1 Maximum ikeihood decoding of treis codes in fading channes with no receiver CSI is a poynomia-compexity probem Chun-Hao Hsu and Achieas Anastasopouos Eectrica Engineering and Computer Science Department University of Michigan, Ann Arbor, MI I. INTRODUCTION The probem of optima decoding of a treis coded sequence transmitted over a frequency non-seective, time-seective fading channe is considered in this paper. It is a we-known fact that when the channe state information (CSI) is known to the receiver, the receiver may use Viterbi s agorithm (VA) to nd the maximum a posteriori probabiity sequence detection (MAPSqD) soution with inear compexity in sequence ength, N. However, when CSI is not avaiabe at the receiver, the MAPSqD soution cannot be obtained using such a simpe dynamic programming technique, due to memory imposed on the observation by the channe. One approach for soving the probem is to transmit reguary spaced piot symbos. The receiver can estimate the channe using the piots and then use VA to decode the sequence based on the estimated CSI. Athough this might be a desirabe approach for high SNR appications, the unreiabe estimated CSI provided by piots may substantiay deteriorate the performance when the operating SNR is ow, e.g., when high-performance codes are used. In this case, joint sequence decoding and channe estimation (i.e., true MAPSqD in the presence of unknown CSI) appears to be the desirabe poicy. There is an extensive iterature on approximate agorithms for soving the joint decoding/estimation probem. The expectation-maximization (EM) agorithm [1, 2] performs a two-step statistica iteration between channe-conditioned sequence decoding and data-conditioned channe estimation. A famiy of agorithms can be constructed by viewing this probem as a hypothesis testing probem with each hypothesis (sequence) being a path in a tree of depth N. Since testing a hypotheses amounts to exponentia compexity, a tree-pruning agorithm, such as the T-agorithm [], the M-agorithm [4], or the per-survivor processing (PSP) [5] agorithm can be empoyed to trade off compexity for performance. In a these works, the underying assumption was that the optima (exact) MAPSqD soution can ony be found with an exponentia compexity in the sequence ength N due to the exponentia growth of the sequence tree. It is our intention to prove that the exact MAPSqD soution can be obtained with ony poynomia compexity in the sequence ength, N. The authors have addressed and soved this probem in the case of uncoded
2 sequences for a cass of channe modes in [6 8]. The basic idea behind this work is that athough the sequence tree (and thus the number of hypotheses) grows exponentiay in N, there is ony a certain number of sequences that are potentia candidates for the MAPSqD soution. This suf cient set of sequences is not known a-priori, but once the noisy observation is obtained at the receiver, there is a poynomia-compexity agorithm to obtain it [6 8]. This agorithm is derived by de ning a new kind of decision regions that partition the channe parameter space, as opposed to the traditionay de ned decision regions that partition the observation space. By studying the structure of these new decision regions the authors showed that their number grows ony poynomiay with N and that there is a poynomia compexity agorithm that constructs them. Unfortunatey, a arguments used in [6 8] rey heaviy on the assumption of uncoded sequence. In this paper, in order to sove the treis coded MAPSqD probem, we adopt the concept of decision regions de ned in the parameter space as in [6 8]. Contrary to the previous works, however, we de ne a set of suf cient survivor sequences and study their evoution in time. In particuar, we show that this set can be updated in a forward recursive fashion and that the resuting set grows ony poynomiay, thus estabishing the poynomia-compexity resut for the coded case. The same ideas can be used to de ne both forward and backward suf cient