Oversampled cosine-modulated filter banks with arbitrary systemdelay

Design methods for perfect reconstruction (PR) oversampled cosine-modulated filter banks with integer oversampling factors and arbitrary delay are presented. The system delay, which is an important parameter in real-time applications, can be chosen independently of the prototype lengths. Oversampling gives us additional freedom in the filter design process, which can be exploited to find FIR PR prototypes for oversampled filter banks with much higher stopband attenuations than for critically subsampled filter banks. It is shown that for a given analysis prototype, the PR synthesis prototype is not unique. The complete set of solutions is discussed in terms of the nullspace of a matrix operator. For example, oversampling allows the design of PR filter banks having unidentical prototypes (of equal and unequal lengths) for the analysis and synthesis stage. Examples demonstrate the increased design freedom due to oversampling. Finally, it is shown that PR prototypes being designed for the oversampled case can also serve as almost-PR prototypes for critically subsampled cosine-modulated pseudo QMF banks     

一类冗余比为2的二维DFT调制滤波器组的设计

本文构建了一类冗余比为2的二维线性相位的双原型离散傅立叶变换（DFT）调制滤波器组。利用原型滤波器的多相位分解,推导出了该滤波器组的完全重构(PR)条件。基于该PR条件,我们将滤波器组的设计归结为一个关于原型滤波器的多相位分量的无约束优化问题。由于原型滤波器是线性相位的,多相位分量之间具有一定的关系,因此我们可以简化该优化问题。仿真结果验证了滤波器组PR条件的正确性。同时,仿真表明了优化算法的有效性,设计所得的滤波器组重构误差很小、频率特性较好,基本满足实际应用的需要。      

Rational sampling filter banks based on IIR filters

The problem of splitting the spectrum of a digital signal by using nonuniform infinite impulse response (IIR) filter banks is addressed. Near perfect reconstruction (NPR) is considered. The method uses the modulation of different IIR prototypes. The cancellation of the main aliasing components constrains the prototypes to be dependent on each other. By using this approach, linear-phase prototypes are needed, and noncausal filtering is required. Numerical examples of filter bank design are given, and the computational complexity is compared with the finite impulse response (FIR) case     

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     

Lowpass minimum phase filter design using IFIR filters

A design procedure for minimum-phase FIR filters using an interpolated FIR (IFIR) filter is proposed. The IFIR technique allows two linear-phase filters of much lower order to be designed thereby making it easier to apply mipizing than is possible in high order FIR prototype filters. In this way, the problem of finding the roots of high order polynomials is overcome     

过采样精确重构余弦调制滤波器组的设计

本文首先推导出过采样滤波器组精确重构的条件,由于此时所需的约束条件数比临界采样时少,因而可以设计出频域衰减特性更好的滤波器用,然后提出了精确重构约束条件下原型概通江波器的一种新的设计方法,采用矢的二次型约束优化算法,该算法优化方便,收剑速度快,与其它方法相比,滤波器的阻带衰减大。     

The theory and design of arbitrary-length cosine-modulated filterbanks and wavelets, satisfying perfect reconstruction

It is well known that FIR filter banks that satisfy the perfect-reconstruction (PR) property can be obtained by cosine modulation of a linear-phase prototype filter of length N=2mM, where M is the number of channels. In this paper, we present a PR cosine-modulated filter bank where the length of the prototype filter is arbitrary. The design is formulated as a quadratic-constrained least-squares optimization problem, where the optimized parameters are the prototype filter coefficients. Additional regularity conditions are imposed on the filter bank to obtain the cosine-modulated orthonormal bases of compactly supported wavelets. Design examples are given     

Design of face-centred orthorhombic filter banks

In this paper, a method to design the two-channel FIR linear-phase (LP) face-centred orthorhombic (FCO) filter banks with equiripple magnitude responses and perfect-reconstruction (PR) is presented. The necessary conditions of lengths of LP FCO filter banks satisfying the PR constraint are derived. An interior-point algorithm is utilized to optimize the peak ripples of the analysis filters and a first-order approximation skill is introduced to satisfy the PR constraint. The simulation example is presented to illustrate the effectiveness of this proposed design technique.     

A new approach to the design of critically sampled M-channeluniform-band perfect-reconstruction linear-phase FIR filter banks

A new approach is presented for the design of uniform-band M-channel perfect-reconstruction (PR) FIR filter banks employing linear-phase analysis and synthesis filters. The technique designs on the impulse responses of the analysis filters directly. The design problem is formulated as an optimization program. The filter bank's PR feature can either be implicitly enforced through a set of mathematical relationships among the analysis filters' coefficients or through a set of constraints in the optimization program. The former approach results in a filter bank whose PR feature's dependency on hardware and software is eliminated or, at least, minimized. The synthesis filters are then obtained by a set of relationships that describe each synthesis filter as a function of the analysis filters. The criterion for optimality is “least-squares,” where the square of the difference between the ideal and actual frequency responses is integrated over the appropriate frequency bands for all M analysis filters and minimized     

