Robust and Computationally Efficient Digital IIR Filter Synthesis and Stability Analysis under Finite Precision Implementations

• Published in: IEEE transactions on signal processing Vol. 68; p. 1
• Main Authors: Ko, Hsien-Ju, Tsai, Jeffrey J.P.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.01.2020; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Computational efficiency; Computer simulation; Estimation; finite word-length (FWL); Fixed point arithmetic; fixed-point; IIR filters; Impulse response; L2-sensitivity; limit-cycle-free; Mathematical models; Numerical stability; Optimization; pole sensitivity; Poles and zeros; Portable equipment; Robustness (mathematics); Sensitivity; Sparse matrices; sparse normalform; Stability analysis; state-space; Synthesis; zero sensitivity
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2020.2977848

In this paper, an analytical synthesis method for obtaining an optimal infinite impulse response (IIR) state-space realization, say minimal pole-zero and pole- L_2 sensitivity realizations, based on minimizing zero/ L_2 sensitivity measures subject to sparse normal-form state transition matrix is proposed. The proposed n^{th} order realization possesses strong robustness and at most only 4n + 1 multiplications per output sample to prevent instability and output distortion, as well as to keep computational efficiency under finite word-length (FWL) effects. This is important for the IIR filter implemented in fixed-point and portable digital devices. In this paper, the proposed pole- L_2 sensitivity minimization method, an alternative approach of pole-zero sensitivity minimization, may solve the issue of zero multiplicity. A normal-form realization has been proven that it can achieve a global minimum of un-weighted pole sensitivity measure as well as zero-input limit-cycle free property. The sparse normal-form realization can be synthesized from an arbitrary initial realization with distinct poles by using an analytical similarity transformation. Based on the derived fixed-point arithmetic model in state-space realizations, the Bellman-Gronwall Lemma, and normal-form transformation, a new word-length estimation to guarantee stability is derived. Finally, numerical simulations are performed to verify the correctness and the effectiveness of the theoretical results.