tentative decisions at the Viterbi decoder output. More recently, reduced state decoding algorithms such as DFSE and RSSE have been proposed, in which this residual ISI component is estimated in a per-survivor fashion. Historically, these algorithms have first applied the PSP concept. The intuitive rationale for this type of approximate realization of MLSE is straightforward: whenever the incomplete knowledge of some quantities prevents us from calculating a particular transition metric in a precise and predictable form, we use estimates of those quantities based on the data sequence associated with the survivor leading to that transition. If any particular survivor is correct (an event of high probability under normal operating conditions), the corresponding estimates are evaluated using the correct data sequence. Since at each stage of decoding we do not know which survivor is correct (or the best), we extend each survivor based on estimates obtained using its associated data sequence. Roughly speaking, the best survivor is extended using the best data sequence available (which is the sequence associated to it), regardless of our temporary ignorance as to which survivor is the best. This concept, inherent in DFSE and RSSE, can be generalized in the above sense to the approximation of MLSE algorithms in many types of uncertain environments. Assuming that the adopted data-aided estimator has the property that, in the absence of noise and for the correct data-aiding sequence, it produces correct estimates of the unknown parameters, then the resulting PSP-based approximate MLSE algorithm provides a correct estimation of the data sequence in the absence of noise. Based on the above, we note that a PSP algorithm results in an asymptotically optimum decoding algorithm for vanishing noise. IV. APPLICATIONS OF PSP Let us first introduce a more explicit expression for the transition metrics. In a classic arrangement [1], the received signal is passed through a Whitened Matched Filter (WMF), with the results that a white sampled noise sequence is obtained at the Viterbi decoder input. Although other realizations are possible [3], this model will be employed in the following. Fig. 1 shows the overall communications system employing a suboptimal MLSE receiver based on tentative decisions. In this figure, y( k ) is the time discrete input to the Viterbi decoderat epoch k, in general dependent on the parameter vector k, and D denotes the final decoding delay of the Viterbi processor. A discrete-time white-noise equivalent of the overall transmitter/channel/wmf/sampler cascade completely characterizes the system that precedes the Viterbi decoder [1]. Denoting by ff i ( k )g L i=0 the impulse response of this equivalent channel, in general dependent on the parameter vector k with L representing the channel memory, the noiseless signal components at the discrete channel output, for each state transition k! k+1,

4 n = k 1 n = k n = k + 1 fˆ ( µ k 1 ) µ n = ( 0, 0) µ k 1 fˆ µk ( ) µ n = ( 1, 0) µ k µ n = ( 0, 1) fˆ ( µk + 1 ) µ n = ( 1, 1) µ k + 1

5 B. Joint Maximum Likelihood Estimation of Channel and Data The system model is here identical to that of the previous section, except for the assumption that the transmission begins at epoch 0. Given the observation sequence fy n g k n=0, it is wellknown [37] that the joint Maximum Likelihood (ML) estimate of data sequence and discrete channel response is obtained by maximizing the likelihood function over fa n g k n=0 and ff n g L n=0 according to max fa n g k n=0 max ff n g L n=0 0 n=0 y n? LX 1 2 f i a n?i A : (1) This operation can be accomplished through a maximization over ff n g L n=0 for each possible data sequence, followed by a maximization of the result over the discrete set of all data sequences. An exhaustive search over all possible data sequences yields the desired joint ML estimate, but is not practical due to the large number of possible sequences. We here propose an approximate implementation of the above maximization problem based on PSP, in which the channel impulse response is derived by maximizing (1) over ff n g L n=0 for each survivor sequence. In this approach, the survivor metrics represent the value of the likelihood function for the associated survivor