On the eigenfilter design method and its applications: a tutorial

The eigenfilter method for digital filter design involves the computation of filter coefficients as the eigenvector of an appropriate Hermitian matrix. Because of its low complexity as compared to other methods as well as its ability to incorporate various time and frequency-domain constraints easily, the eigenfilter method has been found to be very useful. In this paper, we present a review of the eigenfilter design method for a wide variety of filters, including linear-phase finite impulse response (FIR) filters, nonlinear-phase FIR filters, all-pass infinite impulse response (IIR) filters, arbitrary response IIR filters, and multidimensional filters. Also, we focus on applications of the eigenfilter method in multistage filter design, spectral/spacial beamforming, and in the design of channel-shortening equalizers for communications applications

Eigenfilters: A new approach to least-squares FIR filter design and applications including Nyquist filters

A new method of designing linear-phase FIR filters is proposed by minimizing a quadratic measure of the error in the passband and stopband. The method is based on the computation of an eigenvector of an appropriate real, symmetric, and positive-definite matrix. The proposed design procedure is general enough to incorporate both time- and frequency-domain constraints. For example, Nyquist filters can be easily designed using this approach. The design time for the new method is comparable to that of Remez exchange techniques. The passband and stopband errors in the frequency domain can be made equiripple by an iterative process, which involves feeding back the approximation error at each iteration. Several numerical design examples and comparisons to existing methods are presented, which demonstrate the usefulness of the present approach

Critical analysis of the eigenfilter method for the design of FIR filters and wideband beamformers

The least squares based eigenfilter method has been applied to the design of both finite impulse response (FIR) filters and wideband beamformers successfully. It involves calculating the resultant filter coefficients as the eigenvector of an appropriate Hermitian matrix, and offers lower complexity and less computation time with better numerical stability as compared to the standard least squares method. In this paper, we revisit the method and critically analyze the eigenfilter approach by revealing a serious performance issue in the passband of the designed FIR filter and the mainlobe of the wideband beamformer, which occurs due to a formulation problem. A solution is then proposed to mitigate this issue, and design examples for both FIR filters and wideband beamformers are provided to demonstrate the effectiveness of the proposed method

A new class of two-channel biorthogonal filter banks and wavelet bases

We propose a novel framework for a new class of two-channel biorthogonal filter banks. The framework covers two useful subclasses: i) causal stable IIR filter banks. ii) linear phase FIR filter banks. There exists a very efficient structurally perfect reconstruction implementation for such a class. Filter banks of high frequency selectivity can be achieved by using the proposed framework with low complexity. The properties of such a class are discussed in detail. The design of the analysis/synthesis systems reduces to the design of a single transfer function. Very simple design methods are given both for FIR and IIR cases. Zeros of arbitrary multiplicity at aliasing frequency can be easily imposed, for the purpose of generating wavelets with regularity property. In the IIR case, two new classes of IIR maximally flat filters different from Butterworth filters are introduced. The filter coefficients are given in closed form. The wavelet bases corresponding to the biorthogonal systems are generated. the authors also provide a novel mapping of the proposed 1-D framework into 2-D. The mapping preserves the following: i) perfect reconstruction; ii) stability in the IIR case; iii) linear phase in the FIR case; iv) zeros at aliasing frequency; v) frequency characteristic of the filters

Unified eigenfilter approach: with applications to spectral/spatial filtering

The eigenfilter approach is extended to solve general least-squares approximation problems with linear constraints. Such extension unifies previous work in eigenfilters and many other filter design problems, including spectral/spatial filtering, one-dimensional or multidimensional filters, data independent or statistically optimal filtering, etc. With this approach, various filter design problems are transformed into problems of finding an eigenvector of a positive definite matrix that is determined by filter design specifications. This approach has the advantage that many filter design constraints can be incorporated easily. A number of design examples are presented to show the usefulness and flexibility of the proposed approach

On the design of real and complex fir filters with flatness and peak error constraints using semidefinite programming

This paper studies the problem of designing digital finite duration impulse response (FIR) filters with prescribed flatness and peak error constraints using semidefinite programming (SDP). SDP is a powerful convex optimization method, where linear and convex quadratic inequality constraints can readily be incorporated. This property is utilized for the optimal minimax and least squares (LS) design of linear-phase and low-delay FIR filters with prescribed magnitude flatness and peak design error, which are formulated as a set of linear equality and convex quadratic inequality constraints, respectively. A method for structurally imposing these equality constraints in the SDP formulation is also proposed. Using these results, the design approach is further extended to the design of constrained complex coefficient FIR filters and variable digital filters (VDFs). Design examples are given to demonstrate the effectiveness of the approach.published_or_final_versio

Revisit of the eigenfilter method for the design of FIR filters and wideband beamformers

The least squares-based eigenfilter method has been applied to the design of both finite impulse response (FIR) filters and wideband beamformers successfully. It involves calculating the resultant filter coefficients as the eigenvector of an appropriate Hermitian matrix, and offers lower complexity and less computation time with better numerical stability as compared to the standard least squares method. In this paper, we revisit the method and critically analyse the eigenfilter method by revealing a serious performance issue in the passband of the designed FIR filter and the mainlobe of the wideband beamformer, which occurs due to a formulation problem. A solution is then proposed to mitigate this issue by imposing an additional constraint to control the response at the passband/mainlobe, and design examples for both FIR filters and wideband beamformers are provided to demonstrate the effectiveness of the proposed method