Linear-phase Filters Research Articles

Estimates for the stability of low-pass filters

A measure of stability is defined for digital filters. It is shown that the stability of low-pass filters goes to zero as the filter characteristics approach that of an ideal low-pass filter. More precisely, estimates for the rate of convergence are given. It is shown that, except for constants, the stability of a low-pass filter is no greater than the square root of the reciprocal of the logarithm of the transition bandwidth. For linear phase filters the square root may be removed. These estimates are shown to be optimal for the case of linear phase filters.

A class of linear-phase FIR filters for decimation, interpolation, and narrow-band filtering

A new class of linear phase finite impulse response (FIR) digital filters for decimation and interpolation is discussed. The transfer function of these filters is of the form H(z) = A (zD) B(z)where D is the decimation or interpolation ratio and A (zD) is realized at the lower sampling rate. The polynomial A (zD) is adjusted so that the overall magnitude response presents an equiripple behavior in the pass-band, whereas B(z) provides an equiripple stopband response. An efficient iterative procedure for the design of the filters of this type is presented and the optimal selection of the orders of A(zD) and B(z) is considered. Several examples show how the new decimators and interpolators require significantly fewer multiplications per second than equivalent conventional linear phase FIR and elliptic designs. The performance of these linear phase filters is about the same as that of the recursive filters of Martinez and Parks with only powers of zDin the denominator. Some characteristics of the new filters, such as zero positions, coefficient sensitivity, and output noise variance due to roundoff errors, are discussed and compared with conventional FIR designs. Furthermore, it is shown that by cascading multistage new decimators and interpolators, we obtain efficient narrow-band filters requiring considerably fewer multiplications per output sample than equivalent elliptic designs.

Statistical design of nearly linear-phase ARMA filters

This paper proposes a method for statistical design of approximately linear-phase autoregressive-moving average (ARMA) digital filters. The key idea of our method is that a time-delayed ARMA filter is used to approximate a high-order FIR filter that meets the prescribed amplitude spectrum sufficiently well. Computer simulations show that the resulting ARMA low-pass, bandpass, and high-pass filters not only meet the prescribed amplitude response specification well, but yield excellent linear phase with much lower order and smaller delay compared to the associated FIR filters.

Rise times of attenuated seismic pulses detected in both empty and fluid‐filled cylindrical boreholes

Fourier‐Bessel theory is used to derive filters representing the influence of both empty and fluid‐filled cylindrical boreholes on particle motion induced in rock by a plane P-wave incident perpendicular to the borehole axis. For wavelengths greater than 10 times the borehole circumference, the effect of the borehole on particle motions is shown to be negligible; thus the results have little relevance for the long wavelengths commonly encountered in earthquake seismology. The results are, however, relevant to the study of stress wave propagation at ultrasonic frequencies in rock masses. For small wavelengths (αa > 3.0) the filter representing particle motion on the wave incident site of an empty borehole reduces to a linear phase filter which increases all amplitudes by a factor of 2 while the filter representing fluid stress at the center of a fluid‐filled borehole may be reduced to simple mathematical expressions. Experimental results were obtained for the interaction of a stress wave with either accelerometers mounted in an empty borehole or a hydrophone located centrally in a fluid‐filled borehole. Both theory and experiment show a similar distortion in the rise time of the pulse traveling past the borehole.

Perceptual models and the filtering of high-contrast achromatic images

A model based on research in the psychophysics of vision is developed for use in the design of image processing filters in order to quantify the results of image processing as perceived by a human observer. An alternate formulation of the classical frequency-domain filter design problem with both space and frequency-domain specifications is developed. New experimental results concerning the masking effect in the vicinity of edges are reported, and a structure of a model for detection of distortion in complex images is proposed. The ability of the model to predict the visibility of one-dimensional patterns in the vicinity of edges is tested. A distortion measure based on the model is introduced, and practical filter-design criteria are developed. Both FIR linear phase filters and IIR filters are used to demonstrate the applicability of the methods.

