Frequency Selective KYP Lemma and its Applications to IIR Filter Bank Design

For a transfer function/filter F(e^jω) of order n, Kalman-Yakubovich-Popov (KYP) lemma characterizes the intractable semi-infinite programming (SIP) condition F(e^-jω)1 Θ [F(e^jω)1]^T ≥ 0 ∀ ω in frequency domain by a tractable semi-definite programming (SDP) in state-space domain. Some recent results generalize this lemma to SDP for SIP of frequency selectivity (FS-SIP). All these SDP characterizations are given at the expense of the introduced Lyapunov matrix variable of dimension n × n, making them impractical for high order problem. Moreover, the existing SDP characterizations for FS-SIP do not allow to formulate synthesis/design problems as SDPs. In this paper, we propose a completely new SDP characterization of general FS-SIP, which is of moderate size and is free from Lyapunov variables. Extensive examples are provided to validate the effectiveness of our result.