Design of two-channel low-delay IIR nonuniform-division filter banks using L/sub 1/ error criteria

The design of a two-channel nonuniform-division filter (NDF) bank with infinite impulse response (IIR) analysis/synthesis filters and low group delay in the sense of L/sub 1/ error criteria is considered. The problem formulation results in a nonlinear optimisation problem. Based on a variant of Karmarkar's algorithm, the optimisation problem is solved through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1, 1). Simulation results are provided for illustration and comparison.     

Design of two-channel linear-phase QMF banks based on real IIR all-pass filters

The design of two-channel linear-phase quadrature mirror filter (QMF) banks constructed by real infinite impulse response (IIR) digital all-pass filters is considered. The design problem is appropriately formulated to result in a simple optimisation problem. Using a variant of Karmarkar's algorithm, the optimisation problem can be efficiently solved through a frequency sampling and iterative approximation method to find the real coefficients for the IIR digital all-pass filters. The resulting two-channel QMF banks possess an approximately linear phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of the desired phase response and then finding its solution from a linear subspace in a few iterations. Finally, several simulation examples are presented for illustration and comparison     

Minimax design of two-channel low-delay perfect-reconstruction FIRfilter banks

The authors deal with the design problem of low-delay perfect-reconstruction filter banks for which the FIR analysis and synthesis filters have equiripple magnitude response. Based on the minimax error criterion, the design problem is formulated in such a manner that the coefficients for the FIR analysis filters can be found by minimising the weighted peak error of the designed analysis filters, subject to the perfect-reconstruction constraints. A design technique based on a modified dual-affine scaling variant of Karmarkar's (1989) algorithm, in conjunction with approximation schemes, is then developed for solving the resulting nonlinear optimisation problem. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Design of two-channel IIR filter banks with arbitrary group delay

The nonlinear optimisation problem that results from considering the design of a two-channel nonuniform division filter bank is solved. This is through a frequency sampling and iterative approximation technique to find the tap coefficients and the reflection coefficients for the numerator and the denominator of the IIR analysis filters. An efficient stabilisation procedure ensures that the reflection coefficients lie in (-1,1). Simulation examples are provided for illustration     

Analysis and design of periodically time-varying IIR filters, withapplications to transmultiplexing

Fundamental constraints on the form of infinite impulse response (IIR) periodically time-varying (PTV) filters are identified, and a design technique with well-defined error and stability characteristics based on those constraints is presented. The design technique is based on the selection of poles and zeros within the time-invariant filter banks of equivalent PTV filter analysis structures. A simple example is presented to illustrate the design method, which implements the IIR PTV as a time-invariant all-feedback IIR filter of the form 1/D(z^P) cascaded with an finite impulse response (FIR) PTV filter. An application of IIR PTV filters to telecommunications transmultiplexing is presented to illustrate the design method and for comparison to an existing PTV design method. The computational complexity of the resulting system compares favorably with that of existing transmultiplexers     

近似完全重构IIR余弦调制滤波器组的优化设计

提出一种新的近似完全重构因果稳定的IIR余弦调制滤波器组的设计方法。基于预先给定的极点值,IIR原型滤波器的设计问题可以简化成一个凸极大值极小化的优化问题,从而采用二阶锥规划法求解。所得余弦调制滤波器组具有良好的频率特性和合理的完全重构误差。所设计的原型滤波器是因果稳定的,并且其多相因子分母相同,简化了完全重构条件,可以用来进一步优化得到的完全重构系统。     

Design of stable IIR digital filter based on least P-power error criterion

In this paper, the least p-power error criterion is presented to design digital infinite impulse response (IIR) filters to have an arbitrarily prescribed frequency response. First, an iterative quadratic programming (QP) method is used to design a stable unconstrained one-dimensional IIR filter whose optimal filter coefficients are obtained by solving the QP problem in each iteration. Then, the proposed method is extended to design constrained IIR filters and two-dimensional IIR filters with a separable denominator polynomial. Finally, design examples of the low-pass filter are demonstrated to illustrate the effectiveness of the proposed iterative QP method.     

Minimax design of two-channel nonuniform-division filterbanks using IIR allpass filters

The design of two-channel linear-phase nonuniform-division filter (NDF) banks constructed by infinite impulse response (IIR) digital allpass filters (DAFs) in the sense of L/sub /spl infin// error criteria is considered. First, the theory of two-channel NDF bank structures using two IIR DAFs is developed. Then, the design problem is appropriately formulated to result in a simple optimization problem. Utilizing a variant of Karmarkar's algorithm, we can efficiently solve the optimization problem through a frequency sampling and iterative approximation method to find the coefficients for the IIR DAFs. The resulting two-channel NDF banks can possess approximately linear-phase response without magnitude distortion. The effectiveness of the proposed technique is achieved by forming an appropriate Chebyshev approximation of a desired phase response and then to find its solution from a linear subspace in a few iterations. Several simulation examples are presented for illustration and comparison.     

