Yong Ching Lim

A weighted least squares algorithm for quasi-equiripple FIR and IIR digital filter design

It has been demonstrated by several authors that if a suitable frequency response weighting function is used in the design of a finite impulse response (FIR) filter, the weighted least squares solution is equiripple. The crux of the problem lies in the determination of the necessary least squares frequency response weighting function. A novel iterative algorithm for deriving the least squares frequency response weighting function which will produce a quasi-equiripple design is presented. The algorithm converges very rapidly. It typically produces a design which is only about 1 dB away from the minimax optimum solution in two iterations and converges to within 0.1 dB in six iterations. Convergence speed is independent of the order of the filter. It can be used to design filters with arbitrarily prescribed phase and amplitude response. >

Discrete coefficient FIR digital filter design based upon an LMS criteria

An efficient method optimizing (in the least square response error sense) the remaining unquantized coefficients of a FIR linear phase digital filter when one or more of the filter coefficients takes on discrete values is introduced. By incorporating this optimization method into a tree search algorithm and employing a suitable branching policy, an efficient algorithm for the design of high-order discrete coefficient FIR filters is produced. This approach can also be used to design FIR filters on a minimax basis. The minimax criterion is approximated by adjusting the least squares weighting. Results show that the least square criteria is capable of designing filters of order well beyond other approaches by a factor of three for the same computer time. The discrete coefficient spaces discussed include the evenly distributed finite wordlength space as well as the nonuniformly distributed powers-of-two space.

Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude

Four methods are presented for optimizing filters in the normalized peak ripple magnitude (NPRM) sense. Two of the methods being to the passband gain sectioning technique. The other two methods make use of the objective function f= delta - alpha b. Several heuristic methods for determining alpha are also presented. The NPRM is an important performance measure; the absolute peak ripple magnitude and passband gain are less important. In these applications, the passband gain need not be fixed at unity but should be a continuous variable to the optimized. Nevertheless, an upper and a lower bound on the passband gain should be imposed to satisfy overflow and roundoff noise performance requirements. >

Signed power-of-two term allocation scheme for the design of digital filters

It is well known that if each coefficient value of a digital filter is a sum of signed power-of-two (SPT) terms, the filter can be implemented without using multipliers. In the past decade, several methods have been developed for the design of filters whose coefficient values are sums of SPT terms. Most of these methods are for the design of filters where all the coefficient values have the same number of SPT terms. It has also been demonstrated recently that significant advantage can be achieved if the coefficient values are allocated with different number of SPT terms while keeping the total number of SPT terms for the filter fixed. In this paper, we present a new method for allocating the number of SPT terms to each coefficient value. In our method, the number of SPT terms allocated to a coefficient is determined by the statistical quantization step-size of that coefficient and the sensitivity of the frequency response of the filter to that coefficient. After the assignment of the SPT terms, an integer-programming algorithm is used to optimize the coefficient values. Our technique yields excellent results but does not guarantee optimum assignment of SPT terms. Nevertheless, for any particular assignment of SPT terms, the result obtained is optimum with respect to that assignment.

Design of cascade form FIR filters with discrete valued coefficients

The authors show that by cascading two direct-form FIR filters, each with coefficients that are the sum or difference of two power-of-two terms, it is possible to achieve very small peak ripple. An iterative equalization strategy is used in the design of the cascaded filter. The success of the method depends on the initial prototype filter being used. Two equally effective methods are presented for selecting the prototype filter; each yields a final design with good roundoff noise property. >

Frequency-response masking approach for digital filter design: complexity reduction via masking filter factorization

It has been reported in several recent publications that the frequency response masking technique is eminently suitable for synthesizing filters with very narrow transition-width. The major advantages of the frequency response masking approach are that the resulting filter has a very sparse coefficient vector and that the resulting filter length is only slightly longer than that of the theoretical (Remez) minimum. The system of filters produced by the frequency response masking technique consists of a sparse coefficient filter with periodic frequency response and one or more pairs of masking filters. Each pair of the masking filters consist of two filters whose frequency responses are similar except at frequencies near the band-edges. In this paper, we present three methods for reducing the complexity of the masking filters. The success of our technique is due to the fact that each pair of the masking filters can be realized as a cascade of a common subfilter and a pair of equalizers. >

