maximalelikelihooddecoding

Maximal likelihood decoding, often referred to as maximum likelihood decoding (MLD), is a decoding rule used in digital communications and coding theory. It selects the transmitted symbol, sequence, or codeword that maximizes the likelihood of the received signal given the candidate. Formally, if y denotes the received vector and x denotes a possible transmitted symbol or codeword, ML decoding chooses x_hat = argmax_x p(y|x). When the prior probability p(x) is uniform, this coincides with maximum a posteriori decoding; otherwise MAP uses p(y|x)p(x).

In a channel with additive white Gaussian noise (AWGN) and a finite constellation, p(y|x) ∝ exp(-||y - x||^2 / N0). Therefore ML decoding reduces to choosing the constellation point closest in Euclidean distance to y. For complex signaling, this rule generalizes to minimizing the squared error with respect to the received vector after the channel effects.

In coding theory, ML decoding over a codebook C selects the codeword c_hat that maximizes p(y|c). This is the optimal decision rule for the assumed channel model, but computationally it is typically expensive because the search space equals the codebook size. For linear codes with Gaussian noise, ML decoding is equivalent to minimum-distance decoding; practical schemes such as Viterbi decoding, BCJR, or sphere decoding approximate ML performance.

ML decoding provides a benchmark for optimal performance under the channel model and is central to information-theoretic analyses. In many real systems, suboptimal decoders are employed to balance decoding accuracy with computational resources.