Measures for the Degree of Time Variance: Application to Filter Banks

Quantifying a system’s time-invariance deviation may reveal how it represents and processes the signals. The main objective of this paper is to quantify the departure of a time-varying system from being time-invariant. In particular, the paper quantifies the time variance of filter banks using kernel representation. We define the $\ell ^{2}$ measure for the degree of time variance ( $\mathrm {MDTV}^{\ell _{2}}$ ) based on the discrete-time domain form of the kernel representation and the $\ell ^{\infty }$ measure ( $\mathrm {MDTV}^{\ell _{\infty }}$ ) based on the matrix representation of the LPTV kernel. We show that off-diagonal matrix elements can be used to deduce the time-varying behavior of an LPTV system, proving that time-invariant and time-varying components can be separated. The matrix form of the LTI system, which is closest to an LPTV system, has been derived. This research offers a new kernel-based definition for time variance for any time-varying system and provides explicit expressions for MDTV for three distinct scenarios, namely, a single channel (SC), a uniform filter bank (UFB), and a non-uniform filter bank (NUFB). For each scenario, we determine the LPTV kernel and impulse response of the nearest LTI system to plot the MDTV versus decimation factor. Also, MDTV generated by an SC and a UFB with identical low pass filters in each channel is identical. We also observe that two-channel perfect reconstruction (PR) filter banks result in zero MDTV. Finally, we have compared the proposed approach with earlier work and demonstrated its better efficiency in terms of computational complexity.