survivor sets. This essentiay means that exact symbo-by-symbo soft decisions (more speci cay, the messages corresponding to the minsum agorithm [9]) can aso by generated with poynomia compexity. Appications that can potentiay bene t from this deveopment incude seriay concatenated convoutiona codes through at-fading channes, where now the entire system consisting of the inner treis code and the channe has an exact soft inverse [10] [11, p. 85] that is computationay feasibe. II. SYSTEM MODEL AND PROBLEM FORMULATION Consider the transmission of a sequence of information symbos a = [a 1, a 2,..., a N ] T, with a k A {0, 1,..., K 1}. The sequence is encoded by a nite state machine (FSM) de ned by its state s k S {1,..., I} at time k. For a given state s k 1 and a current input a k, de ne e k (s k 1, a k ) E {1,..., IK} to be a treis edge. The FSM is de ned by the next-state function ns : E S s k = ns(e k ), initia state s 0 is known (1) and the output function out : E O {0, 1,..., M 1}, such that the transmitted M-ary phase shift keying (M-PSK) signa is y k = E s e j 2π M out(ek), (2)
with E s being the symbo energy. It wi aso be usefu to de ne the previous-state function ps : E S, and the input function in : E A as foows s k 1 =ps(e k ) () a k =in(e k ). (4) The sequence y = [y 1, y 2,..., y N ] T is transmitted through a frequency-non-seective/time-seective fading channe. Assuming that the channe remains constant for the entire sequence transmission 1, the observation mode can be expressed as z = cy + n (5) where z = [z 1, z 2,..., z N ] T is the received signa, c CN (0, 1) is a compex Gaussian random variabe, n = [n 1, n 2,..., n N ] T is zero-mean circuary symmetric compex additive white Gaussian noise with covariance matrix K N = N 0 I N. When no CSI is avaiabe at the receiver, i.e., when the reaization of c is unknown, the MAPSqD soution to this probem is â MAP SqD = arg max p(z a)p(a) = arg max a A N a A N { n p(a) + } 1 N 0 (N 0 + NE s ) zh y 2 It is fairy we known fact (see for instance [12, p. ]) that, due to the inear and Gaussian nature of the observation mode, this probem can be expressed in a doube maximization form as } â MAP SqD = arg max max {n p(a) 1N0 z cy 2 c 2. (7) a A N c C Since there is an one-to-one correspondence between e and a, nding â MAP SqD is equivaent to nding ê MAP SqD as foows where ê MAP SqD = arg max max e ẼN c C (6) N L k (e k, c) = L N (e, c) (8) k=1 L k (e k, c) n p(in(e k )) 1 N 0 z k c E s e j 2π M out(e k) 2 c 2 N, (9) 1 The poynomia compexity resut that we wi estabish is vaid even if the channe is time varying, as ong as the degrees of freedom in the channe, i.e., the non-zero eigenvaues of the channe covariance matrix, do not increase ineary with N.
4 and L k (e k, c) k L i (e i, c) (10) is the set of vaid paths up to time k. Ẽ k {e k : ns(e i ) = ps(e i+1 ), i = 1, 2,..., k 1} (11) III. SUFFICIENT SURVIVOR MATRICES For a given c, if we de ne the survivor that ends in state i at time k obtained by the VA as we have the foowing emma. Lemma 1: ˆV k (i c) = arg max e k Ẽk :ns(e k )=i L k (e k, c), i S (12) ê MAP SqD = V N (i c) for some i S and c C (1) Proof: See Appendix I. Therefore, if we de ne ˆV k (c) [ ˆV k (1 c), ˆV k (2 c),..., ˆV k (I c)] T to be the survivor matrix consisting of a survivors that end in different states with parameter c at time k and coect a such survivor matrices in a set ˆD k { ˆV k (c) : c C} (14) then instead of searching ê MAP SqD in Ẽ N, we need ony search through a rows of a survivor matrices in a potentiay smaer suf cient set ˆD N. However, since C is an in nite set, constructing ˆD N by searching through a c C wi require in nite compexity. We make the foowing observation: it is sensibe to expect that the function ˆV k (c) remains constant for a range of c s. More rigorousy, we can partition C in such a way that for a c s in each set of the partition, ˆVk (c) is the same. This eads us to the de nition of the parameter-space decision regions de ned for any vaid survivor matrix V k D k, where T k (V k ) {c C : ˆV k (c) = V k } (15) D k {[e (1)k, e (2)k,..., e (I)k ] T : e (i)k Ẽ k, ps(e (i) 1 ) = s o i S, and e (i) e (j) ns(e (i) ) ns(e (j) ) i, j S, 1 k} (16)
