General analysis of two-band QMF banks

The two-channel perfect-reconstruction quadrature-mirror-filter banks (PR QMF banks) are analyzed in detail by assuming arbitrary analysis and synthesis filters. Solutions where the filters are FIR or IIR correspond to the fact that a certain function is monomial or nonmonomial, respectively. For the monomial case, the design problem is formulated as a nonlinear constrained optimization problem. The formulation is quite robust and is able to design various two-channel filter banks such as orthogonal and biorthogonal, arbitrary delay, linear-phase filter banks, to name a few. Same formulation is used for causal and stable PR IIR filter bank solutions     

Technique for design of two-channel approximately linear phase QMFfilter bank and its application to image compression

The two-channel QMF filter bank based on allpass sections is one of the best known circuits for building up a multi-channel filter bank for signal compression. An analysis-synthesis combination can satisfy two of the three perfect reconstruction (PR) conditions. The third, the phase condition, can be met to any desired accuracy. However, when applied to images, PR is possible because non-causality can be allowed in the synthesis bank. The development of explicit formulae for the coefficients of such filters is considered. The resulting QMF designs can be used in a wavelet structure for image compression problems using rectangular or diamond symmetry. Several design examples are given, including a comparison of performance with FIR filter banks for such problems     

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     

Two-Channel FIR Filter Banks Utilizing the FRM Approach

The frequency-response masking (FRM) approach has been introduced as a means of generating narrow transition band linear-phase finite impulse response (FIR) filters with a low arithmetic complexity. This paper proposes an approach for synthesizing two-channel maximally decimated FIR filter banks utilizing the FRM technique. For this purpose, a new class of FRM filters is introduced. Filters belonging to this class are used for synthesizing nonlinear-phase analysis and synthesis filters for two types of two-channel filter banks. For the first type, there exist no phase distortion and aliasing errors, but this type suffers from a small amplitude distortion as for the well-known quadrature mirror filter (QMF) banks. Compared to conventional QMF filter banks, the proposed banks lower significantly the overall arithmetic complexity at the expense of a somewhat increased overall filter bank delay in applications demanding narrow transition bands. For the second type, there are also small aliasing errors, allowing one to reduce the arithmetic complexity even further. Efficient structures are introduced for implementing the proposed filter banks, and algorithms are described for maximizing the stopband attenuations of the analysis and synthesis filters in the minimax sense subject to the given allowable amplitude and/or aliasing errors. Examples are included illustrating the benefits provided by the proposed filter banks.     

Design of non-uniform filter bank transmultiplexer with canonical signed digit filter coefficients

Design of non-uniform filter bank transmultiplexer (NUFB TMUX) with canonical signed digit (CSD) coefficients is presented. NUFB TMUX is preferred in a multicarrier communication system when applications with different data rates are to be multiplexed. If the filter coefficients are represented in CSD format, the hardware complexity of the NUFB TMUX can be reduced. A continuous coefficient NUFB TMUX is designed and the coefficients of the filters are synthesised in CSD format using genetic algorithm (GA). Separate objective functions are formulated for the fitness evaluation of the filters. Chromosomes are encoded as ternary digit strings. New crossover and mutation techniques are introduced to preserve the canonical property of the signed power of two (SPT) representations. For the fast convergence of the GA, positiondependent probability of mutation is used. Simulation results show that the CSD coefficient NUFB TMUX designed using the proposed algorithm has better signal-to-interference ratio (SIR) than that of continuous coefficient NUFB TMUX and CSD coefficient NUFB TMUX obtained by rounding. Frequency responses of its filters are better than that of the filters in CSD coefficient NUFB TMUX obtained by rounding and comparable with that of continuous coefficient NUFB TMUX.     

A direct approach to the design of QMF banks via frequency domainoptimization

This paper studies the design of quadrature mirror filter (QMF) banks via frequency domain optimization. A direct approach is adopted that gives the necessary and sufficient condition for perfect reconstruction (PR). While analysis filter banks are designed to achieve frequency domain specifications required for subband coding, synthesis filter banks are designed to minimize the reconstruction error in frequency domain. The criterion used to measure the reconstruction error is H_∞ or Chebyshev norm (sup-norm). State-space solutions are derived for the H_∞ optimization, and numerical algorithms are developed to obtain the optimal synthesis filter bank. Moreover, the asymptotic PR property is established for optimal H_∞ solution of the synthesis filter bank     

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.     

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)     

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     

用于信号逼近的最佳正交子波选择方法

离散子波变换将离散时间信号分解为一系列分辨率下的离散逼近和离散细节。紧支的正交规范子波与完全重建正交镜象滤波器组相对应。本文提出一种用于信号最佳逼近的正交子波选择方法,即选择满足一定条件的滤波器的方法。通过对滤波器参数化,可以将带约束的最优化问题转化为无约束最优化问题,通过对参数在一定范围内的搜索,得到最优解。文中给出了计算机模拟的结果。     

Designs and architectures of filter bank trees for spectrally efficient multi-user communications: review, modifications and extensions of wavelet packet filter bank trees

