线性相位余弦调制滤波器组研究

图像子带编码要求滤波器组具有线性相位特性，非线性相位特性可对图像编译码带来影响。现研究了一种具有线性相位的最大抽取FIR余弦调制滤波器组，该滤波器组中每一个滤波器都具有线性相位，且由原型滤波器经余弦序列调制得到。经推导这种滤波器组可以设计成具有近似准确重构（NPR）特性或准确重构（PR）特性，文中分别给出了两种情况下滤波器组应满足的条件。通过对一幅图像的二维可分离滤波实验说明了当原型滤波器满足一定条件时滤波器组是准确重构的。     

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     

Implementation and perfect reconstruction of a maximally decimatedFIR filter bank using parallel module decomposition

This paper discusses an implementation and the perfect reconstruction (PR) of an M-channel maximally decimated FIR fitter bank. Using the polynomial module arithmetics, the filter bank is decomposed into a set of module filter banks of size M, independent of the filter length. When the filter bank is uniform, the computational cost is the same as the polyphase/FFT implementation. When it is not uniform, in which case the polyphase/FFT implementation is not applicable, the computational cost is still reduced by sharing among channel filtering computations. The parallel module configuration is favorable for hardware implementation because decomposing a large system into small subsystems is generally advantageous for many realizations. The PR analysis is greatly simplified by working on the module filter banks as well     

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)     

Some results in the theory of crosstalk-free transmultiplexers

The crosstalk-free transmultiplexer (CF-TMUX) focuses on crosstalk cancellation (CC) rather than on suppressing it. The authors present an analysis of the CF-TMUX based on the polyphase component matrices of the filter banks used in TDM→FDM and FDM→TDM conversions, respectively. Thus a necessary and sufficient condition for complete CC is obtained. It is shown that the filters for a CF-TMUX are the same as the filters for a 1-skewed alias free QMF bank. In addition, if the QMF bank satisfies the perfect reconstruction (PR) property, then the TMUX also satisfies PR. The relation between CF-TMUX filters and alias-free QMF banks is used to obtain a direct design procedure for CF-TMUX filters (both FIR and IIR). It is also shown that approximately crosstalk-free TMUX filters can be obtained from any approximately alias-free QMC bank. Design examples and comparison tables are included     

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.     

Design of selective M-channel perfect reconstruction FIR filterbanks

A new method is presented for the optimal design of M-channel perfect reconstruction FIR quadrature mirror filter (QMF) banks which is based on the orthogonal matrix concept. The parameters of the analysis filters are adjusted so that they have a linear phase and minimum stopband energy. Comparison with a recently published method reveals that the resulting design procedure gives better stopband characteristics. A numerical design example is included to validate the theory and to demonstrate the superiority of the designed filters from selectivity, reconstruction and distortion points of view     

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。      

The complex subband decomposition and its application to the decimation of large adaptive filtering problems

We show that a near perfect reconstruction (NPR) M-channel filterbank with a diagonal system inserted between the analysis and synthesis filterbanks may be used to decompose a finite impulse response (FIR) system of order L into M complex subband components, each of order L/K, where K is the downsampling rate. This decomposition is at the expense of using complex arithmetic for the subband processing. The theory surrounding the proposed filterbank structure leads to a new understanding of subbanded adaptive filtering implementations. It also leads naturally to a delayless subbanded adaptive filter scheme. Using conditions on the analysis and synthesis filters, the formulas for the subband components and their respective properties are developed. Simulation results for an acoustic echo cancellation (AEC) example are given to support the developed theory.     

Two-channel IIR QMF banks with approximately linear-phase analysisand synthesis filters

Perfect linear-phase two-channel QMF banks require the use of finite impulse response (FIR) analysis and synthesis filters. Although they are less expensive and yield superior stopband characteristics, perfect linear phase cannot be achieved with stable infinite impulse response (IIR) filters. Thus, IIR designs usually incorporate a postprocessing equalizer that is optimized to reduce the phase distortion of the entire filter bank. However, the analysis and synthesis filters of such an IIR filter bank are not linear phase. In this paper, a computationally simple method to obtain IIR analysis and synthesis filters that possess negligible phase distortion is presented. The method is based on first applying the balanced reduction procedure to obtain nearly allpass IIR polyphase components and then approximating these with perfect allpass IIR polyphase components. The resulting IIR designs already have only negligible phase distortion. However, if required, further improvement may be achieved through optimization of the filter parameters. For this purpose, a suitable objective function is presented. Bounds for the magnitude and phase errors of the designs are also derived. Design examples indicate that the derived IIR filter banks are more efficient in terms of computational complexity than the FIR prototypes and perfect reconstruction FIR filter banks. Although the PR FIR filter banks when implemented with the one-multiplier lattice structure and IIR filter banks are comparable in terms of computational complexity, the former is very sensitive to coefficient quantization effects     

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.     

