Alias-Free Wigner Distribution Function and Complex Ambiguity Function for Discrete-Time Samples

Abstract
If an arbitrary complex continuous waveform s(t) with finite overall frequency extent F Hertz is sampled with time increment Delta 1/F, the aliasing can be controlled and the continuous time waveform s(t) reconstructed exactly at any desired time instant from waveform samples s(kDelta). On the other hand, it is commonly believed that aliasing of the corresponding Wigner distribution function (WDF) can only be avoided by sampling twice as fast; i.e., Delta 1/(2F) is thought to be required. Alternatively, interpolation of the time data has been suggested as a means of circumventing aliasing of the WDF; however, the computational burden has proven excessive if done by sinc function interpolation. It is demonstrated here that this conjecture is false, and that the usual sampling criterion, Delta 1/F, suffices for exact reconstruction of the original continuous WDF, as well as the complex ambiguity function (CAF), at all time, frequency locations, without an excessive amount of computational effort. Keywords: Interspersed aliasing lobes; Temporal correlation; Spectral correlation; Ambiguity function; Aliasing elimination; Discrete time sampling; Wigner distribution; Bandlimited spectrum; Diamond gating function; Interspersed sampling.

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