Linear-phase Filters Research Articles

Efficient structurally symmetric algorithms for least squares FIR filters with linear phase

Two novel least-squares algorithms are derived for the computation of the FIR coefficients of multichannel filters with linear (affine) phase. One is for block processing and requires 2 Np+4.25 p/sup 2/ additions and 2 Np+3p/sup 2/ multiplications, with N being the number of input data samples and p the order of the filter. The second algorithm computes the linear (affine) phase filter's coefficients in an adaptive manner with a computational complexity of 8.5 p additions and 6.5 p multiplications per time recursion. Thus, a reduction in computational complexity of the order of 40-50% is obtained compared to previously derived algorithms. A Fortran subroutine is presented for the time recursive algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The design of multiplierless FIR filters for compensating D/A converter frequency response distortion

Designs are presented for multiplierless linear-phase FIR digital filters for compensating the sinc(x) frequency-response distortion resulting from D/A conversion. The filter coefficients are represented by a simple canonic single-digit code, and thus the filter can be implemented using only a few adders and delay registers. A 7-tap design is given which compensates the D/A distortion to within +or-0.045 dB from DC through 0.41 f/sub s/ and an 11-tap design is given which compensates the D/A distortion to within +or-0.028 dB from DC through 0.045 f/sub s/.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of selective linear phase bandpass switched-capacitor filters with equiripple passband amplitude response

A novel approach is presented to the design and synthesis of switched-capacitor (SC) filters. An efficient iterative algorithm is described for the construction of a class of selective linear-phase bandpass filter with equiripple passband amplitude response. The solution is obtained in terms of digital linear phase bandpass polynomials. A novel SC implementation is performed using an approach based on the reflection filter concept. The synthesis of the SC filter follows the steps used in microwave filter synthesis. The resulting SC circuits are completely insensitive to parasitic capacitances. Design tables and an illustrative example are also given.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Design of N-dimensional multiplierless spherically symmetric FIR digital filters by transformation

The paper describes a procedure for designing N-dimensional (ND) hyperspherically symmetric FIR linear phase digital filters. The method generalises the well known transformation approach of McClellan for designing 2D FIR linear phase filters from optimal minimax ID filters. The advantages of such an approach are that very large ND filter arrays can be designed in an amount of time comparable to that required to do the ID design, and the Chebyshev structure proposed by McClellan and Chan for implementing the 2D filter, which uses the impulse response samples of the ID design directly, can also be used for implementing the resulting ND filter.

Design of minimum phase FIR digital filters

Minimum phase FIR filters are known to have certain practical advantages, e.g. minimum group delay, reduced order for given gain specifications, and lower coefficient sensitivity, when compared with the more common linear phase filters. In the paper, a new adaptive method for designing such filters is presented. The algorithm requires a gain specification in a manner identical to the linear phase case. Conditions for optimality are investigated in terms of the zeros' configuration, equiripple characteristics, minimax and LMS criteria. The result is a very flexible minimum phase FIR filter design technique, with the potential to accommodate a number of different filter structures and types. It is thought to be the second, after the transformation of linear to minimum phase approach, able to guarantee optimum Chebyshev solutions.

Design of FIR filters as a tapped cascaded interconnection of identical subfilters

A general theory is presented for the design of linear-phase FIR digital filters as a tapped cascaded interconnection of identical FIR subfilters. The approach is an extension of the Kaiser-Hamming procedure [1] proposed for sharpening the response of an FIR filter. The new approach allows the subfilter and the tap coefficients to be simultaneously optimized to minimize either the number of subfilters for the given order of the subfilter or the subfilter order for the given number of subfilters. The optimization is based on the use of standard FIR filter design algorithms. Several examples demonstrate how the new approach leads to implementations requiring significantly fewer distinct multipliers than equivalent direct-form minimax FIR designs at the expense of a slight increase in the overall filter order. The number of distinct multipliers can be reduced to approximately <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sqrt{2.6L}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</tex> is the order of the direct-form minimax design. Alternatively, the design of the subfilter and tap coefficients can be separated. This makes it possible to construct the subfilter so that it roughly meets the overall specifications with a highly reduced number of arithmetic operations. In this case, the tap coefficients are optimized to minimize the required number of subfilters to meet the given criteria. Even multiplier-free designs can be obtained by carefully constructing the subfilter and determining the tap coefficients. Several structures are discussed for implementing the overall filter. These structures are compared with each other and with equivalent directform minimax designs in terms of the number of distinct multipliers, overall filter order, overall multiplication rate, number of delay elements, coefficient sensitivity, and output noise variance.

