Shift-Variance Analysis of Generalized Sampling Processes

This paper is concerned with quantifying shift-variance of linear systems with continuous-time input and discrete-time output. We first introduce a notion of -shift-invariance for the system. It specifies how the system should respond when the input signal is shifted. For generalized sampling processes, the property is characterized by the sampling kernel, and is shown to be equivalent to lack of aliasing and to shiftability. We then define a shift-variance level which describes how far the system can be possibly away from the set of systems that are -shift-invariant. The level is defined to be the maximum of the induced norms of the commutators of the system and the shift-operators (both continuous-time or discrete-time are necessarily involved). A shift-variance measure is then defined to be the ratio of the shift-variance level to the system norm, further divided by two so that the measure is between zero and unity. For generalized sampling, we obtain analytical formulas for the shift-variance level and the shift-variance measure. The results allow us to analyze the shift-variance of, among others, the discrete-time wavelet transform (DWT) and the short-time Fourier transform (STFT). We obtain a simple relation between the shift-variance levels at all scales of the DWT, and show that the shift-variance measures are identical at all scales. We calculate the shift-variance level and the shift-variance measure of some typical DWTs and the STFTs.