On the Application of Factor Graphs and the Sum–Product Algorithm to ISI Channels

Abstract

Abstract-In this paper, based on the application of the sum-product (SP) algorithm to factor graphs (FGs) representing the joint a posteriori probability (APP) of the transmitted symbols, we propose new iterative soft-input soft-output (SISO) detection schemes for intersymbol interference (ISI) channels. We have verified by computer simulations that the SP algorithm converges to a good approximation of the exact marginal APPs of the transmitted symbols if the FG has girth at least 6. For ISI channels whose corresponding FG has girth 4, the application of a stretching technique allows us to obtain an equivalent girth-6 graph.For sparse ISI channels, the proposed algorithms have advantages in terms of complexity over optimal detection schemes based on the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. They also allow a parallel implementation of the receiver and the possibility of a more efficient complexity reduction. The application to joint detection and decoding of low-density parity-check (LDPC) codes is also considered and results are shown for some partial-response magnetic channels. Also in these cases, we show that the proposed algorithms have a limited performance loss with respect to that can be obtained when the optimal "serial" BCJR algorithm is used for detection. Therefore, for their parallel implementation, they represent a favorable alternative to the modified "parallel" BCJR algorithm proposed in the literature for the application to magnetic channels.

SISO Detection Over Linear Channels With Linear Complexity in the Number of Interferers

“…The optimal MAP symbol detection algorithm for this application is the so-called BCJR algorithm [40] whose complexity is linear in K and exponential in L. This algorithm has been originally devised for the Forney model (17) and only recently extended to the Ungerboeck observation model [41]. Several algorithms have been proposed in the literature to reduce its complexity (see [42], [43] and references therein). In particular, the algorithm in [42], was derived still using the FG/SPA framework.…”

“…Several algorithms have been proposed in the literature to reduce its complexity (see [42], [43] and references therein). In particular, the algorithm in [42], was derived still using the FG/SPA framework. From (2), (3), and (17) we may write [42] …”

“…In particular, the algorithm in [42], was derived still using the FG/SPA framework. From (2), (3), and (17) we may write [42] …”

“…adopted [42]. In any case, the complexity is linear in K and exponential in the number of non-zero interferes [42], since this number gives the degree of nodes J i .…”

“…In any case, the complexity is linear in K and exponential in the number of non-zero interferes [42], since this number gives the degree of nodes J i . On the contrary, the factorization we adopted and based on the matched filter output always leads to a graph with cycles of length 6 and algorithms with a complexity per iteration which is linear in L.…”