A technique to truncate IIR filter impulse response and its application to real-time implementation of linear-phase IIR filters

Abstract

A technique for realizing linear-phase infinite impulse response (IIR) filters has been proposed by Powell and Chau and gives a real-time implementation of ( 1 ) ( ), where ( ) is a causal IIR filter function. In their system, the input signal is divided into -sample sections, time-reversed, section convolved with ( ), and time-reversed again. The signal is then filtered by ( ) to give the system output with a processing delay of 3 + 1 samples. However, the group delay response of the system exhibits a minor sinusoidal variation superimposed on some constant value. This variation will degrade image quality in image processing and signal quality in signal transmission applications. Furthermore, the output of the system contains harmonic distortion for a sinusoidal input. The main drawbacks of Powell and Chau's technique are the large processing delay of 3 + 1 samples and the accompanying phase and harmonic distortions. A smaller processing delay increases the phase and harmonic distortions, yet the amplitude response remains acceptable. Previously, the present authors presented a method of reducing the processing delay by shortening the section length by an integer factor using a structure with increased number of paths for the time-reversed signal. In this paper, the authors consider how to reduce the phase and harmonic distortions. We examined the operation of the sectioned convolution and analyzed it based on a state-space representation. Then, we found that the cause of the distortions is a periodic variation of the impulse response length in the sectioned convolution. To overcome this problem, a technique is devised to realize a recursive circuit having a truncated impulse response with a fixed-length . A system applying this technique to the Powell-Chau system is demonstrated to exhibit perfect linear-phase characteristic and produce virtually no harmonic distortion. Therefore, the section length can be reduced without limitation due to phase and harmonic distortions. Two methods for reducing the increased computational complexity of this technique assuming fixed are developed, and simulations are performed for the proposed system to confirm the expected improvements.

All About Audio Equalization: Solutions and Frontiers

“…Willson and Orchard described a variation on this approach that yields higher performance (more stopband loss, less passband ripple and/or narrower transition band) [116]. Kurosu et al described performance issues in Powell and Chau's original design; a sinusoidal variation in the group delay, and a harmonic distortion with sinusoidal input [117]. To alleviate these issues, they reduce the filter's overall processing delay by using shorter sections with truncated impulse response [117].…”

“…Kurosu et al described performance issues in Powell and Chau's original design; a sinusoidal variation in the group delay, and a harmonic distortion with sinusoidal input [117]. To alleviate these issues, they reduce the filter's overall processing delay by using shorter sections with truncated impulse response [117]. Azizi proposed an efficient arbitrary sample rate converter using zero phase IIR filtering, which later led to a patented signal interpolator [118].…”

Linear Phase Low-Pass IIR Digital Differentiators

“…Another famous approach is the Powell and Chau approach, which implements a linear phase IIR filter as a tandem connection of an arbitrary transfer function and a time-reversal version of the same function [14]- [16]. Maeng and Lee [15], state in their concluding remarks that the fundamental limitation of the approach's "LIFO based implementation is in the group delay…the application of this method could be limited in real-time interactive signal processing.…”

“…The gain that minimizes is given by (16) The Fletcher and Powell method used in the iteration is described as follows.…”

Design of Exactly Linear Phase $K$-Regular IIR Half-Band Filter

“…Although exactly linear phase IIR filters are either unstable or noncausal, low complexity techniques for handling noncausality in block-based applications are well documented (see, e.g., [8]- [10]). They make IIR filters with exactly linear phase practicable in various applications, especially in image processing [11]- [13].…”