A detection rule that selects the transmitted symbol sequence (or symbol vector) that maximizes the likelihood of the observed received signal under a specified channel and noise model; for linear channels with additive white Gaussian noise this reduces to choosing the sequence minimizing Euclidean distance between predicted and received signals.

Principle

Under the assumed channel and noise model, ML detection minimizes the probability of selecting an incorrect transmitted sequence (sequence‑error probability) because it picks the sequence with highest conditional probability of producing the observation; ML is model‑dependent and becomes computationally intractable for large symbol alphabets or sequence lengths due to the exponential search space.

Demonstration

Situation: MIMO block transmission with known channel matrix H and AWGN. Recognition: ML detector computes the likelihood p(y|x,H) for each candidate transmit vector x. Action: evaluate metric ||y−Hx||^2 for all x in the constellation Cartesian product and select x minimizing the metric. Consequence: the chosen vector is the sequence‑ML solution and yields the lowest sequence‑error probability under the assumed model, but the exhaustive search cost grows exponentially with antenna count and constellation size.

Misapplication

Applying ML detection while treating the detector's objective as robust to model errors—e.g., using ML derived for AWGN when the noise is impulsive, or using an incorrect channel matrix—leads to suboptimal or misleading results. The error is assuming model‑independence: ML optimality holds only under the stated model assumptions.

Consequence

ML detection establishes a performance benchmark (minimum achievable sequence‑error probability under the model) and motivates approximate or reduced‑complexity algorithms (sphere decoding, branch‑and‑bound, successive interference cancellation). Practically, ML may be infeasible for real‑time systems with high dimensionality, forcing tradeoffs between optimality and complexity.

Reversal

If prior probabilities over sequences are nonuniform or priors and costs differ (bitwise error vs sequence error), MAP or other decision rules may be preferred; when partial soft information is required (bitwise probabilities) or channel/model uncertainty exists, Bayesian or robust detectors can outperform naive ML. Also, ML performance claims fail if the assumed channel/noise model is incorrect.

Boundary

Clearly within: block or sequence detection with a fully specified channel and noise model where exhaustive evaluation over the discrete symbol space is possible. Boundary case: constrained searches (e.g., per‑symbol ML vs sequence ML) that reduce complexity but change the decision criterion. Clearly outside: linear suboptimal detectors (ZF, MMSE) and heuristic receivers that do not maximize the likelihood under the exact model.

Semantic Tension

Optimality versus tractability: ML delivers the best sequence‑error performance under model assumptions but incurs exponential computational cost, forcing practical systems to trade off between detection accuracy and implementation complexity.

Synthesis

Maximum likelihood detection is the formal optimal detector under an explicit model and thus defines the performance target; in practice its computational burden and model sensitivity compel practitioners to use approximate algorithms or alternative decision criteria depending on complexity, prior information, and robustness requirements.