Timothy N. Davidson and Jos F. Sturm,
A "Primal"
Positive Real Lemma for FIR Systems, with Applications to
Filter Design and Spectral Factorization
Technical Report,
Department of Electrical and Computer Engineering,
McMaster University,
15 February 2000
The Positive Real Lemma (PRL) has long been an effective tool in the analysis of feedback systems, but recently it has also found applications in the design of finite impulse response (FIR) filters. In this note an alternative form of the PRL for FIR systems is established. It is shown that the alternative form of the PRL facilitates an order of magnitude reduction in the computational cost of some convex optimization techniques for FIR filter design and spectral factorization.
This technical report deals directly with the case of FIR systems with at least one input and at least one output. However,
an independently derived version of Proposition 1 that applies to single-input, single-output systems appears in Dumitrescu, Tabus and Stoica, "On the parameterization of positive real sequences and MA parameter estimation", IEEE Transactions on Signal Processing, 49(11):2630-2639, November 2001;
and a single-input, single-output statement of Proposition 1 in this technical report appears in our work on Linear matrix inequality formulation of spectral mask constraints.
A paper that applies essentially the same analysis to the bounded real lemma is: B. Dumitrescu, "Bounded Real Lemma for FIR MIMO Systems", IEEE Signal Processing Letters, 12(7):496-499, July 2005.
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