Design of Linear Phase FIR Filters in Subexpression Space Using Mixed Integer Linear Programming

In this paper, a novel optimization technique is proposed to optimize filter coefficients of linear phase finite-impulse response (FIR) filter to share common subexpressions within and among coefficients. Existing approaches of common subexpression elimination optimize digital filters in two stages: first, an FIR filter is designed in a discrete space such as finite wordlength space or signed power-of-two (SPT) space to meet a given specification; in the second stage, an optimization algorithm is applied on the discrete coefficients to find and eliminate the common subexpressions. Such a two-stage optimization technique suffers from the problem that the search space in the second stage is limited by the finite wordlength or SPT coefficients obtained in the first stage optimization. The new proposed algorithm overcomes this problem by optimizing the filter coefficients directly in subexpression space for a given specification. Numerical examples of benchmark filters show that the required number of adders obtained using the proposed algorithm is much less than those obtained using two-stage optimization approaches.

A polynomial-time algorithm for designing FIR filters with power-of-two coefficients

This paper presents a polynomial-time algorithm for designing digital filters with coefficients expressible as sums of signed power-of-two (SPT) terms. Our proposal is based on an observation that under certain circumstances, the realization cost of a filter with SPT coefficients depends only on the total number of SPT terms, regardless of how the terms distribute among the coefficients. Therefore, the number of SPT terms for each coefficient is not necessarily limited to a fixed number. Instead, they should be allowed to vary subject to a given number of total SPT terms for the filter. This provides the possibility of finding a better set of coefficients. Our algorithm starts with initializing all the quantized coefficient values to zero. It chooses one SPT term at a time and allocates it to the currently most deserving coefficient to minimize the L/sup /spl infin// distance between the SPT coefficients and their corresponding infinite wordlength values. This process of allocating the SPT terms is repeated until the total number of SPT terms for the filter is equal to a prescribed number. For each filter gain, the time complexity is a second-order polynomial in the number of coefficients to be optimized and is a first-order polynomial in the filter wordlength.

Fast filter bank (FFB)

The sliding fast Fourier transform (FFT) filter bank has an exceedingly low complexity of one multiplication per channel per sample. However, its frequency selectivity and passband response are poor. It is shown that the sliding FFT filter bank is in fact a particular member of a new family of fast filter banks (FFBs). In the case of FFT, each cluster of butterflies can in fact be derived from a pair of complementary two-tap (i.e. first-order) prototype FIR filters. The poor selectivity and degraded passband response of the FFT filter bank is a direct consequence of the poor frequency response of the prototype first-order filter. It is shown that by increasing the order of the prototype filters, it is possible to implement a filter bank with arbitrarily good selectivity and flat passband response. The FFB retains the low-complexity feature of the FFT. Because of its very much improved frequency response characteristics, the FFB be suitable for use in many applications where the FFT filter bank is unsuitable. >

A polynomial-time algorithm for designing digital filters with power-of-two coefficients

An algorithm is presented for designing digital filters with coefficients expressible as sums of signed power-of-two (SPT) terms. For each filter gain, the time complexity of the algorithm is a second-order polynomial in the filter order and is a first-order polynomial in the filter wordlength. Unlike conventional methods where each coefficient is allocated a fixed number of SPT terms, the authors method allows the number of SPT terms for each coefficient to vary subject to the number of SPT terms for the entire filter. This provides the possibility of finding a better filter without increasing the number of adders, which determines the realization cost for a given filter length. Application of the algorithm to finite impulse response (FIR) filter designs shows that it achieves up to 8.9 dB improvement over simulated annealing and mixed integer linear programing on the normalized peak ripples of example filters.<<ETX>>