A system function representation by scale-shift transform-the case of Doppler effect

Wavelet transforms give a time-frequency representation of a signal. But many applications show that more research is needed for characterizing a time-varying system. In this paper, we describe a type of impulse response function with scale parameter a and shift parameter b. We also define a convolution operation with variable a and b. For this type of system, we can prove that the scale-shift transform of output is given by a product of scale-shift transform of input and a system function. As an application of scale-shift transform, we show the system function of Doppler effect