720 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 8, AUGUST 2005 On MAP Symbol Detection for ISI Channels Using the Ungerboeck Observation Model Giulio Colavolpe and Alan Barbieri Abstract—In this letter, the well-known problem of a trans- mission over an additive white Gaussian noise channel affected by known intersymbol interference is considered. We show that the maximum a posteriori (MAP) symbol detection strategy, usually implemented by using the Forney observation model, can be equivalently implemented based on the samples at the output of a filter matched to the received pulse, i.e., based on the Ungerboeck observation model. Although interesting from a conceptual viewpoint, the derived algorithm has a practical relevance in turbo equalization schemes for partial response signalling, where the implementation of a whitening filter can be avoided. Index Terms— Factor graphs (FG), intersymbol interference (ISI), sum-product algorithm (SPA). I. I NTRODUCTION W HEN the maximum a posteriori (MAP) sequence de- tection strategy is considered for transmissions over known intersymbol interference (ISI) channels, two equivalent approaches for linear modulations can be adopted. The first one is the so-called Ungerboeck approach [1]. In this case, the branch metrics of the Viterbi algorithm (VA) implementing this strategy are based on the samples at the output of a filter matched to the received pulse. The second approach is the Forney approach [2] and it is based on the output of a whitened matched filter (WMF). Although the relevant VA branch metrics are different, the number of trellis states is the same for both equivalent approaches. When the MAP symbol detection strategy is adopted, as an example to perform turbo equalization [3]–[5], these two observation models do not seem to be equivalent. All the well- known materialization of the MAP symbol detection strategy, such as for example the BCJR algorithm [6] (see also [7], [8]) have been obtained by using a probabilistic derivation based on the chain rule and the properties of a Markov source observed through a discrete memoryless channel. Hence, this derivation cannot be directly extended to the case of the Ungerboeck observation model. In this letter, we solve this problem by using a properly defined factor graph (FG) and the sum-product algorithm (SPA) [9]. II. PRELIMINARIES In this section, we briefly describe the FGs and the SPA, and summarize the results on MAP sequence detection for ISI channels. Manuscript received December 14, 2004. The associate editor coordinating the review of this letter and approving it for publication was Prof. Jing Li. This paper was presented in part at the IEEE Symposium on Information Theory and its Applications (ISITA’04), Parma, Italy, October 2004. The authors are with the Universit` a di Parma, Dipartimento di In- gegneria dell’Informazione, Parma, Italy (e-mail: giulio@unipr.it, barbi- eri@tlc.unipr.it). Digital Object Identifier 10.1109/LCOMM.2005.08003. A. Factor Graphs and the Sum-Product Algorithm A FG is a bipartite graph which expresses the way a complicated joint probability mass function (pmf) or a joint probability density function (pdf) of many variables factors into the product of local functions (not necessarily pmfs or pdfs) [9]. Let V = {v 1 ,...,v N } denote a set of variables and F (V ) a multivariate function. Let V 1 ,...,V m denote subsets of V . We say that F (V ) admits a factorization with supports V 1 ,...,V m , if F (V ) can be written as the product of the functions {F j : j =1,...,m}, where F j has the variables in V j as arguments. The FG representing the factorization F =  j F j is a bipartite graph G = {V , F , E}, where nodes in V (variable nodes) are associated with the variables v i ∈ V , nodes in F (factor nodes) are associated with the functions F j , and there exists an edge e ∈E joining v i and F j if and only if v i ∈ V j (i.e., if v i is an argument of F j ). Let F (V ) be a pmf. Then, if the FG corresponding to the factorization of F has no cycles, 1 the marginal pmfs can be computed exactly in a finite number of steps by the SPA [9]. The SPA is defined by the computation rules at variable and at factor nodes, and by a suitable node activation schedule. Denoting by μ vi→Fj (v i ) a message sent from the variable node v i to the factor node F j , by μ Fj →vi (v i ) a message in the opposite direction, and by A i the set of functions F j having v i as argument, the message computations performed at variable and factor nodes are, respectively [9] μ vi→Fj (v i )=  H∈Ai\{Fj } μ H→vi (v i ) (1) μ Fj →vi (v i )=  ∼{vi} ⎡ ⎣ F j ({w ∈ V j })  w∈Vj \{vi} μ w→Fj (w) ⎤ ⎦ (2) where, following the notation of [9], we indicate by ∑ ∼{vi} the summary operator, i.e., a sum over all variables excluding v i . In the absence of cycles in the graph, the computation usually starts at the leaves of the graph and proceeds until two messages have been passed over every edge, one in each direction. B. MAP Sequence Detection: Ungerboeck and Forney Ap- proaches The Ungerboeck approach for MAP sequence detection [1] is based on the samples {x k } at the output of a filter matched to the received pulse p(t). The sample at time kT , can be 1 A cycle is a closed path in the graph. 1089-7798/05$20.00 c  2005 IEEE