Shift variance, cyclostationarity and expected shift variance in multirate LPSV systems

Abstract

The shift variant properties of linear periodically shift variant (LPSV) systems are generally analysed using deterministic signals. Shift variance is, however, closely related to cyclostationarities introduced by the LPSV system into originally wide sense stationary (WSS) random signals passing through the system. We first develop a Hilbert space distance measure for shift variance of LPSV operators with deterministic input signals. For an LPSV system with WSS random input signals, we examine the covariance operator associated to its wide-sense cyclostationary (WSCS) output signal. The output signal is WSCS if its covariance operator is LPSV, and WSS if its covariance operator is shift invariant. Applying the shift variance measure to the covariance operator therefore provides a direct measure of the cyclostationarity generated by the LPSV system. To comprehensively capture the shift variant effects of the LPSV system on WSS random signals, we analyze the changes in its output signal when shift operator and system are interchanged. These changes are WSCS, and can be quantified by the expected shift variance over one cycle. We apply the framework to various two-channel filter banks.

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

“…1) can be represented by a kernel g (m, r) satisfying the condition g (m, r)= g (m + P, r + P ) ∀m, r [25]. The output of such a system is y [m] = r∈Z g [m, r] x [r] [25], [31]. Let M P ×P be the space of P × P matrices.…”

“…In (Fig. 2 of [24]), [25], the authors have obtained the expression for the closest LTI system impulse response to an LPTV system as v cl (n) = 1 P P −1 r=0 g (n + r, r), by minimizing the squared distance ∥G − V∥ 2 between the LPTV operator G and the LTI operator V. Since every LTI system is also an LPTV system ∀P , the Theorem 2 provides the corresponding MIMO matrix form (18) for the LTI system as : Theorem 2: Let MIMO matrix (18) for an LTI system having impulse response v (n), denoted here as is given by the diagonal matrix V LTI (z) with diagonal elements V LTI k,k (z) and zero non-diagonal elements where…”

“…Numerous publications on non-uniform filter banks can be found in [9], [22], [23]. Multirate filter banks are linear periodically time-varying (LPTV) systems, as the authors demonstrate in [24], [25]. Multirate systems are inherently time-varying due to rate fluctuations.…”