Almost Linear-Phase Polyphase IIR Lowpass/Highpass Filter Approach Artur Krukowski and Izzet Kale University of Westminster School of Electronic and Manufacturing System Engineering 115 New Cavendish Street, London W1M 8JS, UK. E-mail: krukowa@wmin.ac.uk, kalei@wmin.ac.uk 1. ABSTRACT The direct design of a linear phase FIR one allows perfectly linear phase, but at the expense of the high filter order, N, and consecuently high computational burden for the given specification. IIR filters, however, can be designed with much smaller orders than their FIR couterparts, but at the expense of the non-linear phase. It is, however, possible to design almost linear- phase IIR filters for a given specification of the allowed phase or group delay ripples. There are few approaches for the design of such filters [1], [2]. This paper provides an improvement on the existing algorithms and gives a different viewpoint on these design methods showing a considerable improvement in both stopband attenuation (as well as passband ripples) and maximum group delay ripples over the methods reported in [1] and [2]. 2. INTRODUCTION Ensuring the same time differences between signal spectrum components before and after the filtering is very important, especially for wide-band signals. This requirement is met if phase response of the filter is linear and it has no constant factor. Then, the group delay function is constant. Such a condition is enough to assure the linearity of the digital filter phase response. One of the ways of designing approximately linear- phase IIR filters is to use a two-path polyphase structure [3], [4]. This standard structure can be modified by replacing the allpass filter in the lower branch with a bulk delayor, z -K , where K is the order of the allpass filter in the top branch as in Figure 1. The lowpass polyphase structure works on the principle of the allpass filters from both branches being in-phase at the low frequencies and distant by π at the frequencies close to Nyquist. Having a bulk delayor, z -K , in the lower branch, the allpass filter, A(z), has to be designed to follow the linear phase response of the delayor to achieve the lowpass characteristics. A(z -1 ) z -K Σ 0.5 x(z) y(z) +/- Figure 1. The structure of an approximately linear- phase polyphase two-path filter. The idea as such is not new and was suggested and used by Curtis [6] and employed in a design routine recently published by Lawson [1] and Lu [2]. The current design methods are based on the standard idea of composing two identical IIR (non-linear phase) filters to achieve an approximately linear-phase characteristic [1] or applying iterative quadrature programming methods [2]. Such an approach does not allow much flexibility, limiting the number of points of freedom to half of what would be available when standard IIR filters are used. Thus the resulting filters posses larger stopband ripples as well as larger group delay ripples than what the structure is really capable of achieving. The additional advantage of the presented structure is that it is capable of providing a complimentary set of lowpass and highpass filters, simply by replacing the adder with a subtractor [1], [3], [4]. This feature is very attractive for applying such filters in almost linear-phase IIR quadrature filter banks. 3. THE DESIGN ALGORITHM Our design routine uses the Matlab discrete filter least squares fit to the frequency response data (‘invfreqz’) to approximate the phase of the filter in the passband to be z -K and z -K-0.5 in the filter stopband.