Improving Translation Invariance by Complex Wavelet using Phase Information

In this study, we examine the necessity for the complex discrete wavelet transform (CDWT) from the viewpoint that the signal's position shifting corresponds to the signal's phase change and discuss the effectiveness of the improved shift invariability of the CDWT based on linear translation invariant system theory. The results are: 1) the DWT is sensitive to the signal's shift (the phase change) and correspondingly the wavelet coefficients change sharply as the signal shifts. On the other hand, the CDWT is insensitive to the signal's phase change and the obtained wavelet coefficients are stable. 2) The output obtained by the CDWT is unique to the input signal having an arbitrary shift. That is, the CDWT system is not dependent on the position of the input signal and can be considered a linear translation invariant system