Perfect-reconstruction filter banks having linear-phase FIR filterswith equiripple response

The design of equiripple linear-phase analysis and synthesis FIR filters of two-channel perfect-reconstruction (PR) filter banks is formulated as the minimization of a weighted peak-error under both linear inequality (arising from the desired responses of the analysis filters) and nonlinear equality (PR) constraints. The effectiveness of a proposed method to solve the design problem (a modified dual-affine scaling variant of Karmarkar's (1989) algorithm and an approximation scheme) is illustrated through several design examples     

Wordlength Optimization of Linear Time-Invariant Systems With Multiple Outputs Using Geometric Programming

This paper proposes two new methods for optimizing hardware resources in finite wordlength implementation of multiple-output (MO) linear time-invariant (LTI) systems. The hardware complexity is measured by the exact internal wordlength used for each intermediate data. The first method relaxes the wordlength from integer to real-value and formulates the design problem as a geometric programming, from which an optimal solution of the relaxed problem can be determined. The second method is based on a discrete optimization method called the marginal analysis method, and it yields the desired wordlengths in integer values. By combining these two methods, a hybrid method is also proposed, which is found to be very effective for large scale MO LTI systems. To illustrate the effectiveness of the proposed methods, wordlength optimization problems of two-channel structural perfect reconstruction filter banks and multiplier-less fast Fourier transforms are studied in detail. Design results show that the proposed algorithms offer better results and a lower design complexity than conventional methods     

Design and Factorization of Two-Channel Perfect Reconstruction Filter Banks with Causal-Stable IIR Filters

In this paper, new design and factorization methods of two-channel perfect reconstruction (PR) filter banks (FBs) with casual-stable IIR filters are introduced. The polyphase components of the analysis filters are assumed to have an identical denominator in order to simplify the PR condition. A modified model reduction is employed to derive a nearly PR causal-stable IIR FB as the initial guess to obtain a PR IIR FB from a PR FIR FB. To obtain high quality PR FIR FBs for carrying out model reduction, cosine-rolloff FIR filters are used as the initial guess to a nonlinear optimization software for solving to the PR solution. A factorization based on the lifting scheme is proposed to convert the IIR FB so obtained to a structurally PR system. The arithmetic complexity of this FB, after factorization, can be reduced asymptotically by a factor of two. Multiplier-less IIR FB can be obtained by replacing the lifting coefficients with the canonical signal digitals (CSD) or sum of powers of two (SOPOT) coefficients.     

Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR Perfect Reconstruction Filter Banks

We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a “predesigned” analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method.      

Design and multiplier-less implementation of a class of two-channel PR FIR filterbanks and wavelets with low system delay

A new method for designing two-channel PR FIR filterbanks with low system delay is proposed. It is based on the generalization of the structure previously proposed by Phoong et al. (1995) Such structurally PR filterbanks are parameterized by two functions (/spl beta/(z) and /spl alpha/(z)) that can be chosen as linear-phase FIR or allpass functions to construct FIR/IIR filterbanks with good frequency characteristics. The case of using identical /spl beta/(z) and /spl alpha/(z) was considered by Phoong et al. with the delay parameter M chosen as 2N-1. In this paper, the more general ease of using different nonlinear-phase FIR functions for /spl beta/(z) and /spl alpha/(z) is studied. As the linear-phase constraint is relaxed, the lengths of /spl beta/(z) and /spl alpha/(z) are no longer restricted by the delay parameters of the filterbanks. Hence, higher stopband attenuation can still be achieved at low system delay. The design of the proposed low-delay filterbanks is formulated as a complex polynomial approximation problem, which can be solved by the Remez exchange algorithm or analytic formula with very low complexity. In addition, the orders and delay parameters can be estimated from the given filter specifications using a simple empirical formula. Therefore, low-delay two-channel PR filterbanks with flexible stopband attenuation and cutoff frequencies can be designed using existing filter design algorithms. The generalization of the present approach to the design of a class of wavelet bases associated with these low-delay filterbanks and its multiplier-less implementation using the sum of powers-of-two coefficients are also studied.     

Design of perfect-reconstruction nonuniform recombination filter banks with flexible rational sampling factors

This paper studies the design of a class of perfect-reconstruction (PR) nonuniform filter banks (FBs) called recombination nonuniform FBs (RNFBs). They are constructed by merging subbands in a uniform FB with sets of transmultiplexers (TMUXs). It generalizes the RNFBs previously proposed by the authors to allow more general choice of the sampling factors. The spectral inversion and spurious response suppression problems of these new RNFBs using cosine modulation are analyzed, and a simple design method based on a matching condition is proposed. It is also found that the FB and the TMUX in the recombination structure can be designed separately to satisfy the matching condition. In addition, real-time adaptive merging of the channels to provide dynamic nonuniform frequency partitioning is feasible. Another advantage of the RNFBs is that the recombination and processing of the subband signal can be done at the decimated domain of the uniform FB, which greatly reduces its implementation complexity. Design examples show that high quality nonuniform PR FBs with low implementation complexity and variable time-frequency resolution can be obtained by the proposed method.     

