CONSTRAINT TWO-PATH POLYPHASE IIR FILTER DESIGN USING DOWNHILL SIMPLEX ALGORITHM Dr. Artur Krukowski (krukowa@wmin.ac.uk) and Dr. Izzet Kale (kalei@wmin.ac.uk) University of Westminster, Applied DSP and VLSI Research Group, 115 New Cavendish Street, London W1W 6UW, UK. ABSTRACT The paper addresses the design of the polyphase IIR filters based on the N th -order single-coefficient allpass sub-filters in the constraint coefficient space using the Constraint Downhill Simplex Algorithm. Incorporating the bit-flipping algorithm into its core engine allowed the optimisation routine to converge to better target designs without affecting the high speed of the original algorithm. Establishing the boundaries of the search space required by the Downhill Simplex Algorithm for the two- path polyphase IIR filter is also presented. 1. INTRODUCTION Fractional band filters based on the polyphase IIR structure as given in [1][2][3] characterizes with much lower calculation budget than other ones for the same specification. Their structure is a parallel combination of digital allpass recursive filters, which is very attractive for silicon implementation. As the coefficients control only the phase of allpass filters, the overall structure is much less sensitive to coefficient inaccuracy than other standard filter implementations. This allows shortening the coefficient lengths to, in many cases, few bits only without much damage to the filter response. All this makes the polyphase filter very suitable for fixed-point arithmetic. However, in general finding the correct combination of minimum word-length coefficients to meet a given specification in the lowest order is not a trivial task. Direct floating-point design algorithms and techniques for this class of filters, employing an elliptic approximation and reported in [2], [3] cannot be used in their original form to calculate the constraint filter coefficients. An appropriate optimisation algorithm needs to be used. Methods based on derivatives do not work very well due to significant errors in calculating derivatives especially when the bit length is small. Therefore the idea is to use some type of the binary search algorithm [4][5] or genetic algorithm [6]. Most of algorithms design coefficients in Canonic Signed-Digit (CSD) form to minimize the shift-add operations, but do not take much care of the coefficient length leading to increase in arithmetic wordlength and more costly implementation. The proposed algorithm is intended to design signed-binary coefficients. If it is required the binary coefficients can be easily converted o the CSD form with the number of bits set less or equal the total coefficient bit wordlength. It should be noted that the convergence of many combinatorial algorithms, like the suggested here Downhill Simplex method [7], is sensitive to initial conditions implying that if the seed filter coefficients and search space is chosen with care, then the algorithm will converge more efficiently. For the polyphase IIR structure the desirable search space contains coefficient sets for which all the filter zeros are on the unit circle. Such a condition ensures that the filter has the highest stopband attenuation. This ensures also the best passband performance, as passband ripples are in direct relationship to the stopband ripples [1][2][3]. The trade-off between the stopband depth and the transition band is also most effectively achieved when the zeroes are on the unit circle. We have established that the space of coefficients for which filter zeros are on the unit circle is self contained and unitary, which simplifies the optimisation task of the search algorithm [8]. 2. POLYPHASE TWO-PATH FILTER The basic numerator-denominator form recursive (IIR) all-pass sections, when deployed in a polyphase two-path configuration [1] with a unit delay in one of the branches and having the appropriate coefficients result in a very high performance and relatively easily implementable half-band filter: ( ) Hz a z a z z a z a z i i i K i i i K odd even - + - + - = - - - - = - = + + + + +         ∏ ∏ 1 2 1 2 2 1 2 0 1 1 2 2 2 2 0 1 1 2 1 1 (1) The phases of the allpass filters are designed such that they are very close to each other for frequencies below half-Nyquist and displaced by approximately π above it. With the appropriate coefficients all the zeros are on the left real plane on the unit circle, resulting in very high stopband attenuation and a very flat passband. The magnitude response ripples in the passband PR are related to the stopband attenuation A by PR=A 2 /2 [1][2], thus allowing the optimisation to concentrate on minimizing only one of these parameters. The transition band can be flexibly adjusted between 0.5 and almost zero. 2.1 Finding coefficients for given zero locations Determining the coefficient values from its zero locations is especially useful for specifying the bounded area of filter coefficients for which the zeros are on the unit circle [8]. This information can be used to define the starting point for such floating-point filter design algorithms as Powell or Simulated Annealing as well as by any constraint filter design algorithm like bit-flipping [9] or constraint Downhill Simplex in which the design is based on the search within a bounded space. It is also extremely useful for design algorithms based on filter pole and zero movement. Analytic methods of finding the area of coefficient values for which the polyphase filter zeros are on the unit circle, thus allowing one to achieve the maximum attenuation for a given filter order, proved not to be practical for more than three coefficients [8]. The equation manipulations involved made such an approach computationally impractical and very difficult to solve analytically. Alternatively, the required range of coefficients can be found by a direct numerical method.