Breadth-first maximum-likelihood sequence detection: geometry

SA(B, C) is an algorithm penetrating a tree (or trellis) breadth first. It performs maximum-likelihood sequence detection (MLSD) under that structural constraint, and also under the complexity constraints determined by the parameters B and C. First, C is the number of partitions into which the states are distributed, and B denotes the number of paths in each partition. Recursively selecting paths which are closest to the received signal in the Euclidean distance (Hamming distance) sense guarantees constrained MLSD for the additive white Gaussian (binary symmetric) channel. The previously presented vector Euclidean distance (VED) is an important tool for analyzing SA(B, C) over the additive white Gaussian noise (AWGN) channel. A geometric interpretation of the signals involved clarifies the basic properties of this VED and other relevant general results (invariance, monotonicity). These results also form a basis for the construction of an algorithm for the efficient and fast calculation of minimum VEDs (of interest for large signal-to-noise ratio, SNR, detection performance). This, in turn, reveals the necessary complexity requirements to meet specified performance requirements for concrete trellis-coded systems. Here, the simple example of convolutionally coded (rate 1/2) with Gray-coded quaternary phase-shift keying over the AWGN channel is considered. When C=1 and choosing B∼√S (S being the number of states in the code trellis) gives the same asymptotic detection performance (large SNR) as unconstrained MLSD (e.g., implemented using the Viterbi algorithm).