Structural characterization of locally optimum detectors in terms of locally optimum estimators and correlators

Explicit formulas for locally (SNR
\rightarrow 0)
optimum (MMSE) signal estimators (smoother, filter, and predictor) for discrete-time observations of a random signal in additive random noise are derived and used to characterize the locally optimum (likelihood ratio) signal detector for on-off signaling. The characterizations are canonical (distribution-free) detector structures involving estimator-correlators. These structural characterizations provide new interpretations of known detectors for various special cases. If the one-step signal predictor is recursive and the noise is white (possibly non-Gaussian or nonstationary), there is a canonical structure that admits recursive computation. The primary motivation for these structural characterizations is to render the estimator-correlator design philosophy applicable for the purpose of simplifying implementations and enhancing adaptability. Unlike the known esfimator-correlator structural characterizations for continuous-time globally optimum detectors, the new characterizations apply for non-Gaussian as well as Gaussian noise, and the estimators are explicit rather than implicit.