sequences, in which the maximization over the channel parameters has already been performed. Consequently, the survivor with the largest metric has associated to it the desired joint estimate of channel and sequence. The maximization of the likelihood function with respect to the channel vector ˆf( k ) for a given survivor sequence (i.e., the inner maximization in (1)) may be performed by a Recursive Least Square (RLS) algorithm [5]. At the k-th epoch, ˆf( k+1 ) is estimated by recursively minimizing ( k+1 ) = i=0 kx w k?n yn? ˆf T ( k+1 ) a( n! n+1 ) n=0 2 (17) in which the state sequence f n g k n=0 is the survivor of state k+1. The weighting factor 0 < w < 1 is introduced to limit the memory of the algorithm to allow for possibly time-varying channels. Accordingly, the resulting algorithm is the following. At the k-th step, for all possible transitions k! k+1 the errors and branch metrics are computed according to (13) and (14). One step of the Viterbi algorithm () is then performed. For the transitions that extend the survivors, the Kalman gain vectors, inverse of the correlation matrices, and channel impulse responses, respectively, are updated according to k( k+1 ) = P( k )a ( k! k+1 ) w + a T ( k! k+1 )P( k )a ( k! k+1 ) P( 1 k+1 ) = w P( k )? k( k+1 )a T ( k! k+1 )P( k ) ˆf( k+1 ) = ˆf( k ) + k( k+1 )e( k! k+1 ) : (18) C. Adaptive Whitened Matched Filter In the applications so far presented, we concentrated on the estimation of the discrete channel. In a more general situation, this task is only a part of the channel estimation problem because the discrete channel is obtained as the output of a WMF which must be estimated as well. It is well-known that the WMF is also the optimum feedforward filter in a Zero-Forcing (ZF) DFE [38,39]. A possible approximate approach to the estimation of the WMF is obtained by using the optimum feedforward section of a MSE-DFE, which, in the limit of vanishing noise, equals the ZF-DFE. For a digital implementation, the filter can be realized through a fractionally spaced transversal filter. As an example, this approach was followed in [] where the adaptive control of the WMF coefficients was based on zero-delay tentative decisions. To introduce a PSP-based adaptive version of the algorithm, we denote as r k the content at epoch k of the tapped delay line of a T =Ω-spaced transversal filter, with Ω being a small integer representing the oversampling factor (with respect to the baud rate) r k = r(kt ); r(kt? T Ω ); r(kt? 2T Ω ); : : : ; r(kt? (N? 1) T Ω ) T (19) in which r(t) is the received signal (see Fig. 1) and N is the number of taps. Denoting the coefficients of the WMF associated to the survivor of state k by the vector b( k ), the per-survivor baud-rate signals ỹ( k ) at the output of the WMFs are ỹ( k ) = r T k b( k) : (20) These signals can be used in the computation of the errors (13) and branch metrics (14). Finally, the tap vectors b( k ) are updated according to b( k+1 ) = b( k )? e( k! k+1 )r k (21) over the transitions that extend the survivors, with being a suitable constant. D. Joint TCM Decoding and Phase Synchronization We present here an application of PSP techniques to the estimation of only one channel parameter, namely, the carrier phase. ISI is assumed absent (L = 0), a typical situation of wide-band channels (such as satellite or certain microwave links), and TCM is incorporated. In this case, the Viterbi decoder operates for TCM decoding purposes only. Let the unknown scalar parameter be the demodulation phase k. The input to the Viterbi decoder is then y( k ) = y k e?j k (22) where y k is the TCM-encoded noise-corrupted signal. Associated with the state transition k! k+1 there is now a symbol subset, which we denote as A( k! k+1 ). The branch metrics

