Two Polynomial FIR Filter Structures With Variable Fractional Delay and Phase Shift

This paper introduces two polynomial finite-length impulse response (FIR) digital filter structures with simultaneously variable fractional delay (VFD) and phase shift (VPS). The structures are reconfigurable (adaptable) online without redesign and do not exhibit transients when the VFD and VPS parameters are altered. The structures can be viewed as generalizations of VFD structures in the sense that they offer a VPS in addition to the regular VFD. The overall filters are composed of a number of fixed subfilters and a few variable multipliers whose values are determined by the desired FD and PS values. A systematic design algorithm, based on iteratively reweighted ℓ₁-norm minimization, is proposed. It generates fixed subfilters with many zero-valued coefficients, typically located in the impulse response tails. The paper considers two different structures, referred to as the basic structure and common-subfilters structure, and compares these proposals as well as the existing cascaded VFD and VPS structures, in terms of arithmetic complexity, delay, memory cost, and transients. In general, the common-subfilters structure is superior when all of these aspects are taken into account. Further, the paper shows and exemplifies that the VFDPS filters under consideration can be used for simultaneous resampling and frequency shift of signals.

I. Introduction

Variable (reconfigurable) digital signal processing (DSP) algorithms are gaining an increasing interest with the increasing demand for reconfigurable systems. For example, variable DSP algorithms are required in emerging communication systems supporting different standards and operation modes [1]–[5]. This paper deals with finite-length impulse response (FIR) digital filters having both a variable fractional delay (VFD), say , and a variable phase shift (VPS),¹

A PS filter is also referred to as a fractional Hilbert transformer [6], [7].

say . We refer to such filters as variable fractional delay and phase shift (VFDPS) filters. As will be shown, VFDPS filters find applications in simultaneous resampling (sampling rate conversion) and frequency shift of signals, hereafter abbreviated SRFS. The resampling corresponds to VFD filtering whereas the frequency shift corresponds to VPS filtering. It is well known that VFD filters can be used to efficiently implement resampling alone, especially when regular integer- or rational-factor converters cannot be used or when they imply an unacceptably high computational complexity [8], [9]. This paper thus considers the extension to SRFS which requires VFDPS filters instead of VFD filters.