The Discrete-Time Bounded-Real Lemma in Digital Filtering

I N THE continuous-time world, the description of passivity properties of linear time-invariant systems in terms of minimal state-space descriptions is well known. The application of such results in system theory, network synthesis, and in stability analysis is also well established [1],[2]. For example, given a Lossless Positive Real (LPR) matrix Z(s), any minimal state-space description satisfies the LPR Lemma [l]. A generalization for lossy impedance matrices is the Positive Real Lemma or PR Lemma, which states the PR property in terms of the state-space matrices (A, B, C, D). Scattering domain versions of these descriptions are also known, and are contained in the BoundedReal Lemma (BR Lemma) and the Lossless Bounded-Real Lemma. The proofs of the above lemma are known to be tedious, and an excellent survey can be found in [l]. A typical procedure is to establish the PR lemma, and then obtain the LPR lemma as a special case. The scattering domain versions are then obtained by translating the statements of the (Lossless) PR lemma into the scattering domain by simple wave-transformations, even though a direct development, not based on the PR counterpart, is possible. Discrete-time versions of these results are easily obtained by careful application of the bilinear transformation. For example, the discrete-time LPR lemma is developed in [2], based on this approach. The purpose of this paper is to reconsider the discretetime BR lemma. We outline an independent proof of the lemma without invoking the PR counterpart or any related results in the continuous-time world. We begin by proving the discrete-time LBR (DTLBR) lemma, based only on energy-balance arguments. We then extend this to the case of BR matrices (DTBR lemma) by invoking a classic result on spectral factorization [3]. Note that this approach is unlike the conventional approach, where the BR (or PR)