Scale-Invariant Wigner Distribution And Ambiguity Functions

Abstract

The Wigner distribution (WD) function is a two-dimensional representation that displays the space and spatial frequency content of a one-dimensional signal. Its two-dimensional Fourier transform, the radar ambiguity function, displays the space and spatial frequency shifts of the same signal. The WD has the property that a space or spatial frequency shift of the signal leads to a corresponding shift of the Wigner distribution function. This represents a translation invariance of the WD, a property that is useful for impulse response characterization of a space-invariant system. Similarly, if one signal is shifted with respect to the other, magnitude of their cross-ambiguity function is shifted by the same amount. There are many optical systems that are space-variant. A particular space-variant system, encountered in many imaging applications, is the so-called scale-invariant system. In this paper, scale-invariant Wigner distribution and ambiguity functions are defined. These new functions represent local scale-frequency spectrum of the signal, and scale correlation in space and scale frequency. Properties of these functions are described and the differences and similarities to translation-invariant WD and ambiguity function are pointed out. Potential applications and analog optical implementations of these functions aie also discussed.