FFT-based incremental refinement of suboptimal detection

In the context of FFT-based maximum-likelihood (ML) detection of a complex sinusoid in noise, we consider the result of terminating the FFT at an intermediate stage of computation and applying the ML detection strategy to its unfinished results. We show that detection performance increases monotonically with the number of FFT stages completed, converging ultimately to that of the exact ML detector. The receiver operating characteristic associated with the completion of each FFT stage is derived. This enables the calculation of the minimum number of FFT stages that must be completed in order for desired detection and false alarm probabilities to be obtained.