Designing two-channel causal stable IIR PR filter banks and wavelet bases by model order reduction and constrained optimization

In this paper, two methods for designing two-channel causal stable IIR PR filter banks are introduced. The first method makes use of model reduction and constrained optimization to obtain a causal stable IIR filter bank from the structural PR FIR filter bank proposed in [2]. It yields better frequency characteristics than the original FIR filter bank and avoids the dump at π/2 when allpass filters are used. Using the 1-D to 2-D transformation proposed in [2], two dimensional PR IIR filter bank can readily be obtained from these prototypes. The second method is based on constrained optimization technique using the general PR condition. Using this technique, filter banks with low system delay and flexible frequency characteristics can be designed. The technique can also be modified to design causal stable IIR dyadic wavelet bases with added regularity conditions. A number of design examples are used to demonstrate the usefulness of the proposed design methods.published_or_final_versio

Design of Two-Channel Low-Delay FIR Filter Banks using Constrained Optimization

This paper shows the efficiency of using constrained optimization for designing two-channel low-delay finite impulse response filter banks. The filter banks under consideration are quadrature mirror filter (QMF) banks and perfect reconstruction (PR) biorthogonal filter banks. The design problems for both types of banks are stated as constrained minimization problems in forms that enable us to minimize the maximum of the stopband energies of the analysis filter(s) subject to the given passband and transition band constraints of the filter(s) as well as subject to the given allowable reconstruction error for QMF banks or the PR property for biorthogonal filter banks. For solving the given optimization problems a modified Dutta-Vidyasagar optimization technique has been used. The efficiency of the proposed design methods is illustrated by means of some examples

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

Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having the perfect-reconstruction property

Based on the concept of losslessness in digital filter structures, this paper derives a general class of maximally decimated M-channel quadrature mirror filter banks that lead to perfect reconstruction. The perfect-reconstruction property guarantees that the reconstructed signalhat{x} (n)is a delayed version of the input signal x (n), i.e.,hat{x} (n) = x (n - n_{0}). It is shown that such a property can be satisfied if the alias component matrix (AC matrix for short) is unitary on the unit circle of the z plane. The number of channels M is arbitrary, and when M is two, the results reduce to certain recently reported 2-channel perfect-reconstruction QMF structures. A procedure, based on recently reported FIR cascaded-lattice structures, is presented for optimal design of such FIR M-channel filter banks. Design examples are included

Improved technique for design of perfect reconstruction FIR QMF banks with lossless polyphase matrices

A technique is developed for the design of analysis filters in an M-channel maximally decimated, perfect reconstruction, finite-impulse-response quadrature mirror filter (FIR QMF) bank that has a lossless polyphase-component matrix E(z). The aim is to optimize the parameters characterizing E(z) until the sum of the stopband energies of the analysis filters is minimized. There are four novel elements in the procedure reported here. The first is a technique for efficient initialization of one of the M analysis filters, as a spectral factor of an Mth band filter. The factorization itself is done in an efficient manner using the eigenfilters approach, without the need for root-finding techniques. The second element is the initialization of the internal parameters which characterize E(z), based on the above spectral factor. The third element is a modified characterization, mostly free from rotation angles, of the FIR E(z). The fourth is the incorporation of symmetry among the analysis filters, so as to minimize the number of unknown parameters being optimized. The resulting design procedure always gives better filter responses than earlier ones (for a given filter length) and converges much faste

The theory and design of a class of perfect reconstruction modified DFT filter banks with IIR filters

The 47th Midwest Symposium on Circuits and Systems Conference, Salt Lake City, Utah, USA, 25-28 July 2004This paper proposes a theory and design method for a class of PR causal-stable modified discrete Fourier transform (MDFT) filter bank (FB) with IIR filters. The prototype filter of the MDFT FB is assumed to have identical denominator in order to simplify the PR condition. A new model reduction technique is proposed for deriving a nearly PR (NPR) MDFT FB from a PR MDFT FB with FIR prototype filter. With these NPR IIR MDFT FBs as initial guess, PR IIR MDFT FBs with very good frequency characteristics can be obtained by solving a constrained nonlinear optimisation problem. Because the location of the poles can be approximately determined through model reduciton, the efficiency and reliability of the design method is significantly improved. Design examples are given to demonstrate the effectiveness of the proposed method.published_or_final_versio

Efficient method for designing two-channel PR FIR filter banks with low system delay

In this paper, an efficient method for designing perfect reconstruction (PR) two-channel finite impulse response (FIR) filter banks with low system delay is proposed. It is based on the use of nonlinear-phase FIR function in a structure previously proposed by Phoong et al. The design problem is formulated as a complex polynomial approximation problem and is solved effectively using the Remez exchange algorithm with very low design complexity. Design examples show that filter banks with flexible stopband attenuation and system delay can be readily obtained by the proposed algorithm.published_or_final_versio

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

Factorization of a class of perfect reconstruction modified DFT filter banks with IIR filters

This paper proposed a new factorization of a class of perfect reconstruction (PR) causal-stable modified discrete Fourier transform (MDFT) filter bank (FB) with IIR filters, whose prototype filter has identical denominator in their polyphase components. This factorization technique, which is based on the lifting scheme, is also complete for the PR FIR MDFT FB. It can be applied to convert a nearly PR MDFT FBs to a structural PR system, which is very useful to their multiplier-less realization because the PR property in these structural FBs is unaffected by coefficient quantization. Therefore, it is possible to employ canonical signed digits (CSD) or sum of powers of two coefficients to approximate the coefficients in the factored form without changing the PR property. © 2005 IEEE.published_or_final_versio

On the design and multiplierless realization of perfect reconstruction triplet-based FIR filter banks and wavelet bases

This paper proposes new methods for the efficient design and realization of perfect reconstruction (PR) two-channel finite-impulse response (FIR) triplet filter banks (FBs) and wavelet bases. It extends the linear-phase FIR triplet FBs of Ansari et al. to include FIR triplet FBs with lower system delay and a prescribed order of K regularity. The design problem using either the minimax error or least-squares criteria is formulated as a semidefinite programming problem, which is a very flexible framework to incorporate linear and convex quadratic constraints. The K regularity conditions are also expressed as a set of linear equality constraints in the variables to be optimized and they are structurally imposed into the design problem by eliminating the redundant variables. The design method is applicable to linear-phase as well as low-delay triplet FBs. Design examples are given to demonstrate the effectiveness of the proposed method. Furthermore, it was found that the analysis and synthesis filters of the triplet FB have a more symmetric frequency responses. This property is exploited to construct a class of PR M-channel uniform FBs and wavelets with M = 2 L, where L is a positive integer, using a particular tree structure. The filter lengths of the two-channel FBs down the tree are approximately reduced by a factor of two at each level or stage, while the transition bandwidths are successively increased by the same factor. Because of the downsampling operations, the frequency responses of the final analysis filters closely resemble those in a uniform FB with identical transition bandwidth. This triplet-based uniform M-channel FB has very low design complexity and the PR condition and K regularity conditions are structurally imposed. Furthermore, it has considerably lower arithmetic complexity and system delay than conventional tree structure using identical FB at all levels. The multiplierless realization of these FBs using sum-of-power-of-two (SOPOT) coefficients and multiplier block is also studied. © 2004 IEEE.published_or_final_versio