An approximation procedure for selective linear phase filters

A method is proposed for the simultaneous approximation of attenuation and group delay characteristics of analog and digital filters. The attenuation is approximated to be equal ripple in the passband and the stopband, while the group delay is equalized in the passband using a least-mean-square norm. The number of attenuation ripples in the passband, and the degrees of freedom for the group delay equalization can be traded off for the best possible approximation.

A fast algorithm for block LS design of FIR filters with linear phase and optimum lag

This correspondence is concerned with a fast algorithm for the design of FIR least squares filters with linear phase and optimum lag between the input signal and the desired response. Both linear phase cases with phase delay and group delay are considered. Based on the special form of the resulting equations, an efficient algorithm is developed that recursively computes the linear phase filters of given order for all possible lag delays.

A new structure for adaptive linear-phase filtering

In many applications, including time-varying modeling and adaptive noise canceling, it is desirable to adaptively perform linear-phase filtering to prevent any phase distortion in the observed data. Much attention has been devoted so far to the purely algorithmic aspect of this problem, i.e., how to design an appropriate adaptation algorithm. It is also of interest, however, to measure the extent of the structural influence, i.e., what system structure to choose to optimize the performance-to-cost ratio. This paper presents a different parameterization of the linear-phase filtering problem, naturally leading to a new, very efficient filter implementation. The resulting realization is then used as the system structure for an adaptive linear-phase filter. When associated with suitable stochastic gradient algorithms, this new adaptive structure is found to exhibit a tracking performance competitive with that of linear-phase lattice/ladder realizations, for a computational complexity 33 to 40 percent lower. It can therefore be used advantageously in virtually all applications of adaptive filtering. Simulations performed in the context of adaptive linear-phase modeling show good agreement with the theoretical analysis.

Design of FIR filters in the complex domain

The design of FIR digital filters with a complex-valued desired frequency response using the Chebyshev error is investigated. The complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem. A standard linear programming algorithm for the Chebyshev solution of overdetermined equations is used to solve the real approximation problem. Additional constraints are introduced which allow weighting of the phase and/or group delay of the approximation. Digital filters are designed which have nearly constant group delay in the passbands. The desired constant group delay which gives the minimum Chebyshev error is found to be smaller than that of a linear phase filter of the same length. These filters, in addition to having a smaller, approximately constant group delay, have better magnitude characteristics than exactly linear phase filters with the same length. The filters have nearly equiripple magnitude and group delay.

New approach to the design of fir digital filters

This paper presents a new approach to designing finite impulse response (FIR) digital filters. The design algorithm is based on the least mean square (LMS) criterion in the time domain to calculate the filter coefficients using the weighted gain peak errors to adjust the LMS cost function. The filter responses are optimum in the sense that the maximum gain error is minimised. The design procedure accommodates the entire range of linear phase FIR filter specifications. The flexibility and optimality of the LMS approach is demonstrated with a wide variety of examples, including the classic extraripple, scaled extraripple and equiripple cases of linear phase and examples of nonlinear phase. To date the only two algorithms that have been able to design the full range of linear phase filters are based on the Cheby̅chev polynomial approach, namely Remez exchange and linear programming.

Filter transient response to EEG waveforms

The response of two types of linear filters to sinusoidal bursts was calculated to demonstrate how filters can distort EEG waveforms. Results show that the wider the filter bandwidth the less is the distortion, and for a given bandwidth, the higher the filter order the greater the distortion. The response of a linear phase filter was also calculated to demonstrate that this type of filter can also cause waveform distortion, although it is normally less than that caused by Butterworth, Tchebychev and elliptic filters.

Recursive Nth-band digital filters- Part I: Design and properties

We investigate the properties of a special class of recursive polyphase filters whose subfilters are distinct allpass filters. On the basis of the discovered properties, a Remez-type algorithm is constructed for designing these filters. The proposed algorithm is faster and gives higher filter selectivities than other existing methods. Better understanding of the filter properties allows us to select the branch filter orders such that the filter complexity is minimized. In addition to nonlinear-phase filters, the algorithm is applicable to the design of approximately linear-phase filters. Several examples are included illustrating the efficiency of the proposed design scheme. The characteristics of the resulting filters are exposed by means of experimental results, and the implementation of the allpass branch filters is discussed. In a companion paper [33], we use these polyphase filters as basic building blocks in constructing efficient filters for sampling rate alteration. In addition, it is shown how the don't care bands, which cannot be avoided in filters of this type, can be suppressed by using an additional correction filter stage.

