Optimal Approximate Inverse of Linear Periodic Filters

Abstract

We propose a method for constructing optimal causal approximate inverse for discrete-time single-input single-output (SISO) causal periodic filters in the presence of measurement noise. The analysis is based on block signals and multi-input multi-output (MIMO) time-invariant models for periodic filters. The objective function to be minimized is the asymptotic block mean square error. The optimization problem is formulated in terms of transfer matrices as an optimal model-matching problem with nonsquare model and plant. Based on an inner-outer factorization on the transpose of the plant rational matrix, it is shown that the problem can be further reduced to one with a lower dimensional square model and plant, which is then solved in the time-domain, and a closed-form solution is obtained. A lower bound on the objective function is given. It is shown that the lower bound can be asymptotically achieved as the order of the optimal transfer matrix increases. The proposed method is extended to MIMO periodic systems. Numerical examples are used to illustrate the performance of the proposed approximate inverse.

Designing the inverse of an LPTV system using &#x210B;<inf>&#x221E;</inf> optimization

“…Our design criterion imposes no conditions on the LPTV for inversion. The main advantage of our method is that we can design inverse for any given LPTV system where as earlier methods [4], [5], [6], [7] are imposing conditions on the LPTV system for which we have to design inverse. We present simulation results for an LPTV of order 4.…”

“…The task of finding the inverse of an LPTV is an important requirement in many applications like scrambled signal recovery [2] and equalization based on modulation induced cyclo-stationarity [1]. So, the problem of LPTV inversion has drawn a lot of research attention [3], [4], [5], [6].…”

Error Analysis of the Anti-Causal Realization of Periodic Inverse Systems

“…This inverse system error was not considered in the existing literature based on the inner-outer factorization such as [1,2]. Moreover, it is different from the system error studied in [11] which is caused by the constrained causal system inverse realization but not the system unknown internal state values of the anti-causal exact inverse system considered in this paper. The analysis in this paper shows the system internal mechanisms that generate the system inverse error and, hence, provides understanding and quantification in computing the system inverse error.…”

“…(LPTV) system has attracted considerable attentions; see for example, [5][6][7][8][9][10][11][12]. However, the inverse system may not be stable in causal realization.…”

“…In [11], a method is proposed to construct an optimal causal approximate inverse for discrete-time single-input single-output (SISO) causal periodic filters in the presence of measurement noise. The inverse system obtained is constrained to be a causal and stable system, so it may not be an exact inverse for some systems, and optimization is introduced to minimize the inversion error in terms of 2-norms.…”