5 is the set of a vaid survivor matrices at time k. The constraints in (16) impy that once survivors merge, they have to stay merged for their entire past. With the introduction of parameter-space decision regions it is now cear that the suf cient set can be constructed as ˆD k ={V k D k : T k (V k ) φ}. (17) In the next section we wi show that ˆDN and T N (V N ), V N ˆD N can be generated with poynomia compexity, and furthermore, that the size of the resuting suf cient set ˆD N is aso poynomia in N. IV. RECURSIVE CONSTRUCTION OF ˆD N AND T N (V N ) De ne the set of a possibe extensions of V k as ext(v k ) {W k+1 D k+1 : W k+1 (i, k + 1) = e W k+1 (i) = [V k (ps(e)), e]} (18) where W k (i, j) is the j-th eement of the i-th survivor in W k. Aso de ne the fowoing set of channe parameters P k (V k, e) {c C : L k+1 ([V k (ps(e)), e], c) = max L k+1 ([V k (ps(e )), e ], c)}. (19) e E:ns(e )=ns(e) Observe that the set P k (V k, e) is a convex poytope since its boundaries are straight ines in C. It shoud be cear from this de nition that for any c P k (V k, e), and if the survivor matrix at time k is V k the VA (conditioned on c) wi choose [V k (ps(e)), e] as the survivor extension. Then as a consequence of the VA, we have the foowing emma. Lemma 2: T k+1 (V k+1 ) = Proof: See Appendix II. W k :V k+1 ext(w k ) { T k (W k ) i S P k (W k, V k+1 (i, k + 1)) Lemma 2 essentiay suggests a recursive agorithm for constructing T k+1 (V k+1 ). In addition, the set can be easiy obtained by coecting the V k+1 s with nonempty T k+1 (V k+1 ). However, since the sets P k (W k, e) need to be generated for a W k ˆD k, the size of ˆD k+1 can grow (in the worst case) as β ˆD k for some β > 1, and thus the size of the suf cient set ˆD N wi be exponentia in N. To overcome this probem, we modify the above agorithm by observing that severa survivor matrices W k resut in the same set P k (W k, e). In particuar, if W k and U k are such that W k (i, j) U k (i, j) W k (i, j) = W k (, j) and U k (i, j) = U k (, j) i, S, (21) } (20) ˆD k+1
6 i.e., if they ony differ in positions at which a the survivors merge together, then P k (W k, e) = P k (U k, e), e E. Therefore, we can partition ˆD k into groups G k 1, G k 2,..., G k α k such that any two survivor matrices that satisfy (21) beong to the same group. Fig. 1 shows an exampe of this partitioning. With this modi cation, G G4 G5 G6 G 1 G 2 G 7 G8 G9 G10 Fig. 1. An exampe of groups for I = K = M = 2 and k =. Athough in this exampe ony G 1 and G 2 have more than one eements, for arger k the sets G k i have a arge number of eements. instead of constructing P k (W k, e) for a W k ˆD k, we can construct P k (W k, e) for ony one W k G k i for each group i = 1, 2,..., α k, thus reducing compexity. This eads to the foowing modi ed agorithm: 1) Construct T 1 (V 1 ) for a V 1 D 1. If T 1 (V 1 ) φ, put V 1 into ˆD 1. 2) Given ˆD k and T k (V k ) for a V k ˆD k, construct groups G k 1, G k 2,..., G k α k. ) For each i = 1, 2,..., α k, do the foowing. a) Choose an arbitrary matrix W k G k i, and construct Pe k (W k ), e E. b) For a V k+1 = ext(u k ), U k G k i, if T k (U k ) i S P k V k+1 (i,k+1) (Wk ) φ, i) Union this set with T k+1 (V k+1 ), where T k+1 (V k+1 ) is initiay set to be φ. (Note that T k+1 (V k+1 ) is exacty what was de ned in Lemma 2) ii) Put V k+1 into ˆD k+1. 4) Iterativey do steps 2) and ) unti we get ˆD N and T N (V N ). At this stage, the proof of the poynomia compexity of the agorithm is ony avaiabe for the case of I = 2. Athough this is the simpest case, it has important appications, such as a differentiay encoded BPSK system. The authors are currenty working on the generaization to the case I > 2. Lemma : The number of groups α k is at most poynomia in k. Proof: See Appendix III.
