Frequency-Selective KYP Lemma, IIR Filter, and Filter Bank Design

Abstract—For a transfer function of order, Kalman– Yakubovich–Popov (KYP) lemma characterizes a general in-tractable semi-infinite programming (SIP) condition by a tractable semidefinite programming (SDP) for the entire frequency range. Some recent results generalize this lemma for a certain frequency interval. All these SDP characterizations are given at the ex-pense of the introduced Lyapunov matrix variable of dimension. Consequently, formulation and design of high dimensional problem is challenging. Moreover, existing SDP characterizations for frequency-selective SIP (FS-SIP) do not allow to formulate synthesis problems as SDPs. In this paper, we propose a completely new SDP characterization of general FS-SIP involving SDPs of moderate size and free from Lyapunov variables. Furthermore, a systematic IIR filter and filter bank design is developed in a similar vein, with several simulations provided to validate the effectiveness of our SDP formulation. Index Terms—Filter and filter bank, Kalman–Yakubovich– Popov (KYP) lemma, semidefinite programming. I