6 (10) for this special case become ( k! k+1 ) = min a k 2A( k! k+1) y k e?j k? a k 2 : (23) As shown in [4], a carrier recovery decision-directed loop may be used, which uses tentative decisions from the Viterbi decoder. An analysis of the optimal delay to be used in such tentative decisions showed that a delay of a few symbol intervals is preferable a larger delay improves the quality of the decisions but increases excessively the loop time constant. PSP is then perfectly suitable to this situation. Following [4] and [40], an iterative solution to the problem of estimating the carrier phase is obtained according to o ˆ k+1 = ˆ k + Im ny k â k?d e?j ˆ k (24) in which, tentative decisions â k?d from the Viterbi decoder are used and is a suitable constant. Alternatively, using PSP the algorithm modifies as follows: At the k-th epoch, the branch metrics are evaluated according to ( k! k+1 ) = min a k 2A( k! k+1) y k e?j ˆ ( k )? ak 2 (25) and the per-survivor phase estimates ˆ( k ) are updated for the transitions that extend the survivors according to ˆ( k+1 ) = ˆ( k ) + Im n y k a k e?j ˆ ( k ) o (2) in which the symbol a k 2 A( k! k+1 ) minimizing (25) is used. V. NUMERICAL RESULTS Some of the PSP-based suboptimal MLSE algorithms presented in the previous sections are here simulated and compared with suboptimal MLSE receivers based on tentative decisions of identical state complexity. Receivers based on tentative decisions will be termed conventional in the following. In all the simulation work, the final decoding delay D (see Fig. 1) is large enough to avoid any performance degradation due to unmerged survivors. For completeness, we have also considered conventional MLSE receivers in which the parameter estimation process is decoupled from data decision by assuming that the correct sequence is available with zero delay at the receiver for data-aided parameter estimation only. In fact, the performance of these idealized receivers represents the ultimate lower bound of any specific joint estimator of data and channel parameters. In the sequel, we will refer to these idealized cases as knowndata MLSE receivers and the corresponding lower bounds will be termed known-data bounds (an even lower bound is the performance of the known-parameter receiver). Preliminary work has shown that adaptive receivers based on PSP always attain the known-data bound on stationary channels [20,21]. Conventional adaptive receivers practically attain the Preamble Information symbols Fig. 3. Data format. Tail symbols bound only for values of d approaching the decoding depth of the Viterbi processor. This fact suggests that a significant difference between the conventional and PSP proposed receivers should be expected on time-varying channels, whereby a large delay in the parameter estimation loop cannot be tolerated. A. Tracking of a Rayleigh Fading Channel We assume here a time-varying Rayleigh fading discrete channel. The elements of the impulse response f k;i are mod- L i=0 eled as independent low-pass, zero-mean complex random processes with Rayleigh distributed amplitude and uniformly distributed phase in the interval [?; ]. Each element is generated by passing a white complex Gaussian noise through a digital second-order low-pass Butterworth filter. The cutoff frequency f D (3 db frequency) of this filter is representative of the Doppler band of the fading channel. Full and reduced state adaptive MLSE receivers using the conventional and persurvivor LMS tracking algorithms described in Section IV-A are simulated. Their performance is expressed in terms of probability of symbol error versus E s =N 0, E s being the received signal energy per information symbol averaged over the data and channel statistics. The tentative decision delay d used in conventional algorithms is selected in each considered case in order to optimize the performance. The assumed modulation is uncoded Quadrature Phase Shift Keying (QPSK). Thus, M = 4 and a k 2 f1+j; 1?j;?1+j;?1?jg (here, j denotes the imaginary unit). To reproduce a typical cellular mobile communications environment, a Time Division Multiple Access (TDMA) data frame is assumed, where each user transmits a block of data symbols preceded by a known preamble and followed by a known tail as shown in Fig. 3 [41,42]. Specifically, each data block is formed by 0 information symbols. The information rate is set to Kbits/s. At this stage, we are interested in channel tracking and thus ignore the channel acquisition aspects. We then initialize correctly the channel estimates at the beginning of each information block. For this reason the preamble sequence consists of only K known symbols, so that the correct initial state can be provided to the Viterbi processor. Specifically, the path metrics are initialized