The design and multiplier-less realization of software radio receivers with reduced system delay

This work studies the design and multiplier-less realization of a new software radio receiver (SRR) with reduced system delay. It employs low-delay finite-impulse response (FIR) and digital allpass filters to effectively reduce the system delay of the multistage decimators in SRRs. The optimal least-square and minimax designs of these low-delay FIR and allpass-based filters are formulated as a semi-definite programming (SDP) problem, which allows zero magnitude constraint at /spl omega/=/spl pi/ to be incorporated readily as additional linear matrix inequalities (LMIs). By implementing the sampling rate converter (SRC) using a variable digital filter (VDF) immediately after the integer decimators, the needs for an expensive programmable FIR filter in the traditional SRR is avoided. A new method for the optimal minimax design of this VDF-based SRC using SDP is also proposed and compared with traditional weight least squares method. Other implementation issues including the multiplier-less and digital signal processor (DSP) realizations of the SRR and the generation of the clock signal in the SRC are also studied. Design results show that the system delay and implementation complexities (especially in terms of high-speed variable multipliers) of the proposed architecture are considerably reduced as compared with conventional approaches.     

Discrete wavelet transform based on cyclic convolutions

A filterbank in which cyclic convolutions are used in place of linear convolutions will be referred to as a cyclic convolution filterbank (CCFB). A dyadic tree-structured CCFB can be used to perform a discrete wavelet transform suitable for coding based on symmetric extension methods. This paper derives two types of efficient implementation techniques for the tree-structured CCFB: one using complex arithmetic and one using only real arithmetic. In addition, the present paper analyzes in detail the perfect reconstruction (PR) condition for the two-channel CCFB and shows that this condition is much less restrictive than that of usual two-channel filterbanks. When each of the two-channel CCFBs constituting a tree-structured CCFB is designed to be PR, the whole system is PR. A quadrature mirror filter (QMF) CCFB having the PR property is demonstrated to be easily designed using a standard filter design subroutine. In contrast, designing a linear-phase PR FIR QMF bank that has good frequency response is not possible when filters are realized by linear convolutions.     

Design of Time–Frequency Localized Filter Banks: Transforming Non-convex Problem into Convex Via Semidefinite Relaxation Technique

We present a method for designing optimal biorthogonal wavelet filter banks (FBs). Joint time–frequency localization of the filters has been chosen as the optimality criterion. The design of filter banks has been cast as a constrained optimization problem. We design the filter either with the objective of minimizing its frequency spread (variance) subject to the constraint of prescribed time spread or with the objective of minimizing the time spread subject to the fixed frequency spread. The optimization problems considered are inherently non-convex quadratic constrained optimization problems. The non-convex optimization problems have been transformed into convex semidefinite programs (SDPs) employing the semidefinite relaxation technique. The regularity constraints have also been incorporated along with perfect reconstruction constraints in the optimization problem. In certain cases, the relaxed SDPs are found to be tight. The zero duality gap leads to the global optimal solutions. The design examples demonstrate that reasonably smooth wavelets can be designed from the proposed filter banks. The optimal filter banks have been compared with popular filter banks such as Cohen–Daubechies–Feauveau biorthogonal wavelet FBs, time–frequency optimized half-band pair FBs and maximally flat half-band pair FBs. The performance of optimal filter banks has been found better in terms of joint time–frequency localization.     

基于遗传算法的两通道完全重构滤波器组的设计

介绍了两通道滤波器组的完全重构条件,利用Euclidean分解算法,将两通道滤波器组的设计问题简化为寻找给定特性的低通滤波器的最佳Euclidean互补滤波器的单变量非线性优化问题,并探讨了采用遗传算法设计此类高度非线性优化问题.最后通过设计例子说明将遗传算法应用到滤波器组的设计中是可行的.     

Two Classes of Cosine-Modulated IIR/IIR and IIR/FIR NPR Filter Banks

This paper introduces two classes of cosine-modulated causal and stable filter banks (FBs) with near perfect reconstruction (NPR) and low implementation complexity. Both classes have the same infinite-length impulse response (IIR) analysis FB but different synthesis FBs utilizing IIR and finite-length impulse response (FIR) filters, respectively. The two classes are preferable for different types of specifications. The IIR/FIR FBs are preferred if small phase errors relative to the magnitude error are desired, and vice versa. The paper provides systematic design procedures so that PR can be approximated as closely as desired. It is demonstrated through several examples that the proposed FB classes, depending on the specification, can have a lower implementation complexity compared to existing FIR and IIR cosine-modulated FBs (CMFBs). The price to pay for the reduced complexity is generally an increased delay. Furthermore, two additional attractive features of the proposed FBs are that they are asymmetric in the sense that one of the analysis and synthesis banks has a lower computational complexity compared to the other, which can be beneficial in some applications, and that the number of distinct coefficients is small, which facilitates the design of FBs with large numbers of channels.     

