The Viterbi algorithm (VA) is modified to deliver the most likely path sequence in a finite-state Markov chain, as well as either the a posteriori probability for each bit or a reliability value. With this reliability indicator the modified VA produces soft decisions to be used in the decoding of outer codes. The inner software output Viterbi algorithm (SOVA) accepts and delivers soft sample values and can be regraded as a device for improving the signal-to-noise ratio, similar to an FM demodulator. Several applications are investigated to show the gain over the conventional hard-deciding VA, including concatenated convolutional codes, concatenation of trellis-coded modulation with convolutional FEC (forward error correcting) codes, and coded Viterbi equalization. For these applications additional gains of 1-4 dB as compared to the classical hard-deciding algorithms were found. For comparison, the more complex symbol-to-symbol MAP, whose optimal a posteriori probabilities can be transformed into soft outputs, was investigated. >

A Viterbi algorithm with soft-decision outputs and its applications

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, the authors concentrate 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. >

Per-Survivor Processing: a general approach to MLSE in uncertain environments

A list Viterbi decoding algorithm (LVA) produces a rank ordered list of the L globally best candidates after a trellis search. The authors present two such algorithms, (i) a parallel LVA that simultaneously produces the L best candidates and (ii) a serial LVA that iteratively produces the k/sup th/ best candidate based on knowledge of the previously found k-1 best paths. The application of LVA to a concatenated communication system consisting of an inner convolutional code and an outer error detecting code is considered in detail. Analysis as well as simulation results show that significant improvement in error performance is obtained when the inner decoder, which is conventionally based on the Viterbi algorithm (VA), is replaced by the LVA. An improvement of up to 3 dB is obtained for the additive white Gaussian noise (AWGN) channel due to an increase in the minimum Euclidean distance. Ever larger gains are obtained for the Rayleigh fading channel due to an increase in the time diversity. It is also shown that a 10% improvement in throughput is obtained along with significantly reduced probability of a decoding failure for a hybrid FEC/ARQ scheme with the inner code being a rate compatible punctured convolutional (RCPC) code. >

List Viterbi decoding algorithms with applications

Concatenated coding schemes with interleavers consist of a combination of two simple constituent encoders and an interleaver. The parallel concatenation known as “turbo code” has been shown to yield remarkable coding gains close to theoretical limits, yet admitting a relatively simple iterative decoding technique. The recently proposed serial concatenation of interleaved codes may offer performance superior to that of turbo codes. In both coding schemes, the core of the iterative decoding structure is a soft-input soft-output (SISO) module. In this article, we describe the SISO module in a form that continuously updates the maximum a posteriori (MAP) probabilities of input and output code symbols and show how to embed it into iterative decoders for parallel and serially concatenated codes. Results are focused on codes yielding very high coding gain for space applications.

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A soft-input soft-output maximum a posteriori (MAP) module to decode parallel and serial concatenated codes

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Quadrature Amplitude Modulation: From Basics to Adaptive Trellis-Coded, Turbo-Equalised and Space-Time Coded OFDM, CDMA and MC-CDMA Systems

The Viterbi algorithm (VA), an optimum decoding rule for a Q -ary trellis code of constraint length K , operates by taking the best survivor from each of Q^{K-1} lists of candidates at each decoding step. A generalized VA (GVA) is proposed that makes comparisons on the basis of a label of length L(L\leq K) . It selects, incorporating the notion of list decoding, the S best survivors from each of Q^{L-1} lists of candidates at each decoding step. Coding theorems for a discrete memoryless channel are proved for GVA decoding and shown to be natural generalizations of those for VA decoding. An example of intersymbol interference removal is given to illustrate the practical benefits that the GVA can provide.

T. Hashimoto

Papers

A list-type reduced-constraint generalization of the Viterbi algorithm