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     

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。     

On Bounds of Shift Variance in Two-Channel Multirate Filter Banks

Critically sampled multirate FIR filter banks exhibit periodically shift variant behavior caused by nonideal antialiasing filtering in the decimation stage. We assess their shift variance quantitatively by analysing changes in the output signal when the filter bank operator and shift operator are interchanged. We express these changes by a so-called commutator. We then derive a sharp upper bound for shift variance via the operator norm of the commutator, which is independent of the input signal. Its core is an eigensystem analysis carried out within a frequency domain formulation of the commutator, leading to a matrix norm which depends on frequency. This bound can be regarded as a worst case instance holding for all input signals. For two channel FIR filter banks with perfect reconstruction (PR), we show that the bound is predominantly determined by the structure of the filter bank rather than by the type of filters used. Moreover, the framework allows to identify the signals for which the upper bound is almost reached as so-called near maximizers of the frequency-dependent matrix norm. For unitary PR filter banks, these near maximizers are shown to be narrow-band signals. To complement this worst-case bound, we derive an additional bound on shift variance for input signals with given amplitude spectra, where we use wide-band model spectra instead of narrow-band signals. Like the operator norm, this additional bound is based on the above frequency-dependent matrix norm. We provide results for various critically sampled two-channel filter banks, such as quadrature mirror filters, PR conjugated quadrature filters, wavelets, and biorthogonal filters banks.     

一种新型伪镜像滤波器组的实现方法

本文提出了一种新型２Ｍ个通道伪镜像滤波器组的结构和设计方法。本设计方法中，低通原型滤波器采用两级级联线性相位ＦＩＲ的结构，并使得整个分析／综合系统性能近于全恢复，无群时延失真。应用此方法设计伪镜像滤波器组，分析和综合滤波器组的结构和设计方法都比较简单，算法效率高，实现比较容易，这类滤波器组可用于语音的子带编码等。本文最后给出了一个实例。     

基于FPGA的高阶高速FIR滤波器设计与实现

提出了一种基于FPGA的高阶高速FIR滤波器的设计与实现方法。通过一个169阶的均方根升余弦滚降滤波器的设计,介绍了如何应用流水线技术来设计高阶高速FIR滤波器,并且对所设计的FIR滤波器性能、资源占用进行了分析。     

Linear-phase perfect reconstruction filter bank: lattice structure,design, and application in image coding

A lattice structure for an M-channel linear-phase perfect reconstruction filter bank (LPPRFB) based on the singular value decomposition (SVD) is introduced. The lattice can be proven to use a minimal number of delay elements and to completely span a large class of LPPRFBs: all analysis and synthesis filters have the same FIR length, sharing the same center of symmetry. The lattice also structurally enforces both linear-phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform (GLBT) with an arbitrary number of channels M and arbitrarily large overlap. The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions, which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the potential of the new transforms     

高速FIR滤波器的流水线结构

通过一个13阶线性相位的平方根升余弦滚降FIR数字滤波器的结构设计，介绍了如何应用流水线技术来设计高速FIR滤波器。考虑到FPGA的容量问题，对采用流水线技术之后的FIR滤波器占用的硬件资源进行了分析，得出一些结论。     

Pade table, continued fraction expansion, and perfectreconstruction filter banks

We investigate the relationships among the Pade table, continued fraction expansions and perfect reconstruction (PR) filter banks. We show how the Pade table can be utilized to develop a new lattice structure for general two-channel bi-orthogonal perfect reconstruction (PR) filter banks. This is achieved through characterization of all two-channel biorthogonal PR filter banks. The parameterization found using this method is unique for each filter bank. Similar to any other lattice structure, the PR property is achieved structurally and the quantization of the parameters of the lattice does not effect this property. Furthermore, we demonstrate that for a given filter, the set of all complementary filters can be uniquely specified by two parameters, namely, the end-to-end delay of the system and a scalar quantity. Finally, we investigate the convergence of the successive filters found through the proposed lattice structure and develop a sufficient condition for this convergence