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Factorability of lossless time-varying filters and filter banks

Phoong, See-May and Vaidyanathan, P. P. (1997) Factorability of lossless time-varying filters and filter banks. IEEE Transactions on Signal Processing, 45 (8). pp. 1971-1986. ISSN 1053-587X. doi:10.1109/78.611189.

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We study the factorability of linear time-varying (LTV) lossless filters and filter banks. We give a complete characterization of all, degree-one lossless LTV systems and show that all degree-one lossless systems can be decomposed into a time-dependent unitary matrix followed by a lossless dyadic-based LTV system. The lossless dyadic-based system has several properties that make it useful in the factorization of lossless LTV systems. The traditional lapped orthogonal transform (LOT) is also generalized to the LTV case. We identify two classes of TVLOTs, namely, the invertible inverse lossless (IIL) and noninvertible inverse lossless (NIL) TVLOTs. The minimum number of delays required to implement a TVLOT is shown to be a nondecreasing function of time, and it is a constant if and only if the TVLOT is IIL. We also show that all IIL TVLOTs can be factorized uniquely into the proposed degree-one lossless building block. The factorization is minimal in terms of the delay elements. For NIL TVLOTs, there are factorable and unfactorable examples. Both necessary and sufficient conditions for the factorability of lossless LTV systems are given. We also introduce the concept of strong eternal reachability (SER) and strong eternal observability (SEO) of LTV systems. The SER and SEO of an implementation of LTV systems imply the minimality of the structure. Using these concepts, we are able to show that the cascade structure for a factorable IIL LTV system is minimal. That implies that if a IIL LTV system is factorable in terms of the lossless dyadic-based building blocks, the factorization is minimal in terms of delays as well as the number of building blocks. We also prove the BIBO stability of the LTV normalized IIR lattice.

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Vaidyanathan, P. P.0000-0003-3003-7042
Additional Information:© Copyright 1997 IEEE. Reprinted with permission. Manuscript received April 7, 1995; revised March 27, 1997. This work was supported by the NSF under Grant MIP 92-15785, by Textronix, Inc., and by Rockwell International. The associate editor coordinating the review of this paper and approving it for publication was Dr. Bruce W. Suter.
Subject Keywords:IIR filters, band-pass filters, circuit stability, delays, filtering theory, lattice filters, matrix algebra, time-varying filters, transforms
Issue or Number:8
Record Number:CaltechAUTHORS:PHOieeetsp97b
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:8702
Deposited By: Archive Administrator
Deposited On:06 Sep 2007
Last Modified:08 Nov 2021 20:52

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