Optimization of recursive digital filters with finite word lengths
01 Dec 1974-IEEE Transactions on Acoustics, Speech, and Signal Processing (IEEE)-Vol. 22, Iss: 6, pp 424-431
TL;DR: The application of the "branch and bound" technique for nonlinear discrete optimization, due to Dakin, to the problem of finding the coefficients of a recursive digital filter with prescribed number of bits, to meet arbitrary response specifications of the magnitude characteristic is investigated.
Abstract: The application of the "branch and bound" technique for nonlinear discrete optimization, due to Dakin, to the problem of finding the coefficients of a recursive digital filter with prescribed number of bits, to meet arbitrary response specifications of the magnitude characteristic, is investigated. Due to the fact that the objective function is nonlinear and the stability constraints are linear with respect to the parameter, the recent algorithm for nonlinear programming due to Best and Ritter is used. Based on the ideas presented, a general computer program has been developed. Numerical experience with the present approach is also presented.
Citations
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TL;DR: In this article, the remaining unquantized coefficients of a FIR linear phase digital filter when one or more of the filter coefficients takes on discrete values are optimized using the least square response error.
Abstract: 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.
240 citations
TL;DR: The application of a general-purpose integer-programming computer program to the design of optimal finite wordlength FIR digital filters is described and an analysis of the approach based on the results of more than 50 design cases is presented.
Abstract: The application of a general-purpose integer-programming computer program to the design of optimal finite wordlength FIR digital filters is described. Examples of two optimal low-pass FIR finite wordlength filters are given and the results are compared with the results obtained by rounding the infinite wordlength coefficients. An analysis of the approach based on the results of more than 50 design cases is presented and the problem of optimal wordlength choice is discussed.
164 citations
TL;DR: In this article, the sensitivities of the transfer function of a digital filter with respect to its coefficients are used to derive lower bounds on the roundoff noise output in the cases of L ∞ and L 1 ∞ scaling for fixed-point arithmetic.
Abstract: The sensitivities of the transfer function of a digital filter with respect to its coefficients are utilized to derive lower bounds on the roundoff noise output in the cases of L_{\infty} and L_{\infty} scaling for fixed-point arithmetic. General bounds are produced which apply to any filter structure if rounding is performed after multiplication and the filter has already been scaled. For the parallel and cascade forms, alternate bounds are derived which apply to rounding after multiplication or summation and which do not require prior scaling. The alternate bounds arethus independent (or nearly so) of pairing, ordering, and transposition. Examples are presented which show that the bounds are reasonably tight.
83 citations
TL;DR: This paper presents a detailed review of the basic design approaches applicable for the synthesis of hardware efficient finite duration impulse response (FIR) filter and the traditional and heuristic search algorithms have been incorporated and properly arranged.
Abstract: Digital signal processing (DSP) is one of the most powerful technologies which will shape the science, engineering and technology of the twenty-first century Since 1970, revolutionary changes took place in the broad area of DSP which has made it an essential tool in many engineering applications Digital filter is considered to be one of the most important components of almost every DSP sub-systems and therefore a number of extensive works had been carried out by researchers on the design of such filter In order to meet the stringent requirements of filter specification, order of the designed filter is generally assumed to be very large and this leads to high power and area consumption during their implementation As a matter of fact, design of hardware efficient digital filter has drawn enormous attention which needs to be addressed by various useful means One popular approach has been to encode the tap coefficients of such filter in the form of sum of signed powers-of-two and thus the operation of multiplication is substituted by simple addition and shifting This paper presents a detailed review of the basic design approaches applicable for the synthesis of hardware efficient finite duration impulse response (FIR) filter Both the traditional and heuristic search algorithms have been incorporated and properly arranged in this review
64 citations
TL;DR: In this paper, a brief overview of that part of the area designated in the title which is well within the traditional scope of these TRANSACTIONS is presented, i.e., circuits and systems whose basic description is by signal flow graphs and which can be implemented by digital means.
Abstract: This paper is devoted to a brief overview of that part of the area designated in the title which is well within the traditional scope of these TRANSACTIONS, i.e., circuits and systems whose basic description is by signal flow graphs and which can be implemented by digital means. Topics of general and fundamental interest are stressed and important particularities sometimes neglected are included. Main attention is focused on a variety of aspects of constant one-dimensional digital filters. The extension to the multidimensional case is briefly considered, and some further areas are touched upon.
56 citations
References
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618 citations
TL;DR: In this article, the problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem using the minimum p -error criterion, which is successfully solved using the Fletcher-Powell algorithm.
Abstract: The problem of designing a stable recursive digital filter to have an arbitrarily prescribed frequency response may be considered as an approximation problem. Using the minimum p - error criterion, a new problem of minimizing a function of n variables results, which is successfully solved using the Fletcher-Powell algorithm. An important theorem guaranteeing the existence of a stable optimum for a large class of synthesis problems is stated, and necessary modifications to the Fletcher-Powell algorithm to assure stability are considered. Finally a number of results of the application of this method are given.
265 citations
TL;DR: This article describes the theoretical validation of Carroll's proposal for the convex programming problem and describes the computational experience with a newly developed algorithm that makes the technique competitive with known methodology.
Abstract: This article is based on an idea proposed by C. W. Carroll for transforming a mathematical programming problem into a sequence of unconstrained minimization problems. It describes the theoretical validation of Carroll's proposal for the convex programming problem. A number of important new results are derived that were not originally envisaged: The method generates primal-feasible and dual-feasible points, the primal objective is monotonically decreased, and a subproblem of the original programming problem is solved with each unconstrained minimization. Briefly surveyed is computational experience with a newly developed algorithm that makes the technique competitive with known methodology. A subsequent article describing the computational algorithm is in preparation.
248 citations
TL;DR: In this paper, a method for designing recursive digital filters with arbitrary, prescribed magnitude characteristics is described, which uses the Fletcher-Powell optimization algorithm to minimize a square-error criterion in the frequency domain.
Abstract: A practical method is described for designing recursive digital filters with arbitrary, prescribed magnitude characteristics. The method uses the Fletcher-Powell optimization algorithm to minimize a square-error criterion in the frequency domain. A strategy is described whereby stability and minimum-phase constraints are observed, while still using the unconstrained optimization algorithm. The cascade canonic form is used, so that the resultant filters can be realized accurately and simply. Design examples are given of low-pass, wide-band differentiator, linear discriminator, and vowel formant filters.
145 citations
TL;DR: Two methods for reducing the necessary word length of a digital filter by choosing a suitable structure for the filter and taking selective filters as a model will be presented.
Abstract: The cost of a digital filter, if implemented as a special-purpose computer, depends heavily on the word length of the coefficients. Therefore, it should be reduced as much as possible. On the other hand, a small word length causes large coefficient deviations that impair the wanted performance of the digital filter. The necessary word length may be reduced by choosing a suitable structure for the filter. Two methods for doing this will be presented, taking selective filters as a model. A further reduction of the word length may be won by optimizing the rounded filter coefficients in the discrete parameter space. A description of a modified univariate search will be given.
104 citations