at the beginning of each information block (i.e., at the end of the preamble) by setting the metric associated with the correct state to zero and all others to a very large value. Similarly, K known tail symbols are used for terminating the Viterbi algorithm in a specific state at the end of the block. In a first example, we assume a channel memory L = 2. The standard deviations of the three complex processes (f k;0 ; f k;1 ; f k;2 ) are set at (0:407; 0:815; 0:407), respectively. We refer to this channel as channel A. Full-state MLSE can be performed in this case because the number of states of the channel model is M L = 4 2 = 1 (K = L = 2). Several values of

7 Probability of symbol error known-data PSP conv. (d=4) non-adaptive E S /N 0 (db) Probability of symbol error known-data PSP conv. (d=2) conv. (d=4) non-adaptive E S /N 0 (db) Fig. 4. Full state MLSE algorithms (1 states) for channel A with f D = 250 Hz. Fig. 5. Full state MLSE algorithms (1 states) for channel A with f D = 500 Hz. Doppler shift f D have been considered, in the range Hz which corresponds to a velocity in the range Km/h for a carrier frequency of 1800 MHz. For f D = 100 Hz, results not reported indicate that there is no appreciable difference between the conventional and PSP receivers. In fact, both exhibit a moderate degradation with respect to the known-data A 1 bound (a fraction of db). Fig. 4 shows the simulated performance for f D = 250 Hz. For this more rapidly varying channel, the PSP-based adaptive receiver shows a degradation of approximately 2:5 db with respect to the known-data bound. A delay d = 4 has been selected for the data-aiding sequence in the conventional case, by trial and error, as the one for which performance is optimum. With this value of d, the conventional receiver shows a significant degradation and presents a floor about 3 10?5. All curves are obtained using a value of the step size = 0:125. The last curve shows the performance B 1 of a non-adaptive receiver and exhibits a floor about 2 10?2. A larger Doppler shift f D = 500 Hz has been assumed in Fig. 5 in order to better investigate the tracking capability limit of the per-survivor LMS algorithm. For this Doppler shift, the PSP algorithm shows a floor about 10?4. Two values of tentative decision delay d = 2 and d = 4 are reported for the conventional algorithm because it is not clear which case exhibits a better performance for this rapidly varying channel. In any case, the conventional receiver exhibits a floor at error probability larger than 10?3. All curves are obtained with = 0:125. The performance of the non-adaptive receiver is also reported for completeness. As a second example, we consider a channel memory L = 5 and assume that the random processes modeling the elements of the discrete impulse response have all the same standard deviation equal to This channel model is similar to the profile for equalization test recommended by the European B 2 C 1 C 2 C 3 C 4

8 Probability of symbol error kn.-data DFSE PSP DFSE conv. DFSE (d=4) kn.-data RSSE PSP RSSE conv. RSSE (d=4) E S /N 0 (db) Fig. 7. Reduced state MLSE algorithms (4 states) for channel B with f D = 250 Hz. Solid line: DFSE. Dotted line: RSSE. Probability of symbol error kn.-phase kn.-data PSP conv. (d=1) E S /N 0 (db) Fig. 9. Joint TCM decoding and phase tracking for coded 8PSK (4 states). Probability of symbol error kn.-data DFSE PSP DFSE conv. DFSE (d=4) kn.-data RSSE PSP RSSE conv. RSSE (d=4) E S /N 0 (db) Fig. 8. Reduced state MLSE algorithms (4 states) for channel B with f D = 500 Hz. Solid line: DFSE. Dotted line: RSSE. while these values are increased to 4 in the case of the RSSE receivers. Fig. 7 shows the performance of the described systems for f D = 250 Hz and = 0:125. From the known-data bounds (kn.