Minimax design of recursive digital filters with a latticedenominator

The authors deal with the problem of minimax recursive digital filter design with a lattice structure for the denominator. The design problem is formulated so that the coefficients for the numerator and denominator of a recursive filter can be found by solving the best linear complex Chebyshev approximation (LCCA). A design technique based on the weighted least-squares algorithm previously proposed by one of the authors is then developed for solving the resulting LCCA problem. During the design process, this technique finds the tap coefficients for the numerator and the reflection coefficients for the denominator simultaneously. The stability of the designed recursive filter is ensured by incorporating an efficient stabilisation procedure to make all of the reflection coefficient values fall between -1 and +1. Computer simulations show that the proposed technique provides better design results than existing techniques     

Minimax deconvolution design of multirate systems with channelnoises: a unified approach

The conventional signal reconstruction problem of multirate systems with channel noises can be cast as a robust multirate deconvolution design problem. We investigate a unified minimax approach for the robust deconvolution design of multirate systems. We discuss two typical multirate systems: the multirate filter bank system and the transmultiplexer system. We consider transmission noises resulting from quantization coding errors or external noises. The deconvolution filters for these systems that we derive are all IIR filters. The keypoint is converting the original robust deconvolution design problem to an equivalent minimax matching problem via polyphase decomposition and noble identities. Then, in spite of the presence of input signals and channel noises, we can solve this minimax matching problem by an optimization technique. The proposed method can be interpreted as designing an optimal multirate deconvolution filter such that the worst-case multirate system reconstruction error is minimized over all possible inputs and noises from the energy perspective. Therefore, our proposed design method is more robust than the conventional design method for multirate systems in the presence of uncertain input signals and channel noises. We present several numerical examples that show the good performance of our design method     

FIR, Allpass, and IIR Variable Fractional Delay Digital Filter Design

This paper presents two-step design methodologies and performance analyses of finite-impulse response (FIR), allpass, and infinite-impulse response (IIR) variable fractional delay (VFD) digital filters. In the first step, a set of fractional delay (FD) filters are designed. In the second step, these FD filter coefficients are approximated by polynomial functions of FD. The FIR FD filter design problem is formulated in the peak-constrained weighted least-squares (PCWLS) sense and solved by the projected least-squares (PLS) algorithm. For the allpass and IIR FD filters, the design problem is nonconvex and a global solution is difficult to obtain. The allpass FD filters are directly designed as a linearly constrained quadratic programming problem and solved using the PLS algorithm. For IIR FD filters, the fixed denominator is obtained by model reduction of a time-domain average FIR filter. The remaining numerators of the IIR FD filters are designed by solving linear equations derived from the orthogonality principle. Analyses on the relative performances indicate that the IIR VFD filter with a low-order fixed denominator offers a combination of the following desirable properties including small number of denominator coefficients, lowest group delay, easily achievable stable design, avoidance of transients due to nonvariable denominator coefficients, and good overall magnitude and group delay performances especially for high passband cutoff frequency ( ges 0.9pi) . Filter examples covering three adjacent ranges of wideband cutoff frequencies [0.95, 0.925, 0.9], [0.875, 0.85, 0.825], and [0.8, 0.775, 0.75] are given to illustrate the design methodologies and the relative performances of the proposed methods.     

Maxflat Fractional Delay IIR Filter Design

Fractional delay (FD) filters are an important class of digital filters and are useful in various signal processing applications. This paper discusses a design problem of FD infinite-impulse-response (IIR) filters with the maxflat frequency response in frequency domain. First, a flatness condition of FD filters at an arbitrarily specified frequency point is described, and then a system of linear equations is derived from the flatness condition. Therefore, a set of filter coefficients can be easily obtained by solving this system of linear equations. For a special case in which the frequency response is required to be maxflat at omega = 0 or pi , a closed-form expression for its filter coefficients is derived by solving a linear system of Vandermonde equations. It is also shown that the existing maxflat FD finite-impulse-response (FIR) and IIR filters are special cases of the FD IIR filters proposed in this paper. Finally, some examples are presented to demonstrate the effectiveness of the proposed filters.     

Minimax design of two-dimensional parallelogram filter banks

The authors present a technique for the minimax design of two-dimensional (2-D) parallelogram filter bank (PFB) systems with linear-phase analysis/synthesis filters. To achieve perfect reconstruction, the required analysis filters must have parallelogram-shaped frequency responses. In general, the original design problem is found to be an optimisation problem with nonlinear constraints. The authors present a linearisation approach to reformulate the design problem. As a result, updating the filter coefficient vector at each iteration for the original design problem can be accomplished by searching the gradient of the linearised optimisation problem. They further present an efficient method based on a modified Karmarkar's algorithm for computing the required gradient vector and finding the required step size analytically. Therefore the filter coefficients can easily be computed by solving only linear equations at each iteration during the design process. The effectiveness of the proposed technique is shown by computer simulations     

Minimax design of two-channel nonuniform-division FIR filter banks

The paper deals with the minimax design of two-channel nonuniform-division filter (NDF) banks. Based on a linearisation scheme, the design problem is formulated as an optimisation problem with linear constraints. The authors present a method to design a two-channel NDF bank using a modified dual-affine scaling variant of Karmarkar's (1984) algorithm. This method provides the optimal results that the linear-phase FIR analysis and synthesis filters have equiripple stopband response and the resulting NDF bank also shows equiripple reconstruction error behaviour. The effectiveness of the proposed design technique is demonstrated by several simulation examples     