7 We summarize the main resut of this paper in the foowing theorem. Theorem 1: For I = 2, the agorithm stated in Lemma 2 and modi ed using the groups de ned in (21) can nd the exact ê MAP SqD soution with worst-case poynomia compexity in N for any signa-to-noise ratio. Proof: A sketch of the proof is now presented. The proof hinges on Lemma, which impies that a sets T k+1 (V k+1 ) are poytopes de ned by a number of equations which is poynomia in N. Since each equation represents a ine in the compex pane, the probem becomes equivaent to nding a partitions generated by a poynomia number of ines in C. This probem has been studied before in [1] where it has been shown that there exists a poynomia-compexity agorithm that can nd the at-most-poynomia number of such poytopes. The nature of the agorithm is not important here; its existence competes the proof. V. DISCUSSION AND CONCLUSION In this paper, the probem of optima MAPSqD of a treis coded data sequence transmitted over a frequencynonseective, time-seective channe is considered. The case when the receiver does not have CSI is addressed. It is shown that, contrary to the traditiona beief, the exact soution can be obtained with poynomia compexity in the sequence ength (the proof for a two-state treis is ony presented). The nove approach we used here to estabish these resuts is to view this detection probem from the channe parameter space as opposed to the observation space and de ne appropriate decision regions. We woud ike to point out that with a sma modi cation one can aso sove the more interesting probem of obtaining symbo-by-symbo soft decisions with poynomia compexity. In particuar, the metric { } 1 { } SbS k (a) = max n{p(z a)p(a)} = max n p(a) + a:a k =a a:a k =a N 0 (N 0 + NE s ) zh y 2 = max max L N (e, c), e:in(e k )=a c which is exacty the metric impied by the min-sum agorithm can be obtained using the foowing idea. Reca that in the proposed agorithm, we have the suf cient sets ˆD k of forward survivor matrices for a time instants k. Simiary, we can construct the suf cient sets ˆB k of backward survivor matrices at a time instants. Therefore for any given edge e k we can aways nd its best past and future evoution and the corresponding soft metric by comparing a the possibe combinations of a the past and future survivors in the forward and backward ˆB k+1 suf cient sets respectivey. Currenty we are working on generaizing Theorem 1 for I > 2. For this we might need a more precise description of the evoution of groups in order to nd α k exacty, or even a different grouping technique. (22) ˆD k 1
8 Another topic of interest is the design of practica approximate agorithms for performing MAPSqD or generating symbo-by-symbo soft decisions. The authors have derived such agorithms for the uncoded case in [6 8] by introducing a suboptima, but sensibe, partitioning of the parameter space. A simiar approach can be foowed for the coded case, thus resuting in performance compexity tradeoffs that are more attractive than the ones obtained by the existing approximate agorithms. De ne to be the best survivor given c, we have APPENDIX I PROOF OF LEMMA 1 î(c) arg max j S L( ˆV N (j c), c) (2) De ne further Since ĉ(e) C e Ẽ N, (8) becomes ˆV N (î(c) c) = arg max e k Ẽk L(e, c) (24) ĉ(e) arg max c C LN (e, c) (25) ê MAP SqD = arg max e ẼN L N (e, ĉ(e)) = ˆV N (î(c) c) for some c C (26) which proves emma 1. APPENDIX II PROOF OF LEMMA 2 By the Viterbi s agorithm we know ˆV k+1 (c) ext( ˆV k (c)) and (27) ˆV k+1 (i, k + 1 c) = e ˆV k+1 (i c) = ( ˆV k (ps(e) c), e) L k+1 (( ˆV k (ps(e) c), e), c) = max e E:ns(e )=i Lk+1 (( ˆV k (ps(e ) c), e ), c), i S (28)