-data, in the figures), we note that a theoretical gain of approximately 1 db is potentially exhibited by the RSSE receiver. However, this gain is actually shown by both suboptimal receivers only in a limited range of values of E s =N 0. The PSP adaptive receiver exhibits a loss of approximately 3 db with respect to the known-data bound. On the other hand, the con- ventional adaptive receiver has a floor about 5 10?5. This simulation has been repeated for f D = 500 Hz and the results are shown in Fig. 8. Basically this figure confirms previous results. It indicates that a clear performance improvement can be obtained by per-survivor LMS adaptation as compared to the conventional implementation based on tentative decisions. B. Joint TCM Decoding and Phase Synchronization In this application, ISI is assumed absent and phase recovery is considered. The time-varying phase rotation introduced by the channel is modeled as a Wiener random process by increasing the channel phase, at each symbol interval, by an independent Gaussian increment with zero mean and variance 0:00125 rad 2. The algorithm described in Section IV-D is used, in its knownparameter, known-data, PSP, and conventional versions, assuming 4-state TCM encoded 8-point Phase Shift Keying (8-PSK) as a signaling format, and an updating step-size = 0:2. Fig. 9 shows the performance of the various decoders in terms of error probability versus E s =N 0, under the assumption of correct initialization of the estimated phase. A tentative decision delay d = 1 was found to optimize the performance of the conventional algorithm. From this figure, it is evident that in this time-varying environment the PSP approach is superior. In fact, a loss of approximately 3 db is exhibited by the conventional algorithm with respect to the PSP algorithm, while the latter shows a loss of about 1 db over the known-data bound. The performance with known phase is also shown for comparison. C. Blind Acquisition of Unknown Parameters The superiority of the PSP approach in the acquisition of the unknown parameters without training is here demonstrated by two simple examples. In a first example, we have con-

9 TABLE I AVERAGE ACQUISITION TIME IN SYMBOL INTERVALS FOR AN LMS ALGORITHM ON A STATIONARY CHANNEL. CONVENTIONAL E s =N 0 (db) KNOWN-DATA PSP d = 0; 2; 4 d = 10 d = 20 d = 30 d = (0%) 5441(19%) (0%) 47028(32%) TABLE II AVERAGE ACQUISITION TIME IN SYMBOL INTERVALS FOR JOINT TCM DECODING AND TRACKING OF A TIME-VARYING PHASE. E s =N 0 = 15 db. 4 3 CONVENTIONAL KNOWN-DATA PSP d = 0 d = 1 d = 2 d = sidered 1-point Quadrature Amplitude Modulation (1-QAM) transmission over a stationary discrete channel whose elements (f 0 ; f 1 ; f 2 ; f 3 ; f 4 ) are set at (0:218; 0:43; 0:54; 0:43; 0:218), respectively. DFSE with K = 2 is performed (25 states). The channel estimates are initialized for any 0 with ˆf( 0 ) = (0; 0; 1; 0; 0) T. We define the acquisition time as the number of signaling intervals that are required for the mean square difference between the estimated and true impulse responses to fall within a specified threshold equal to 0:0015. Table I shows the measured acquisition time for the known-data, PSP, and conventional versions of the algorithm. This measurement is relative to E s =N 0 values of 30 and 35 db, = 0:01, and was performed by averaging over 100 independent runs. In this table, a percentage indicates, wherever necessary, the number of runs in which acquisition took place. It is clear that the acquisition time of the PSP receiver is about one-tenth that of the conventional receiver with a tentative decision delay d 20. In a second example, the acquisition of the phase synchronizers described in Section IV-D is also considered. Table II shows the measurements of the acquisition time for the various algorithms averaged over 100 independent runs. In this test, the acquisition time is defined as the number of signaling intervals that are necessary for the difference between the estimated phase and the true channel phase to fall within a specified value. Specifically, this threshold is set to the tracking range of the S-curve of the estimator, namely =8 [4]. This measurement is relative to an E s =N 0 value of 15 db, the same channel phase model and modulation format described in Section V-B, and = 0:1. An initial offset of 0 degrees between true and estimated phase is assumed. From the table, it is readily seen that the acquisition time of the PSP decoder is halved with respect to the conventional decoder independently of the used value of d. An example of typical phase trajectories of the various algorithms during acquisition is shown in Fig. 10. D. Complexity Evaluation We present here a comparison between the complexity of digital implementations of the PSP-based adaptive LMS algo- phase (radians) 2 true phase 1 kn.