Design of 1-D Stable Variable Fractional Delay IIR Filters

In this brief, a two-stage approach for the design of 1-D stable variable fractional delay infinite-impulse response (IIR) digital filters is proposed. In the first stage, a set of fixed delay stable IIR filters are designed by minimizing a quadratic objective function, which is defined by integrating error criterion with IIR filter stability constraint condition. Then, the final design is determined by fitting each of the fixed delay filter coefficients as a 1-D polynomial. Two design examples are given to show the effectiveness of the proposed design method     

Design of stable, causal, perfect reconstruction, IIR uniform DFTfilter banks

Design procedures for stable, causal and perfect reconstruction IIR parallel uniform DFT filter banks (DFT FBs) are presented. In particular a family of IIR prototype filters is a good candidate for DFT FB, where a tradeoff between frequency selectivity and numerical properties (as measured by the Weyl-Heisenberg frames theory) could be made. Some realizations exhibiting a simple and a massively parallel and modular processing structure making a VLSI implementation very suitable are shown. In addition, some multipliers in the filters (both the analysis and synthesis) could be made; powers or sum of powers of 2, in particular for feedback loops, resulting in a good sensitivity behavior. For these reasons as well as for the use of low order IIR filters (as compared with conventional FIR filters), the overall digital filter bank structure is efficient for high data rate applications. Some design examples are provided     

A new method for the design of IIR filters with flat magnitude response

This paper presents a new direct design of infinite-impulse response (IIR) filters with a flat magnitude response in both passband and stopband (Butterworth filters). The design specifications are passband and stopband frequencies and passband droop and stopband attenuation. The approach is based on an allpass filter with flatness at frequency points /spl omega/=0 and /spl omega/=/spl pi/. Depending on the parity of the IIR filter order, the allpass filter is either real or complex. However, in both cases, the resulting IIR filter is real.     

A direct approach to H2 optimal deconvolution ofperiodic digital channels

This paper studies the H₂ optimal deconvolution problem for periodic finite impulse response (FIR) and infinite impulse response (IIR) channels. It shows that the H₂ norm of a periodic filter can be directly quantified in terms of periodic system matrices and linear matrix inequalities (LMIs) without resorting to the commonly used lifting technique. The optimal signal reconstruction problem is then formulated as an optimization problem subject to a set of matrix inequality constraints. Under this framework, the optimization of both the FIR and IIR periodic deconvolution filters can be made convex, solved using the interior point method, and computed by using the Matlab LMI Toolbox. The robust deconvolution problem for periodic FIR and IIR channels with polytopic uncertainties are further formulated and solved, also by convex optimization and the LMIs. Compared with the lifting approach to the design of periodic filters, the proposed approach is simpler yet more powerful in dealing with multiobjective deconvolution problems and channel uncertainties, especially for IIR deconvolution filter design. The obtained solutions are applied to the design of an optimal filterbank yielding satisfactory performance     

On the Design of Nearly-PR and PR FIR Cosine Modulated Filter Banks Having Approximate Cosine-Rolloff Transition Band

This paper proposes an efficient method for designing nearly perfect reconstruction (NPR) and perfect reconstruction (PR) cosine modulated filter banks (CMFBs) with prototype filters having an approximate cosine-rolloff (CR) transition band. It is shown that the flatness condition required for an NPR CMFB can be automatically satisfied by using a prototype filter with a CR transition band. The design problem is then formulated as a convex minimax optimization problem, and it can be solved by second-order cone programming (SOCP). By using the NPR CMFB so obtained as an initial guess to nonlinear optimizers such as Fmincon in Matlab, high-quality PR CMFBs can be obtained. The advantages of the proposed method are that it does not require a user-supplied initial guess of the prototype filter and bumps in the passband of the analysis filters can be effectively suppressed.     

PCLS IIR digital filters with simultaneous frequency responsemagnitude and group delay specifications

This paper presents the peak-constrained least-squares (PCLS) approach to designing IIR digital filters. PCLS IIR digital filters that meet simultaneous specifications on the frequency response magnitude and the group delay are introduced. As a point of reference, we consider the IIR digital filter design problem that appears in Deczky's (1972) classic paper and in the popular textbook by Oppenheim and Schafer (1989). In addition, the same design problem appears in the IIR filter design chapter by Higgins and Munson (1993) in the Handbook for Digital Signal Processing. By using our new algorithm with simultaneous optimization of the frequency response magnitude and the group delay, we obtain a dramatic improvement in the solution of this classic IIR digital filter design problem. Starting from the same filter structure and the same specifications for the frequency response magnitude as in the works of Deczky, Oppenheim and Schafer, and Higgins and Munson, we are able to reduce the group delay ripple by a factor of 35. In another design problem that originated in Deczky's work, we use PCLS optimization to reduce the group delay ripple by a factor of 40 at the same time we reduce the stopband energy by 6 dB, without sacrificing any other performance measure. The group delay ripple in this IIR digital filter example is reduced to only ±0.002 samples