Approximation Of Digital Filters Research Articles

A new approach for the design of fractional delay by an FIR filter

Fractional delay filters modeling non-integer delays are digital filters which ideally have flat group delays. In this letter, a new, simple, accurate and efficient FIR filter design to implement the digital ideal fractional delay is presented. The design technique is based on the MacLaurin series expansion formula which is applied to a given function to obtain a closed form FIR digital filter approximation of the digital ideal fractional delay operator z−α. The obtained FIR filter approximation formula of the digital ideal fractional delay operator z−α is similar to the well known Grünwald-Letnikov FIR digital filter approximation formula of the ideal analog fractional differentiator sα. Some examples have been presented to illustrate the performance and the effectiveness of this new design method. Some comparisons results show that the proposed design yields better performances than the existing designs in the literature.

Magnitude approximation of digital filters with specified degrees of flatness and constant group delay characteristics

New techniques to approximate the magnitude response of infinite-impulse response digital filters with maximally flat or equiripple delay characteristics are presented. A polynomial with maximally flat or equiripple group delay characteristics is chosen to be the denominator of the filter transfer function. The coefficients of the numerator polynomial are found such that the group delay characteristic is unaffected, and the magnitude response either has desired degrees of flatness at /spl omega/=0 and/or /spl omega/=/spl pi/ or is flat in the passband and equiripple in the stopband. The magnitude response can also be matched to prescribed values at a number of frequencies between /spl omega/=0 and /spl omega/=/spl pi/. In the first case, the solution for the coefficients of the numerator polynomial is shown to be purely analytical. In the second case, the well-known Remez Exchange Algorithm is used to obtain the desired passband and stopband magnitude characteristics simultaneously. It is also shown that the proposed design methods can be used to obtain linear phase finite-impulse response filters with a magnitude response that is flat in the passband and flat or equiripple in the stopband.

Approximation of multivariable linear systems with impulse response and autocorrelation sequences

This paper considers the construction of approximants of multi-input-multi-output, discrete-time linear systems from the finite data of the impulse response and autocorrelation sequences. In the approximation of a multivariable linear system, it is common practice to use a finite portion of its impulse response sequence. This is formally equivalent to the Padé approximation technique, which may produce unstable approximants, even though the original system is stable. Mullis and Roberts proposed a new method, which yields stable approximants, in connection with approximation of digital filters. This is, however, restricted to the single-input-single-output case. This paper extends their method to the multi-input-multi-output case and shows a fast recursive algorithm to construct stable approximants of linear systems.

A method of statistical design of ARMA group‐delay filters

AbstractThis paper proposes group‐delay response approximation method based on the ARMA filter, an extension of digital filter approximation by statistical design. Since the denominator and numerator coefficient sequences are in opposite order in recursive group‐delay filters, obtaining either coefficient sequence leads to the other. the approximation described here replaces group‐delay response with the magnitude‐squared response via the complex cepstrum. It is then approximated by an auto‐regressive (AR) filter to obtain the denominator coefficient sequence by linear prediction; reversing this sequence gives the numerator coefficient sequence. A design procedure using the FFT and Levinson‐Durbin algorithm gives a solution much faster than the previous method of solution by means of iterated calculations, in addition to which it is easy to assure that the approximation is stable. A coefficient series allowing realization of a low‐coefficient‐sensitivity recursive structure is directly obtained in the design procedure. Examples of group‐delay equalization and results of radar signal simulation are presented as applications of the approximation method. Finally, some additional problems are considered in connection with the difficulty of obtaining a high‐order approximation with large group‐delay response.

Approximation of Multivariable Linear Systems with Impulse Response and Autocorrelatlon Sequences

This paper considers the construction of approximants of multi-input-output, discrete-time linear systems from the finite data of the impluse response and autocorrelation sequences (or cross correlation between input and output, and autocorrelation of output). In the approximation of multivariable linear systems, it is common practice to use the impluse response sequenceThis is formally equivalent to the Padé approximation technique, which may produce unstable approximants, even though the original system is stable. Mullis and Roberts proposed a new method, which yields stable appr0ximants, in connection with the approximation of digital filters. This is, however, restricted to the single input-output case. This paper extends their method to the multi-input-output case and shows a fast recursive algorithm to construct stable approximants of linear systems

Uniform rational approximation by differential correction and remes‐differential correlation

AbstractWe present two self‐contained programs which have given excellent results for uniform approximation of functions (or discrete data) by generalized rational functions. The first program is an extension of a program published in 1975 which uses a linear programming approach known as the differential correction algorithm. The present version is more robust, and allows for the use of a multiplicative weight function and restrictions on the values of the approximating function. These features make the program more suitable for applications such as digital filter approximation. The second program uses a combination of the Remes and differential correction algorithms which combines some of the good features of both algorithms. Given here is a discussion of the algorithms together with several examples. A FORTRAN listing can be obtained from the first author.

