Existence of cross terms in the wavelet transform

Document Type

Conference Proceeding

Date of Original Version



Since the wavelet transform, a time-scale representation is linear by definition, it does not have any cross terms. However, since signal processors often use the plots of the quadratic square magnitude, i.e., the energy distribution of the WT, to represent a signal, there exists nonlinear cross terms which could cause problems while analyzing multicomponent signals. In this paper, we show that these WT cross terms do exist and discuss the nature and the geometry of these cross terms by deriving mathematical expressions for the energy distribution of the WT of a multicomponent signal. From the mathematical expressions for the WT cross terms, we can infer that the nature of these `cross terms' are comparable with those found in the Wigner distribution (WD), a quadratic time-frequency representation, and 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, whereas for the WD, cross terms occur midtime and midfrequency. The parameters of the cross terms are a function of the difference in frequency and time of the perpended signals. The amplitude of these cross terms can be twice as large as the product of the magnitudes of the transforms of the two signals in question in all the three cases. We also discuss the significance of the existence of WT cross terms while analyzing a multicomponent signal with representative examples.

Publication Title

Proceedings of SPIE - The International Society for Optical Engineering



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