-data 0-1 Fig. 10. tracking. PSP conv. (d=2) conv. (d=1) conv. (d=0) symbol intervals Phase acquisition trajectories for joint TCM decoding and phase rithm described and simulated in the previous sections and the corresponding conventional realization. A reduced state DFSE algorithm for uncoded transmission is considered, characterized by the parameters M, K, and L previously introduced. Two types of binary representation of the various quantities are assumed. Specifically, we assume that m bits per real and imaginary component are required for storing a complex data symbol a k and b bits per real and imaginary component are necessary for storing a sample of the received signal and an element of the discrete channel response. The required survivor memory in the Viterbi algorithm is the same for both the PSP-based and the conventional algorithms. Usually the decoding depth D is larger than the channel memory L by a few units typically 2L D 5L. In addition to the survivor metrics, storage of the channel estimates is necessary. In this respect, PSP and conventional implementations differ significantly. A worst case situation is obtained by selecting the minimum value D = 2L (for this value, the storage requirement for survivor memory is minimum as compared to that for channel estimates). A straightforward computation of the total number of storage bits required in the PSP and conventional implementations allows us to express a factor representative of the total memory increase of a PSP over conventional implementations. For this worst case, this factor is = 4M K Lm + 2M K (L + 1)b 4M K Lm + 2(L + 1)b : (27) Neglecting the amount of memory required in the conven-

10 TABLE III NUMBER OF REAL OPERATIONS. function operation CONVENTIONAL PSP Metric b addition M K (4M + 4L + 2) M K (M + 4L) Update bm multipl. 4M K (L + 1) 4M K (L + M ) bb multipl. 2M K+1 2M K+1 Channel b addition 8(L + 1) 4M K (L + 1) Estimate bm multipl. 8(L + 1) 4M K (L + 1) Update bb multipl. 2 2M K TABLE IV NUMBER OF REAL OPERATIONS FOR M = 4 AND K = L = 2 (FIRST EXAMPLE IN SECTION V-A). function operation CONVENTIONAL PSP Metric b addition Update bm multipl bb multipl Channel b addition Estimate bm multipl Update bb multipl tional implementation for the channel estimates, can be upper bounded as (L + 1)b < 2Lm + 1 : (28) We now briefly address the computational requirements for a digital implementation by counting the number of necessary operations. For this purpose, we consider 3 types of real operations: additions of b-bit numbers, multiplications of a b-bit number by an m-bit number, and multiplications of a b-bit number by a b-bit number. These operations will termed b addition, b m multiplication, and b b multiplication, respectively. The distinction between these two types of multiplications is of interest when m << b because significant saving can be obtained by separate hardware implementations. We performed a detailed computation of the number of these real operations required at each step of Viterbi processing for both survivor metric and channel estimate update. In this computation, we took into account that all transition metrics stemming from each particular current state have common terms that need only be calculated once. Similarly, in the conventional algorithm all transition metrics into each successor state have common terms that must be calculated once. The results are listed in Table III, where the metric and channel update functions are separately taken into account. In order to provide numerical examples, we have specialized these results to the applications in Section V-A. In the first example, the parameters are M = 4 and L = K = 2 (1-state Viterbi processor). Table IV lists the number of real operations for this case. These results can be summarized by providing three factors representing the increase in the number of operations exhibited by PSP over a conventional implementation. Table V provides these factors for the above case and also for the second example in Section V-A, characterized by M = 4, K = 3, and L = 5. Finally, we remark that m = 1 in these examples. We then observe that each b 1 multiplication is a very simple operation, roughly equivalent to a change of sign. Discarding these b 1 multiplications, we conclude that in both examples the PSP implementation exhibits an overall increase in the number of b additions by a factor 1: 1:75 and an increase in the number of b b multiplications of approximately 1:25. VI. CONCLUSIONS We have introduced a class of algorithms for the approximation of MLSE based on the principle of performing signal TABLE V COMPLEXITY INCREASE FACTORS FOR A PSP IMPLEMENTATION. operation M = 4, K = L = 2 M = 4, K = 3, L = 5 b addition 1: 1:75 bm multipl. 