Approximation of digital filters in one and two dimensions

An algorithm for the time domain approximation of discrete systems with a recursion is described. The algorithm iterates towards a solution minimizing the sum of squared differences between the desired and the actual output. Convergence is guaranteed. The scheme is applied to the design of low-pass filters by time domain approximation. The results compare well with other design strategies. We have extended the algorithm to two dimensions. This algorithm is essentially the same iterative scheme used for the one-dimensional case. The two-dimensional iteration achieves better results than the currently used two-dimensional filter synthesis techniques since the starting point of the iteration is the solution of the latter approach. Usually, a few iterations suffice to improve the solution in a satisfactory amount. Convergence is also guaranteed. An example of two-dimensional impulse response approximation is also given as an illustration.

Approximation theory for digital filters

It is shown that most of the techniques for analog filter design presented by Watanabe [7] and others who have specialized or extended Watanabe's work are applicable, without the usual restrictions (due to undesirablility of mutual coupling, for example), to digital filter approximation problems when the bilinear transformation is employed.

Lowpass digital filters using least-squared-error design

A method is presented of designing digital filters which give a least-squared-error fit to a polynomial expression. An example is given of a second-order digital-filter approximation to a first-order continuous filter.

REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*

AbstractSharp cut‐off frequency filtering is carried out in the discrete time domain on digital computers. A convolution of the digital filter impulse response with the sampled input yields the output. For practical reasons, the length of the filter inpulse response, corresponding to the number of filter coefficients, is limited, and consequently the resulting frequency characteristic will no longer be identical to that originally specified. This is analogous to synthesising some specified frequency characteristic with a finite number of resistive, capacitative and inductive components.In Part I of this paper, we examine the effect of approximating the sharp cut‐off frequency characteristic best in a mean square sense by an impulse response of finite length. The resulting frequency characteristic corresponds to the truncated impulse response of the specified frequency characteristic. It has a cut‐off slope proportional to, and a mean square error inversely proportional to, the length of the impulse response, and is a biassed odd function about the cut‐off frequency point. Because of the Gibbs phenomenon for discontinuous functions, the resulting frequency characteristic will always have a maximum overshoot with respect to the specified characteristic of ± 9%, regardless of the length of the corresponding impulse response. Equal length truncated impulse responses of specified filters with different cut‐off frequencies yield frequency characteristics which are almost identical about their respective cut‐off points. Now on a log frequency scale (as against a linear frequency scale implied previously) such characteristics may be made almost identical about the respective cut‐off points by having the truncated impulse responses composed of an equal number of zero crossings. Results for the low‐pass filter are applicable to the high‐pass and band‐pass characteristics.In the latter case, the mean square error is double that for a single slope characteristic (low‐pass or high‐pass) and the slopes at both edges of the passband are approximately equal in magnitude to the length of the impulse response (linear frequency scale).Part II of this paper is concerned with reducing the ± 9% overshoot that results from the discontinuous nature of the sharp cut‐off frequency characteristic and which is not dependent on the length of the truncated impulse response. The reduction is achieved, at the expense of the steepness of cut‐off for the resulting frequency characteristic, by the use of functions which weight the truncated impulse response of the specified frequency characteristic. These functions are called apodising functions. Among other variables, the length of the truncated weighted impulse response will determine the amount of maximum overshoot since the effective frequency characteristic being approximated is no longer a discontinuous function. The digital realization of the finite length impulse responses of Parts I and II is discussed in Part III, together with the optimum partially specified digital filter approximation to the desired frequency characteristic.

REALIZATION OF SHARP CUT‐OFF FREQUENCY CHARACTERISTICS ON DIGITAL COMPUTERS*

AbstractIn Part I of this paper, we examined the properties of the best mean square approximation to the sharp cut‐off frequency characteristic by an impulse response of finite length. It was found that the sharpness of cut‐off for the resulting frequency characteristic depended on the length of the impulse response–but because of the discontinuous nature of the specified frequency characteristic, this best mean square approximation always had a maximum overshoot of ± 9%, independent of the length of the impulse response (Gibbs phenomenon).In Part II, we investigate ways of reducing this ± 9% overshoot at the expense of a reduced sharpness of cut‐off. The discontinuous frequency characteristic is first approximated by a continuous characteristic with linear or cosine frequency tapering. The impulse response for such tapered characteristics consists of the impulse response of the discontinuous frequency characteristic weighted by a certain function corresponding to the type of tapering employed. The best mean square approximation to the tapered characteristic by an impulse response of finite length M will produce a frequency characteristic whose properties are now dependent on the time‐band width product Mζ, where 2ζ is the tapering range.A trade‐off exists between the maximum overshoot and the sharpness of cut‐off for the resulting characteristic for both forms of frequency tapering. Instead of considering other forms of tapering in the frequency domain, we now investigate arbitrarily chosen weighting functions in the time domain to determine the minimum length of impulse response for a minimum value of maximum overshoot and a maximum value of sharpness of cut‐off.Part III will discuss the digital realization of the above finite length impulse responses together with the optimum partially specified digital filter approximation to the desired frequency characteristic.