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

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.