2:7 2:42 bb multipl. 1:23 1:25 processing operations, necessary for the estimation of unknown parameters, in a per-survivor fashion. Two main overlapping areas of application of PSP may be identified. The first, related to complexity reduction of an MLSE receiver, led several authors to the introduction of per-survivor DFE and materialized in DFSE and RSSE reduced-state receivers. The second, directly related to data decoding in the presence of unknown channel parameters, has been specifically addressed in this paper. 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COM-19, pp , June [41] GSM Recommendations No DCS (version 2.0.0), Physical layer on the radio path: general description, European Telecommunications Standards Institute, Jan [42] GSM Recommendations No DCS (version 3.0.0), Radio transmission and reception, European Telecommunications Standards Institute, Jan Riccardo Raheli (M 87) was born in Alezio (Lecce), Italy, in He received the Dr. Ing. degree (Laurea) from the University of Pisa, Italy, in 1983, the Master of Science degree from the University of Massachusetts at Amherst, U.S.A., in 198, and the Doctoral degree (Perfezionamento) from the Scuola Superiore di Studi Universitari e di Perfezionamento, Pisa, Italy, in All degrees are in electrical engineering. From 198 to 1988 he was with the Radio System Division of Siemens Telecomunicazioni, Milan, Italy. From 1988 to 1991, he was a faculty member in the Scuola Superiore di Studi Universitari e di Perfezionamento S. Anna, Pisa, Italy. In 1990, he was a Visiting Assistant Professor at the University of Southern California, Los Angeles, U.S.A.. Since 1991, he has been a faculty member in the University of Parma, Italy. His current research interests include digital transmission theory, sequence detection techniques, signal processing and adaptive algorithms for communications. Andreas Polydoros (M 78 SM 92 F 95) was born in Athens, Greece, in He was educated at the National Technical University of Athens, Greece (Diploma in EE, 1977), State University of New York at Buffalo (MSEE, 1979) and the University of Southern California (Ph.D., EE, 1982). He has been a faculty member in the Electrical Engineering Department Systems and the Communication Sciences Institute at USC since 1982, becoming a Professor in His general area of scientific interest is statistical communication theory with applications to spread-spectrum systems, signal detection and classification, data demodulation in uncertain environments, and multi-user radio networks. He has over fifteen years of teaching, research and extensive consulting experience on these topics, both for the government and industry. Prof. Polydoros is the recipient of a 198 NSF Presidential Young Investigator Award. He has served as the Associate Editor for Communications of the IEEE Transactions on Information Theory ( ), the Guest Editor of the July 1993 Special Issue on Digital Signal Processing in Communications for Digital Signal Processing: A Review Journal and a designated Area Editor for the International Journal on Wireless Personal Communications. Ching-Kae Tzou (S 92 M 94) was born in Hsinchu, Taiwan on April 30, 194. He received the B.S.E.E and M.S.E.E degrees from the National Chiao-Tung University, Taiwan, in 1984 and 198, and Ph.D. degree from the University of Southern California, U.S.A., in From 198 to 1989 he worked as an assistant scientist in Chung-Shan Institute of Science and Technology, Taiwan. He is now a Design Engineer with the Computer and Communication Research Laboratory at the Industrial Technology Research Institute, Taiwan. His interests include signal processing, channel coding and wireless communication. Dr. Tzou is a member of Phi-Tau-Phi scholastic society.