9 Therefore T k+1 (V k+1 ) {c C : ˆV k+1 (c) = V k+1 } ={c C : V k+1 ext(w k ), W k = ˆV k (c), V k+1 (i, k + 1) = e = which proves emma 2. L k+1 ((W k (ps(e)), e), c) = W k :V k+1 ext(w k ) max L k+1 ((W k (ps(e )), e ), c), i S} e E:ns(e )=i T k (W k ) {c C : V k+1 (i, k + 1) = e L k+1 ((W k (ps(e)), e), c) = max e E:ns(e )=i Lk+1 ((W k (ps(e )), e ), c), i S} { = T k (W k ) } P k V k+1 (i,k+1) (Wk ) i S W k :V k+1 ext(w k ) Suppose I = 2. APPENDIX III PROOF OF LEMMA Let i j denote a transition from state i to state j, E i j E be the set of edges going from state i to state j and ê i j (c) arg max L (e, c) (29) e E i j be the best edge going from state i to state j for a given c at time. For any give i, j and, consider the partition J i j of C created by the foowing set partitioning ines L (e, c) = L (f, c), e f, e, f E i j (0) Since each ine in (0) de nes a boundary of the two possibe resuts of a pairwise comparison in set E i j, ê i j (c), as a resut of a possibe pairwise comparisons in set E i j, is a constant for a c in the same eement of the partition J i j. Now consider the ner partition Ĵ k created by a the ines in (0) i, j S, 1 k. Then it foows directy from the above discussion that ê i j (c) is a constant for a c in the same eement of the partition Ĵ k, i, j S, 1 k. Let another set of partitioning ines L (e 1 2, c) + L (e 2 1, c) = L (e 1 1, c) + L (e 2 2, c) e i j E i j, i, j S (1)
10 de ne the partition J. Then for a c and c in the same eement of J, k, ˆV k (c) and ˆV k (c ), can not be such that ˆV k (1, c) = e 1 2 E 1 2, ˆV k (2, c) = e 2 1 E 2 1 and (2) ˆV k (1, c ) = e 1 1 E 1 1, ˆV k (2, c ) = e 2 2 E 2 2 () since and ˆV k (1, c) = e 1 2 E 1 2, ˆV k (2, c) = e 2 1 E 2 1 1 L i ( ˆV 1 k (1, i c), c) + L (e 1 2, c) L i ( ˆV k (2, i c), c) + L (e 2 2, c), 1 L i ( ˆV 1 k (2, i c), c) + L (e 2 1, c) L i ( ˆV k (1, i c), c) + L (e 1 1, c) L (e 1 2, c) + L (e 2 1, c) = L (e 2 2, c) + L (e 1 1, c) (4) ˆV k (1, c ) = e 1 1 E 1 1, ˆV k (2, c ) = e 2 2 E 2 2 1 L i ( ˆV 1 k (1, i c ), c ) + L (e 1 1, c ) L i ( ˆV k (2, i c ), c ) + L (e 2 1, c ), 1 L i ( ˆV 1 k (2, i c ), c ) + L (e 2 2, c ) L i ( ˆV k (1, i c ), c ) + L (e 1 2, c ) L (e 1 1, c ) + L (e 2 2, c ) = L (e 2 1, c ) + L (e 1 2, c ) (5) contradict with the fact that c and c is in the same eement in J. Therefore for a c in the same eement of J the two-to-two transition at time k for a ˆV k (c) is uniquey de ned. If we construct a partition J k by intersecting a the ines in (1) 1 k and a the ines partitioning Ĵ k, then for a c in the same eement of J k, we have the foowing two facts. 1) ê i j (c) is a constant i, j {1, 2}, 1 k. 2) The two-to-two transitions (1 2, 2 1 or 1 1, 2 2) of survivor matrices ˆV k (c) at a time instants, 1 k is uniquey de ned. As de ned in section IV, a group at time k is uniquey de ned by the foowing information. and and
