Union Bounds on Viterbi Algorithm Performance

Abstract

In the present paper we use transform methods (characteristic function techniques) and contour integrals to derive a closed-form expression for the performance union bound of a general discrete-time system. We show that previously published results may be derived as particular cases of the general formulation developed in this paper. It is well known that the maximumlikelihood Viterbi algorithm may be employed not only for decoding of convolutional codes but also for optimal detection in other situations. Examples include bandwidth-efficient demodulation, optimal accommodation for intersymbol interference and cross-channel coupling, text recognition, simultaneous carrier phase recovery and data demodulation, digital magnetic recording, nonlinear estimation and smoothing. The union bound is a useful measure of the performance of the Viterbi algorithm. Past closed-form expressions for the union bound have usually involved considerable approximation.

Evaluating the Exact Performance of the Viterbi Algorithm

“…The Viterbi bound [3] is much simpler to evaluate but is not very tight. The bound developed in [4] and [5] is tighter and not significantly more complex. We illustrate the utility of matrix probabilities [6] for computations related to HMMs and, in particular, for evaluating the performance of the Viterbi algorithm in channels with memory.…”

“…where q = 1− p. A more accurate bound is given in [4] which can be presented as [5], [6,Appendix 4.3]…”

“…Obviously, upper bounds are useful for comparison of different systems if they are "tight", because a small difference between two upper bounds on some values does not necessarily correspond to the small difference between the values. It has been demonstrated [4], [5] and also in this paper that the bounds of [3], [1] are not tight even for small channel error rates.…”

“…The exact union bound was presented by Post [4] for a binary convolutional code in channels with independent errors. This exact bound was generalized to FSM in [5], [6,Appendix 4.3]. Obviously, upper bounds are useful for comparison of different systems if they are "tight", because a small difference between two upper bounds on some values does not necessarily correspond to the small difference between the values.…”