Two-band wavelets and filterbanks over finite fields with connections to error control coding

Recently, we have developed a new framework to study error-control coding using finite-field wavelets and filterbanks (FBs). This framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones. The paper introduces the theory of wavelet decompositions of signals in vector spaces defined over Galois fields. To avoid the limitations of the number theoretic Fourier transform, our wavelet transform relies on a basis decomposition in the time rather than the frequency domain. First, by employing a symmetric, nondegenerate canonical bilinear form, we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform. Then, we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two-channel paraunitary (PU) FBs. Finally, we describe the application of this transform to the construction of error correcting codes. In particular, we give examples of double circulant codes that are generated by wavelets.     

A study of two-channel complex-valued filterbanks and wavelets withorthogonality and symmetry properties

We investigate two-channel complex-valued filterbanks and wavelets that simultaneously have orthogonality and symmetry properties. First, the conditions for the filterbank to be orthogonal, symmetric, and regular (for generating smooth wavelets) are presented. Then, a complete and minimal lattice structure is developed, which enables a general design approach for filterbanks and wavelets with arbitrary length and arbitrary order of regularity. Finally, two integer implementation methods that preserve the perfect reconstruction property of the filterbank are proposed. Their performances are evaluated via experimental results     

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     

Chebyshev Design of Linear-Phase FIR Filters With Linear Equality Constraints

By using basis transformation, the Chebyshev approximation of linear-phase finite-impulse response (FIR) filters with linear equality constraints can be converted into an unconstrained one defined on a new function space. However, since the Haar condition is not necessarily satisfied in the new function space, the alternating property does not hold for the solution to the resulted unconstrained Chebyshev approximation problem. A sufficient condition for the best approximation is obtained in this brief, and based on this condition, an efficient single exchange algorithm is derived for the Chebyshev design of linear-phase FIR filters with linear equality constraints. Simulations show that the proposed algorithm can converge to the optimal solution in most cases and to a near-optimal solution otherwise. Design examples are presented to illustrate the performance of the proposed algorithm.     

无完全重构约束的两通道自适应FIR滤波器组设计

该文研究了在无完全重构约束情况下两通道自适应FIR无混叠滤波器组的时域设计方法。由于放松了对完全重构的约束,子带编码器的失真由系统失真和量化失真两部分构成。在整体比特数和输入过程给定的情况下,本文通过优化滤波器系数,使得子带编码器的整体失真达到最小,实现提高其编码增益的目的。后面的设计实例验证了该方法的有效性。     

Optimal design of IIR digital filters with robust stability using conic-quadratic-programming updates

In this paper, minimax design of infinite-impulse-response (IIR) filters with prescribed stability margin is formulated as a conic quadratic programming (CQP) problem. CQP is known as a class of well-structured convex programming problems for which efficient interior-point solvers are available. By considering factorized denominators, the proposed formulation incorporates a set of linear constraints that are sufficient and near necessary for the IIR filter to have a prescribed stability margin. A second-order cone condition on the magnitude of each update that ensures the validity of a key linear approximation used in the design is also included in the formulation and eliminates a line-search step. Collectively, these features lead to improved designs relative to several established methods. The paper then moves on to extend the proposed design methodology to quadrantally symmetric two-dimensional (2-D) digital filters. Simulation results for both one-dimensional (1-D) and 2-D cases are presented to illustrate the new design algorithms and demonstrate their performance in comparison with several existing methods.     

Digital filter bank design quadratic-constrained formulation

Formulate the filter bank design problem as an quadratic-constrained least-squares minimization problem. The solution of the minimization problem converges very quickly since the cost function as well as the constraints are quadratic functions with respect to the unknown parameters. The formulations of the perfect-reconstruction cosine-modulated filter bank, of the near-perfect-reconstruction pseudo-QMF bank, and of the two-channel biorthogonal linear-phase filter bank are derived using the proposed approach. Compared with other design methods, the proposed technique yields PR filter banks with much higher stopband attenuation. The proposed technique can also be extended to design multidimensional filter banks     

Design of low-delay FIR QMF banks using the Lagrange-multiplierapproach

A new approach for the design of two-channel perfect reconstruction FIR filter banks with short reconstruction delays is presented. A low-order filter is first designed and the objective function of the filter bank is formulated as a quadratic programming problem with linear constraints. The Lagrange-multiplier method is then used to design a higher-order filter. The method is simple, efficient, and flexible and leads to a closed-form solution. A design example is included to illustrate the advantages of the method     

Orthonormal FBs With Rational Sampling Factors and Oversampled DFT-Modulated FBs: A Connection and Filter Design

Methods widely used to design filters for uniformly sampled filter banks (FBs) are not applicable for FBs with rational sampling factors and oversampled discrete Fourier transform (DFT)-modulated FBs. In this paper, we show that the filter design problem (with regularity factors/vanishing moments) for these two types of FBs is the same. Following this, we propose two finite-impulse-response (FIR) filter design methods for these FBs. The first method describes a parameterization of FBs with a single regularity factor/vanishing moment. The second method, which can be used to design FBs with an arbitrary number of regularity factors/vanishing moments, uses results from frame theory. We also describe how to modify this method so as to obtain linear phase filters. Finally, we discuss and provide a motivation for iterated DFT-modulated FBs.