METEOR: a constraint-based FIR filter design program

It is proposed to specify a filter only in terms of upper and lower limits on the response, find the shortest filter length which allows these constraints to be met, and then find a filter of that order which is farthest from the upper and lower constraint boundaries in a minimax sense. The simplex algorithm for linear programming is used to find a best linear-phase FIR filter of minimum length, as well as to find the minimum feasible length itself. The simplex algorithm, while much slower than exchange algorithms, also allows the incorporation of more general kinds of constraints, such as concavity constraints (which can be used to achieve very flat magnitude characteristics). Examples are given to illustrate how the proposed and common approaches differ, and how the proposed approach can be used to design filters with flat passbands, filters which meet point constraints, minimum phase filters, and bandpass filters with controlled transition band behavior     

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.      

Lagrange multiplier approaches to the design of two-channelperfect-reconstruction linear-phase FIR filter banks

The authors present two approaches to the design of two-channel perfect-reconstruction linear-phase finite-impulse-response (FIR) filter banks. Both approaches analyze and design the impulse responses of the analysis filter bank directly. The synthesis filter bank is then obtained by simply changing the signs of odd-order coefficients in the analysis filter bank. The approach deals with unequal-length filter banks. By designing the lower length filters first, one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect-reconstruction constraint equations. The second approach generalizes the first, and covers the design for all parts of linear phase perfect reconstruction constraint equations     

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     

一种满足精确重建条件的多采样率滤波器组的实现方法

多采样率滤波器组可以通过对线性相位低通原型滤波器进行余弦调制来实现，本文给出了一咱满足重建条件的采用余弦调制结构的系统实现方法，并指出了其多项元素矩阵无损的充分必要条件，在精确重建的前提下，对低通原型滤波器系数进行了优化。     

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     

A programmable FIR digital filter using CSD coefficients

An area-efficient programmable FIR digital filter using canonic signed-digit (CSD) coefficients was implemented that uses a switchable unit-delay to allocate the desired number of nonzero CSD coefficient digits to each filter tap. The prototype chip can allocate up to 16 pairs of nonzero CSD coefficient digits for a linear-phase filter, thus realizing filters with 32 linear-phase taps operating at 180 MHz with two nonzero CSD digits per filter tap. Additional nonzero CSD digits can be allocated to filter taps at the penalty of a reduced filter length and a reduced data-rate. The chip was implemented with 16-bit I/O in a die size of 5.9 mm by 3.4 mm using 1.0-μm CMOS technology     

A unified approach to multistage frequency-response masking filter design using the WLS technique

This paper presents a unified approach to the optimal design of sharp linear-phase finite-impulse-response (FIR) digital filters synthesized using the multistage frequency-response masking (FRM) technique. In this approach, the design of a k-stage FRM filter is achieved in a recursive manner. The minimax design problem arising at each step of the synthesis process is converted into a corresponding weighted least-squares (WLS) problem. The WLS problem is highly nonlinear with respect to the coefficients of the filter. Consequently, it is decomposed into several linear least-squares (LS) problems, each of which can be solved analytically. It is then solved iteratively by using an alternating variable approach. Numerical design examples are included to demonstrate the effectiveness of the method.     

A general formulation of modulated filter banks

This paper presents a general framework for maximally decimated modulated filter banks. The theory covers the known classes of cosine modulation and relates them to complex-modulated filter banks. The prototype filters have arbitrary lengths, and the overall delay of the filter bank is arbitrary, within fundamental limits. Necessary and sufficient conditions for perfect reconstruction (PR) are derived using the polyphase representation. It is shown that these PR conditions are identical for all types of modulation-modulation based on the discrete cosine transform (DCT), both DCT-III/DCT-IV and DCT-I/DCT-II, and modulation based on the modified discrete Fourier transform (MDFT). A quadratic-constrained design method for prototype filters yielding PR with arbitrary length and system delay is derived, and design examples are presented to illustrate the tradeoff between overall system delay and stopband attenuation (subchannelization)     

Minimum-phase filter design from linear-phase startpoint viabalanced model truncation

A new practical design approach for minimum-phase FIR or IIR filters, setting out from a high dimensionality FIR linear-phase prototype is described. The novelty of this technique lies in overcoming the inherent problem of finding the roots of a high order polynomial with repeated and/or very closely clustered roots     

Design of QMF banks based on derivative information

New L₂ objective functions for the design of quadrature mirror filter (QMF) banks are proposed. They are based on the derivative information of the reconstruction error. Simple and explicit matrix-form formulas for the proposed objective functions are derived. Efficient design methods are proposed by incorporating a separability technique into the derived optimality conditions on prototype filters. The proposed design methods need only solve linear equation iteratively without nonlinear optimisation. Design examples demonstrate that good low-delay QMF banks and linear-phase QMF banks can be obtained in only a few iterations. Compared with the conventional approach, the new approach leads to QMF banks with larger stopband attenuation and smaller reconstruction errors