A COMPARISON OF THE EXISTENCE OF CROSS TERMS IN THE WIGNER DISTRIBUTION AND THE SQUARED MAGNITUDE OF THE WAVELET TRANSFORM AND THE SHORT-TIME FOURIER-TRANSFORM

The wavelet transform (WT), a time-scale representation, is linear by definition. However, the nonlinear energy distribution of this transform is often used to represent the signal; it contains "cross terms" which could cause problems while analyzing multicomponent signals. In this paper, we show that the cross terms that exist in the energy distribution of the WT are comparable with those found in the Wigner distribution (WD), a quadratic time-frequency representation, and the energy distribution of the short time Fourier transform (STFT), of closely spaced signals. The cross terms of the WT and the STFT energy distributions occur at the intersection of their respective WT and STFT spaces, while for the WD they occur midtime and midfrequency. The parameters of the cross terms are a function of the difference in center frequencies and center times of the perpended signals. The amplitude of these cross terms can be as large as twice the product of the magnitudes of the transforms of the two signals in question in all three cases. In this paper, we consider the significance of the effect of the cross terms on the analysis of a multicomponent signal in each of these three representations. We also compare the advantages and disadvantages of all of these methods in applications to signal processing.