Gerald D. Cain

Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction

An algorithm for the approximation of finite impulse response (FIR) filters by infinite impulse response (IIR) filters is presented. The algorithm is based on a concept of balanced model reduction. The matrix inversions normally associated with this procedure are notoriously error prone due to ill conditioning of the special matrix forms required. This difficulty is circumvented here by directly formulating a reduced state-space system description which is input/output equivalent to the system that would more conventionally be obtained following the explicit step of constructing an (interim) balanced realization. Examples of FIR by IIR filter approximations are included. >

WLS design of variable frequency response FIR filters

This paper extends the Weighted Least Squares method to designing FIR filters capable of changing, in the real-time, one of their frequency response characteristics (group delay, the width of the passband, resonance frequency or any other). The filter coefficients are polynomial functions of the parameter characterising the variable feature. The computations needed in such designs can be kept at low level if the weight function in the performance criterion is separable. The advantages of the proposed approach are illustrated by a design of a Fractional Sample Delay filter with variable delay. If this filter has to meet demanding specifications then the proposed approach provides a cheaper and more effective solution than traditional approaches based on Lagrange interpolation.

A WISE method for designing IIR filters

The problem of designing optimal digital IIR filters with frequency responses approximating arbitrarily chosen complex functions is considered. The real-valued coefficients of the filters transfer function are obtained by numerical minimization of carefully formulated cost, which is referred here to as the weighted integral of the squared error (WISE) criterion. The WISE criterion linearly combines the WLS criterion that is used in the weighted least squares approach toward filter design and some time-domain components. The WLS part of WISE enforces the quality of the frequency response of the designed filter, while the time-domain part of the WISE criterion restricts the positions of the filters poles to the interior of an origin-centred circle with arbitrary radius. This allows one not only to achieve stability of the filter but also to maintain some safety margins. A great advantage of the proposed approach is that it does not impose any constraints on the optimization problem and the optimal filter can be sought using off-the-shelf optimization procedures. The power of the proposed approach is illustrated with filter design examples that compare favorably with results published in research literature.

Sampling rate conversion using fractional-sample delay

In this paper we demonstrate application of the fractional sample delayor (FSD) to constructing sampling rate converters for incommensurate sampling rates. We use Lagrange polynomials for interpolating the values of the signal between sampling instants. It is shown that to deploy this technique effectively one has to use polynomials of high degree. Otherwise the interpolation error for high frequencies (above half Nyquist) is very big. Increasing the polynomials degree leads inevitably to numerical problems. The interpolation process may be improved when additional samples of the signal are generated before interpolation. It is shown that FSD can be used for achieving this task. The final version of the sampling rate converter comprises two FSDs and a very simple four-point interpolator based on a Lagrange polynomial.<<ETX>>

On an instantaneous frequency estimator with FIR filters having maximally flat frequency response error magnitude

Abstract In this paper an instantaneous frequency estimator (IFE) of a discrete-time base band complex signal is considered. The IFE is built around one-band, maximally flat linear-phase FIR filters, which are used for differentiating and delaying Cartesian components of the complex signal. One of the key features of the estimator is that it avoids problems related to the ambiguity of the instantaneous phase waveform. The quality of the estimator is tested. A closed-form formula for the static characteristic of the IFE is derived and expressed as a function of the frequency responses of the filters used. Two representative test signals: a full band complex linear frequency modulated (LFM) chirp and a three-component complex synthetic signal are used to demonstrate the characteristic features of the estimator. If the chirp is sufficiently long in comparison with the length of the filters, the instantaneous frequency (IF) estimation errors are comparable to those obtained by using the static characteristic. For this case, the IF estimation error plots for the practical versus ideal IFE are presented and a design chart showing the dependence of the IF estimation error magnitude on the input signal bandwidth and the FIR filters’ length is given. This chart can be exploited in, e.g., FM-telemetry applications, where the IF carries a very slowly changing telemetric message. The three-component signal chosen allows demonstration of the ability of the estimator to track the IF which extends beyond the signal spectral range, permitting measurement even beyond the Nyquist frequency. Finally, the power of the proposed IFE to measure the stability of highly precise frequency oscillators is shown.

Stable IIR filters-a new design approach

The paper proposes a new approach to designing digital, causal, time-invariant IIR filters approximating arbitrary frequency response. The design task is formulated as an optimisation problem where the cost is a nonlinear function of the tuneable coefficients of the filters transfer function. The novelty of the approach lies in the way the stability of the filter is tackled. The cost function comprises a combination of frequency-domain and time-domain components. Frequency-domain components are used to enforce the filter to represent desired complex-valued frequency response, while the time-domain components impose its stability. No equality or nonequality constraints are involved in the optimisation problem related to the proposed method.

Optimal Signal Selection for FIR Matched Filtering in Pole-Only Noise

With duration-limited signals there is the opportunity for perfect matched filtering by a suitable FIR filter whenever the accompanying noise has pole-only coloration. The maximal SNR value obtainable via matched filtering is itself sensitive to the signal pulse shape initially given. In some signaling situations (e.g. radar) we are unilaterally free to choose signal shape, and with this come prospects of further SNR improvement. Four distinct perspectives on optimizing signal selection are pursued, including a new single-frequency toneburst windowing approach that becomes a very good approximation for large signal sizes. Three of the approaches are capable of near-optimal SNR performance, but the whitener eigenvalue method introduced is recommended as the best possible way of selecting signals. Discussion is included of special multi-tonality issues which arise for smaller signals and for multiple maximum whitener eigenvalues.

A note on the "Approximation of FIR by IIR digital filters: an algorithm based on balanced model reduction" [Comment and reply]

In the present paper Rudko extends the algorithm presented by Beliczynski et al. (see ibid., vol.40, p.532, 1992), based on balanced order reduction, for the conversion of a narrow-band linear phase FIR filter to an almost linear phase IIR filter. It is shown that the algorithm can be used to reduce the order of wideband FIR filters. Beliczynski et al. reply to the Comment. >

Optimal signal and jamming dynamics embracing digital filter strictures

Dynamic jamming/evasion duels are described purely in terms of re-design iterations for three separate agile FIR digital filters. At the signal generation end of the contest an optimal pulse-shaping filter works in concert with an optimal detection filter. At the jamming end, two strategies are evaluated for the re-design of the noise coloration filter. One of these utilizes an optimal square-rooting filter which (ideally) neutralizes the selectivity of the detection filter, while the other gives slightly less impressive jamming effectiveness but shows an exploitation advantage if it is known that the receiver recklessly uses a simple flip-template filter for detection.

Simultaneous magnitude and group delay approximation for IIR filters via balanced model reduction

A technique for designing IIR (infinite impulse response) filters to simultaneously meet arbitrary magnitude and group delay specifications, without explicit equalizer sections, through the use of balanced model reduction is described. FIR (finite impulse response) design tools which make the approximation task relatively easy are employed prior to an FIR-to-IIR conversion process. The use of this reduction algorithm in a real-time FIR system identification and subsequent IIR compensation scene is also reported.<<ETX>>