A framework using semidefinite programming is proposed to enable the design of sigma delta modulator loop filters at the transfer function level. Both continuous-time and discrete-time, low-pass and band-pass designs are supported. For performance, we use the recently popularized Generalized Kalman–Yakubovič–Popov (GKYP) lemma to place constraints on the $$\mathcal {H}_\infty $$ H ∞ norm of the noise transfer function (NTF) in the frequency band of interest. We expand the approach to incorporate common stability criteria in the form of $$\mathcal {H}_2$$ H 2 and $$\ell _1$$ ℓ 1 norm NTF constraints. Furthering the discussion of stability, we introduce techniques from control systems to improve the robustness of the feedback system over a range of quantizer gains. The performance-stability trade-off is examined using this framework and motivated by simulation results.

中文翻译：

---

具有改进稳定性特性的噪声整形环路滤波器设计的优化框架

提出了一种使用半定规划的框架，以实现在传递函数级别设计 Sigma Delta 调制器环路滤波器。支持连续时间和离散时间、低通和带通设计。为了性能，我们使用最近流行的广义 Kalman–Yakubovič–Popov (GKYP) 引理对频带中噪声传递函数 (NTF) 的 $$\mathcal {H}_\infty $$ H ∞ 范数施加约束出于兴趣。我们扩展了该方法，以 $$\mathcal {H}_2$$ H 2 和 $$\ell _1$$ ℓ 1 范数 NTF 约束的形式合并通用稳定性标准。进一步讨论稳定性，我们引入了来自控制系统的技术，以提高反馈系统在量化器增益范围内的鲁棒性。