An Approach of Realizing a Linear-Phase Filter with a Multiple-Notched Property

There are many applications where signals with relatively low frequencies must be measured under the following conditions: 1) eliminating very low frequencies (such as slow fluctuations) from a reference level, 2) rejecting 50/60 Hz and their harmonics, 3) keeping linear-phase frequency characteristics. A filter based on a conventional microprocessor having the above conditions has been simply implemented by modifying the concept of the Cascaded Integrator Comb (CIC) filter. In this paper, we will describe the hardware, software, and noise analysis of the advocated filter.

Reconstruction of signals from phase: Efficient algorithms, segmentation, and generalizations

In this paper, we develop several new formulations of the phase-only reconstruction problem which lead to new, efficient algorithms for signal reconstruction. We describe a procedure for approximately reconstructing a long duration signal from short segments. Phase-only blind deconvolution for linear phase filters relies on mirror image symmetry in the filter transfer function H(z). We generalize this deconvolution result to include several other types of symmetries for H(z).

An auditory spectral analysis model using the chirp z-transform

In this paper we discuss a new signal-processing approach to implementing an auditory processing model. The model is based on auditory physiology and psychophysics; it uses constant-bandwidth analysis below 500 Hz and constant-Q analysis above 500 Hz, with the analysis filters being implemented with multiple real poles. The constant-Q analysis is performed using the chirp z-transform (CZT). We review the evidence for the auditory model and the properties of constant-Q analysis, and derive the CZT formulation for both causal and linear-phase analysis filters. The paper concludes with an example of speech illustrating the use of the auditory model.

The Resonant Frequency of Rectangular Interdigital Filter Elements

A procedure is given for the computation of the resonant frequency of loosely coupled interdigital resonators with rectangular cross section. The procedure is based on the use of Getsinger's fringing capacitance data [1]. The accuracy of the method was verified experimentally and found to be approximately 1 percent for a 2-percent bandwidth interdigital linear-phase filter.

Linear phase waveguide filters

A survey of linear phase filters and transfer functions having linear-phase properties, along with two different waveguide structures which realize such functions, are given. The design procedure and constructional problems are solved. Two examples, one for a single-mode, direct-coupled structure, and one for a dual-mode realization, are worked out in detail and the crucial stops are pointed out.

Design of FIR Filter Banks for a 24-Channel Transmultiplexer

The design of a bank of FIR bandpass filters for channel filtering and sample rate alteration in a 24-channel transmultiplexer is considered. It is shown that such a filter bank can be realized by the combination of a weighting network and a discrete cosine transform. A minimum phase design is suggested for the low-pass prototype filter in order to solve the long absolute delay problem inherent in linear phase filters. Two approaches to the design of long minimum phase filters are disclosed; one employs frequency sampling design techniques, and the other makes use of the properties of the complex cepstrum of a minimum phase sequence. Filter specifications for the low-pass prototype are discussed and a design example is included. The hardware realization of the filter bank using a multiplier-accumulator as the arithmetic element is also discussed.

Selective linear phase filters possessing a pair of J‐axis transmission zeros

AbstractA design technique is developed for prototype selective linear phase filters which have an equiripple passband amplitude response and a pair of j‐axis transmission zeros in the stopband while maintaining a linear phase characteristic in the passband. Since the basic technique requires numerical optimization techniques, the element values for several cases have been tabulated together with detailed response characteristics.One important result is that for certain characteristics, one internal cross‐coupling element may be designed to be zero. As a result, in the sixth degree case with only one extra cross‐coupling element between elements 1 and 6 a significant improvement in both the amplitude and group delay performance can be achieved as compared to the Chebyshev case. Owing to its importance, this case is investigated in detail and a 2.6 GHz post decoupled combline filter has been constructed based upon this prototype.

Least squares algorithms for adaptive linear-phase filtering

In many applications, it is desirable to use filters with linear-phase characteristics. This paper presents an approach for developing such filters for adaptive signal processing. Several least squares algorithms are derived for adjusting the coefficients of an adaptive linear phase filter. It is shown that smoothing filters, rather than the commonly used prediction filters, are a more natural choice if linear phase characteristics are required.

