On the optimality of the Wigner distribution for detection

A variety of methods have been proposed for the detection of a signal, with unknown signal parameters, in a noisy environment. In this paper, the noise statistics are incorporated to reveal that certain processing of the Wigner distribution (WD) signal representation can lead to an optimal, and often easy to compute, detection scheme. For the special case of linear FM signals in complex white Gaussian noise, it is shown that the optimal detector is equivalent to integrating the WD along the line of instantaneous frequency. If the position and sweep rate of the linear chirp are unknown, then a Generalized Likelihood Ratio Test (GLRT) leads one to integrate the WD along all possible lines in the time-frequency plane and choose the largest integrated value for comparison to a threshold. Simulation examples of the WD detection scheme are given to demonstrate the utility of the proposed method. Finally, some comments concerning the detection of the general phase modulated signal are offered.