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.

中文翻译：

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有限精度实现下稳健且计算效率高的数字 IIR 滤波器合成和稳定性分析

在本文中，一种用于获得最佳无限脉冲响应 (IIR) 状态空间实现的分析综合方法，例如最小极点零和极点$L_2$ 灵敏度实现，基于最小化零/$L_2$提出了基于稀疏范式状态转移矩阵的灵敏度度量。拟议的$n$次阶实现具有很强的鲁棒性，最多只 $4n+1$每个输出样本的乘法，以防止不稳定和输出失真，以及在有限字长 (FWL) 效应下保持计算效率。这对于在定点和便携式数字设备中实现的 IIR 滤波器很重要。在本文中，提出的极点$L_2$灵敏度最小化方法是零极点灵敏度最小化的一种替代方法，可以解决零多重性问题。标准形式的实现已被证明可以实现未加权极点灵敏度测量的全局最小值以及零输入极限循环特性。稀疏范式实现可以通过使用解析相似变换从具有不同极点的任意初始实现合成。基于状态空间实现中导出的定点算术模型、Bellman-Gronwall 引理和范式变换，导出了一种新的字长估计以保证稳定性。最后通过数值模拟验证了理论结果的正确性和有效性。