Design of finite impulse response digital filters with nonlinear phase response

Most of the existing literature on FIR digital filters is concerned with linear-phase (LP) filters. However, several papers have appeared on the subject of nonlinear-phase (NLP) filters, mainly proposing methods for designing minimum-phase filters, or approximating a desired phase response. In this paper, an investigation is made of one such method, based upon a selection of zeros from a prototype LP filter. It is shown that with respect to minimizing the order of a filter subject to given gain response specifications, this is the most efficient method for designing FIR filters. Coefficient quantization error is analyzed for filters generated by this method. A practical comparison is given between the resulting filter and the corresponding minimal order LP filter. It is shown that while most LP filters can be implemented more efficiently than NLP filters by taking into account the symmetry of their coefficients, for filters with very wide passband and for certain special purpose filters such as CCD and those used for filtering a delta-modulated or ADPCM signal, an NLP implementation is usually more efficient. In addition, an alternate design algorithm is proposed for NLP filters which decreases ripple magnitude. The resulting filters, while not of minimal order, can be efficiently implemented by decomposing the filter into LP stopband and NLP passband sections, which is especially attractive for narrow passbands.

Design of FIR digital phase networks

The problem of the minimax design of FIR digital filters with prescribed phase characteristics and unit magnitude is a nonlinear optimization problem. In this paper it is approximated by a linear programming problem, and it is shown that the solution of this linear program is optimal to first order. That is, if δ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> and ε <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> are optimal deviations of magnitude and phase characteristics, then the actual deviations obtained from the linear program solution satisfy <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta \leq \delta_{0} + \epsilon\min{0}\max{2}</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon \leq \epsilon_{0} (1 + \delta_{0})/(1 - \delta_{0})</tex> . Numerical examples are given, including design results for full-band M-term chirp filters which (like linear phase filters) can be implemented with (M + 1)/2 multiplications per point.

Linear phase filter design on a time-domain basis

The maximum error between the input and output signals of filters is bounded. This bound is used to develop low-pass filter design procedures based on time domain errors. This technique is applied to phase-corrected Butterworth and Chebyshev filters.

Filter-length word-length tradeoffs in FIR digital filter design

We give a theorem which shows that there is a lower bound on the Chebyshev approximation error for linear-phase direct-form FIR digital filters, when the coefficients are constrained to be b-bit numbers. We then investigate the tradeoff between filter-length N and coefficient word-length b, using the product Nb as a complexity measure, for both the usual direct form and the sharpening structures of Kaiser and Hamming. The sharpening structures usually provide no overall gain in Nb product, but achieve a given performance with a smaller value of b.

Finite Precision Design of Linear-Phase FIR Filters

The FIR (finite impulse response) filter is an essential tool for a large number of applications in communication. In this paper we consider the design of linear-phase FIR digital filters with finitely precise coefficients. Coefficient inaccuracy is known to degrade the frequency response of band-select FIR filters, especially in the stopband region. We derive a bound on the attainable stopband attenuation, and we also develop techniques for designing FIR filters with finitely precise coefficients. Mixed-integer programming algorithms are presented to select finitely precise coefficients for a filter that best approximates an arbitrary magnitude characteristic in the minimax sense. Our method generates a number of possible solutions including that of simple rounding or truncation and then selects the best finitely precise coefficients from this set. In this way, significant improvement in the filter performance is gained over methods that simply round or truncate the infinitely precise coefficients. We also show how integer programming can be used to design filters with powers of two coefficients. Such filters are easier to mechanize since they do not require multipliers.

An efficient l&amp;lt;inf&amp;gt;p&amp;lt;/inf&amp;gt;optimization technique for the design of two-dimensional linear-phase FIR digital filters

An iterative optimization technique, based on the method of parallel tangents coupled with an efficient line searching technique, is proposed for the design of two-dimensional linear-phase FIR digital filters. The performance index for the optimization procedure is shown to be convex and hence this technique will always converge towards the global minimum. An important feature is that the design time increases slowly as the number of filter coefficients increases. To illustrate the technique several circular low-pass filters were designed. These filters compare favorably with filters designed using the minimax technique that was developed by Harris. The technique can also be used for the design of one-dimensional and multidimensional linear-phase FIR digital filters.

Selective constant delay filters with chebyshev passband amplitude response

AbstractThe aim of this letter is to show that the principle of least squares norm can be utilized to offer flexibility in the design of selective linear phase filters.