[Paper Review] Multirate Digital Signal Processing via Sampled-Data H-infinity Optimization

Masaaki Nagahara|arXiv (Cornell University)|Aug 20, 2013

Stability and Control of Uncertain Systems3 references3 citations

TL;DR

This paper proposes a sampled-data H∞ optimization framework for multirate digital signal processing and communication systems, integrating continuous-time signal characteristics and A/D-D/A conversion effects. By modeling the system in a hybrid continuous-discrete domain, it achieves optimal filters that outperform conventional discrete-time designs, especially when signals are not perfectly band-limited, and provides an LMI-based solution for H∞-optimal FIR approximation of IIR filters with improved performance and trade-off control between passband ripple and stopband attenuation.

ABSTRACT

In this thesis, we present a new method for designing multirate signal processing and digital communication systems via sampled-data H-infinity control theory. The difference between our method and conventional ones is in the signal spaces. Conventional designs are executed in the discrete-time domain, while our design takes account of both the discrete-time and the continuous-time signals. Namely, our method can take account of the characteristic of the original analog signal and the influence of the A/D and D/A conversion. While the conventional method often indicates that an ideal digital low-pass filter is preferred, we show that the optimal solution need not be an ideal low-pass when the original analog signal is not completely band-limited. This fact can not be recognized only in the discrete-time domain. Moreover, we consider quantization effects. We discuss the stability and the performance of quantized sampled-data control systems. We justify H-infinity control to reduce distortion caused by the quantizer. Then we apply it to differential pulse code modulation. While the conventional Delta modulator is not optimal and besides not stable, our modulator is stable and optimal with respect to the H-infinity-norm. We also give an LMI (Linear Matrix Inequality) solution to the optimal H-infinity approximation of IIR (Infinite Impulse Response) filters via FIR (Finite Impulse Response) filters. A comparison with the Nehari shuffle is made with a numerical example, and it is observed that the LMI solution generally performs better. Another numerical study also indicates that there is a trade-off between the pass-band and stop-band approximation characteristics.

Motivation & Objective

• To address the limitations of conventional discrete-time multirate filter design, which assumes ideal band-limited signals and ignores analog characteristics and conversion effects.
• To develop a unified framework for designing interpolators, decimators, and sampling rate converters that accounts for both continuous-time and discrete-time signal spaces.
• To analyze and optimize the performance and stability of quantized sampled-data systems, particularly for differential pulse code modulation (DPCM).
• To provide an LMI-based solution for optimal H∞ approximation of IIR filters by FIR filters, improving upon the Nehari shuffle method.

Proposed method

• Uses sampled-data H∞ control theory to model the entire signal chain, including A/D and D/A conversion, as a hybrid continuous-discrete system.
• Applies lifting techniques to transform the continuous-time system into a discrete-time equivalent for efficient computation and optimization.
• Employs linear matrix inequalities (LMIs) to solve the H∞-optimal approximation problem of IIR filters by FIR filters, enabling convex optimization.
• Models quantization as additive noise in a linearized system to analyze stability and performance, ensuring bounded states and low power gain.
• Iteratively designs transmit and receive filters in communication systems via H∞ optimization, ensuring monotonic improvement of the objective function.
• Utilizes the linear fractional transformation (LFT) framework to structure the optimization problem and derive robust solutions.

Experimental results

Research questions

• RQ1 Can multirate signal processing filters be designed more effectively by considering the original analog signal characteristics and A/D-D/A conversion effects, rather than relying solely on discrete-time models?
• RQ2 How can the stability and performance of quantized sampled-data systems be analyzed and optimized, particularly in the context of DPCM?
• RQ3 What is the performance gain of an LMI-based H∞-optimal FIR approximation of IIR filters compared to the classical Nehari shuffle method?
• RQ4 Is there a fundamental trade-off between passband ripple and stopband attenuation in H∞-optimal FIR approximation, and can it be quantitatively characterized?
• RQ5 Can the proposed sampled-data H∞ framework be extended to time-varying or wireless communication channels, and what are the implications for adaptive filter design?

Key findings

• The optimal multirate filter is not necessarily an ideal low-pass filter when the original analog signal is not perfectly band-limited, a fact undetectable in the discrete-time domain alone.
• The proposed DPCM system is both stable and H∞-optimal, unlike conventional Δ modulators, which are unstable and suboptimal under channel noise.
• The LMI-based H∞ approximation of IIR filters by FIR filters outperforms the Nehari shuffle in numerical examples, with lower H∞ and H² error norms.
• A clear trade-off exists between passband ripple and stopband attenuation in the FIR approximation, with W1-based design yielding the best overall H∞ and H² performance despite moderate stopband attenuation.
• The iterative design algorithm for communication systems ensures monotonic decrease of the objective function, indicating convergence to a suboptimal solution.
• The method enables better performance in image processing applications, such as JPEG/MPEG filter banks, by extending the multirate design framework to multidimensional signals.