Recursive Nth-band digital filters- Part II: Design of multistage decimators and interpolators

In a companion paper [1], an efficient procedure was given for the design of recursive polyphase filters whose subfilters are distinct allpass filters, and the properties of these filters were considered in detail. Both nonlinear-phase and approximately linear-phase filters were discussed. In this paper, we show how these filters can be used for constructing efficient single-stage and multistage decimators and interpolators. The resulting designs suffer from the inherent limitation that aliasing cannot be prevented into the transition band in the decimation case. It is shown how this limitation can be overcome by using an additional filter stage at the output of the decimator. Working at the lowest sampling rate, this filter increases only slightly the overall computation rate. The optimization of this filter stage, together with the actual decimation stages, is discussed in detail in both the nonlinear-phase and approximately linear-phase case. Several examples illustrate the efficiency of the new overall designs compared to other known decimators and interpolators.

Generalized approximation techniques for selective linear-phase digital and nonreciprocal lumped filters

The paper presents expressions for low-pass digital filter transfer functions with finite-band approximation to constant amplitude and delay in the passband, together with an arbitrary number of transmission zeros at half the sampling frequency. The corresponding high-pass filter transfer functions can be obtained by a transformation. The available degrees of freedom can be divided arbitrarily between the passband and stopband responses. The functions are of the nonminimum-phase type, and the corresponding nonreciprocal analog (continuous) filters are also covered; these can be realized in standard active RC structures. The zero-bandwidth cases are obtained, either directly or as limiting cases of the finite-band ones. It is also indicated that there is an upper bound on the number of transmission zeros which may be introduced while maintaining the stability of the filter for a given degree. The technique represents the most comprehensive one available for the solution of the combined amplitude and phase approximation problem, and leads to a large family of stable transfer functions.

ECG baseline wander reduction using linear phase filters

The continuous real time reduction of baseline wander is a considerable problem in electrocardiography during exercises. Our solution consists of spectral filtering. The legitimacy of high-pass filtering of the ECG by means of digital linear phase filters with a low cut-off frequency as high as the heart rate is shown. The specifications of these filters are derived from experimental results. Special hardware is presented that simultaneously performs the desired real-time filter operation in four ECG leads.

Transfer function realization of a class of doubly-terminated two-variable lossless networks and their application in linear-phase 2-dimensional digital filter design

A method is presented for the design of a class of two-dimensional (2-D) stable digital filters satisfying prescribed magnitude and constant group delay specifications. The design method generates a 2-D digital transfer function which is a product of two transfer functions H 1 ( z 1 , z 2 ) and H 2 ( z 1 , z 2 ), corresponding to a recursive filter and a nonrecursive filter, respectively, Component H 1 ( z 1 , z 2 ) ensures a wave-digital realization, that is, the design method guarantees the generation of a corresponding analog function H 1A ( s 1 , s 2 ) which is realizable as the transfer function of a doubly-terminated two-variable lossless network. Thus the design technique ensures that not only is a given frequency response achieved, but also the generated transfer function is realizable as a cascade of a wave-digital filter and a nonrecursive digital filter. The class of filters considered here is one in which the doubly-terminated analog network used to realize the wave digital filter is a cascade of s 1 - and s 2 -variable lossess two-ports with all their transmission zeros at infinity.

Removal of base-line wander and power-line interference from the ECG by an efficient FIR filter with a reduced number of taps.

Linear phase filtering is proposed for the removal of baseline wander and power-line frequency components in electrocardiograms. In order to reduce the large number of computations involved in the digital filtering that are necessary, the desired filter spectrum was defined periodically. Making use of the property that the spectrum period is 50 Hz, the spectrum can be realized with a considerably reduced number of impulse response coefficients. This, in combination with the necessary impulse response symmetry, leads to a reduction in the number of multiplications per output sample by a factor of 10. A suitable impulse response is designed with a pass-band ripple of less than 0.5 dB and a high stop-band attenuation. The applicability is demonstrated by applying the filtering to exercise electrocardiograms.

Efficient block LS design of FIR filters with linear phase

This paper is concerned with the efficient determination of FIR least squares filters with linear phase. The general unknown statistics case is considered, whereby only sample records of the input and the desired response signals are available. Both linear phase cases with phase delay and group delay are examined. Taking advantage of the underlying near-to-Toeplitz structure of the resulting normal equations, a fast algorithm is developed that recursively produces all linear phase filters orders for a single block of data.

Selective linear-phase filters with controllable amplitude response

A method is presented for the construction of transfer functions of analog or digital filters exhibiting linear phase low-pass or bandpass characteristics. The group delay which is determined by the denominator polynomial can be maximally flat or equiripple in the whole or part of the passband. The passband loss as well as the stopband loss can be monotonic or have ripples with arbitrary widths. The iterative procedure involves the solutions to a polynomial and a set of linear equations.