11 1) The merging time m < k before which a the survivors merge together into one tai. 2) The state m s S = {1, 2} to which the ony tai connects at the merging time m. ) The one-to-two transition (triviay m s 1, m s 2) of the survivor matrices at time m + 1. 4) The two-to-two transitions (1 2, 2 1 or 1 1, 2 2) of survivor matrices for a time instants, m + 2 k. 5) The edges corresponding to a the transitions in ) and 4). Since for a c in the same eement of J k, information 4 and 5 are uniquey de ned, the number of different groups for a ˆV k (c) s is at most equa to the number of a possibe combinations of information 1 and 2 (since is a trivia information), which is 2k + 1. Therefore the number of groups α k at time k is at most (2k + 1) J k. To enumerate the size of J k, rst observe that J i j is a partition created by K(K 1) ines i, j {1, 2} 2 1 k. Therefore Ĵ k is a partition created by 2 2 k K(K 1) ines. Moreover, since J 2 is created by at most ( K 2 )4 ines 1 k, we concude that J k is a partition created by at most (2 2 k K(K 1) 2 )(( K 2 )4 k) = k 2 K 2 (K 1) 8 ines, which is poynomia in k. This competes the proof of Lemma. REFERENCES [1] A. P. Dempster, N. M. Laird, and D. B. Rubin, Maximum ikeihood from incompete data via the EM agorithm, J. Roy. Stat. Soc., vo. 9, no. 1, pp. 1 8, 1977. [2] C. N. Georghiades and J. C. Han, Sequence estimation in the presence of random parameters via the EM agorithm, IEEE Trans. Communications, vo. 45, no., pp. 00 08, Mar. 1997. [] S. Simmons, Breadth- rst treis decoding with adaptive effort, IEEE Trans. Communications, vo. 8, pp. 12, 1990. [4] J. B. Anderson and S. Mohan, Sequentia coding agorithms: A survey cost anaysis, IEEE Trans. Communications, vo. 2, pp. 169 176, Feb. 1984. [5] R. Rahei, A. Poydoros, and C.-K. Tzou, Per-survivor processing: A genera approach to MLSE in uncertain environments, IEEE Trans. Communications, vo. 4, no. 2//4, pp. 54 64, Feb/Mar/Apr. 1995. [6] I. Motedayen and A. Anastasopouos, Poynomia-compexity noncoherent symbo-by-symbo detection with appication to adaptive iterative decoding of turbo-ike codes, IEEE Trans. Communications, vo. 51, no. 2, pp. 197 207, Feb. 200. [7] I. Motedayen and A. Anastasopouos, Poynomia compexity ML sequence and symbo-by-symbo detection in fading channes, in Proc. Internationa Conf. Communications, Anchorage, Aaska, May 200, pp. 2718 2722. [8] I. Motedayen and A. Anastasopouos, Optima joint detection/estimation in fading channes with poynomia compexity, IEEE Trans. Information Theory, Sept. 200, (Submitted; can be downoaded from http://www-persona.engin.umich.edu/ anastas/preprints.htm). [9] B. J. Frey, Graphica modes for machine earning and digita communications, MIT Press, Cambridge, MA, 1998. [10] S. Benedetto, D. Divsaar, G. Montorsi, and F. Poara, Soft-input soft-output modues for the construction and distributed iterative decoding of code networks, European Trans. Teecommun., vo. 9, no. 2, pp. 155 172, March/Apri 1998. [11] K. M. Chugg, A. Anastasopouos, and X. Chen, Iterative Detection: Adaptivity, Compexity Reduction, and Appications, Kuwer Academic Pubishers, 2001.
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