A review is presented first of the evolution of transmultiplexers since about 1966, in the context of a long progression of theoretical advances and developments leading to recent proposals to fundamentally improve OFDM type systems using principles of perfect reconstruction filter (PRF) banks. The equivalence of transmultiplexers to OFDM type multi-user systems is discussed. The desirable goals for performance and implementation of transmultiplexers or multiband, multiuser communication systems that are addressed and met in this paper using filter bank trees are set down. Then modifications and extensions are presented of the designs and architectures of wavelet packet based synthesis and analysis pairs of filter bank trees (Sablatash and Lodge in Digital Signal Process 13: 58–92, 2003) that can be used as transmultiplexers. These exhibit a number of advantages over the previous designs and address three shortcomings of the designs used to illustrate basic principles in Sablatash and Lodge (Digital Signal Process 13:58–92, 2003). The first of these is the asymmetry of the magnitude frequency responses of the multiplexer channels, which is addressed using a symmetric design for a lowpass and highpass quadrature mirror filter (QMF) pair described herein. The second is the problem of minimizing the total delay of the signal in passing through the analysis and synthesis filter banks. This is addressed using an architecture involving DFT polyphase synthesis filter banks to replace the wideband VSB filters at the roots of the two identical synthesis filter bank trees, but results in the multiplexer having fewer levels. In this way a tradeoff is effected of lower delay and complexity with fewer levels of bandwidth on demand. At the receiver matching DFT polyphase analysis filters and the other matching analysis filters are implemented. The third shortcoming is the difficulty in designing a synchronization scheme if the filters in the synthesis and analysis filter banks have non-linear phase. This is addressed by designing linear phase filters that do not affect the ISI to any significant degree for communication purposes, although exact perfect reconstruction is lost, but greatly ease and improve the design of the synchronization scheme. Relationships of this paper and its advantages over recent research studies and IEEE 802.22 standards proposals using PR filter banks for multi-user systems to greatly improve on OFDM systems are discussed. Financial support under Industry Canada’s Spectrum Research Funding is gratefully acknowledged.     

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     

M带离散时间子波变换

二带连续时间子波变换可由无限级树形正交镜像滤波器(QMF)组产生,类似地,二带离散时间子波变换可表示为有限级树形QMF组。该文对二带离散时间子波变换进行了推广,给出了M带离散时间于波变换,并研究了M带离散时间子波变换与M带仿酉滤波器组之间的关系。结果表明,在L级M带树形滤波器组中,如果每级滤波器组是仿酉滤波器组,则该树形滤波器组所产生的离散时间子波基是正交基。     

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.     

Second Order Taylor Polyphase Reconstruction of Periodic-Nonuniform Samples in Time-Interleaved ADCs

This paper focuses on a filter bank approach for reconstructing periodic-nonuniform samples using second order Taylor polyphase decomposition for a four-channel time-interleaved ADC (TIADC). The proposed method is based on a parametric linear least squares approach to find the coefficients of synthesis filters used for signal reconstruction of periodic-nonuniform samples. The coefficients of synthesis filters found by linear least squares approach satisfy the general condition for an alias-free filter bank.     

A simple method for designing high-quality prototype filters forM-band pseudo QMF banks

Discusses a new method for designing the prototype filters necessary to implement M-band pseudo QMF banks. This method does not rely on the traditional nonlinear optimization used in past work but rather optimizes a single parameter on a convex error surface, consistently delivering the best equiripple filter possible while minimizing the overlapped passband distortion. A very simple algorithm for designing lowpass prototype filters suitable for use in pseudo QMF banks is described. To illustrate the applicability of this algorithm, two different filters are designed, both for such applications as wideband audio coding that require high quality reconstructed signals     

An iterative approach for minimax design of multidimensional quadrature mirror filters

In this paper, a nest of iterative techniques is proposed for the minimax design of quadrature mirror filter (QMF) banks. The method can be generalized such that multidimensional QMF banks can be designed by the proposed algorithm. For a given weighting function, an iterative method is used to minimize the objective error function in the inner iterations. To further reduce the peak error of overall magnitude response, an iterative weighting-updated technique is adopted in the outer iterations. Comparing with the existing works concern the design of perfect-reconstruction QMF banks, only one of the filters is needed to be designed under the cost of magnitude distortion, but the system complexity can be reduced drastically. Several examples, including design of 2-D and 3-D QMF banks, will be presented to demonstrate the effectiveness of the proposed method.     

Time-varying analysis-synthesis systems based on filter banks andpost filtering

Perfect reconstruction (PR) time-varying analysis-synthesis filter banks are those in which the filters are allowed to change from one set of PR filter banks to another as the input signal is being processed. Such systems have the property that, in the absence of coding, they faithfully reconstruct every sample of the input. Various methods have been reported for the time-varying filter bank design; all of them, however, utilize structures for conventional PR filter banks. These conventional structures that have been applied in the past result in different limitations in each method. This paper introduces a new structure for exactly reconstructing time-varying analysis-synthesis filter banks. This structure consists of the conventional filter bank followed by a time-varying post filter. The new method requires neither the redesign of the analysis sections nor the use of any intermediate analysis filters during transition periods. It provides a simple and elegant procedure for designing time-varying filter banks without the disadvantages of the previous methods     

Polyphase Conditions and Structures for 2-D Quincunx FIR Filter Banks Having Quadrantal or Diagonal Symmetries

In this brief, we derive conditions on the polyphase matrix of 2-D finite-impulse response (FIR) quincunx filter banks, for the filters in the filter bank to have quadrantal or diagonal symmetry. These conditions provide a framework for synthesizing polyphase structures which structurally enforce the symmetry. This is demonstrated by constructing examples of small parameterized matrix structures which satisfy the above conditions, thus giving perfect reconstruction FIR quincunx filter banks with quadrantal or diagonally symmetric short-kernel (i.e., short-support) filters. It is also shown that cascades of the above constructed small structures can be used to construct filters of higher order.     

A Multimode Transmultiplexer Structure

This paper introduces a multimode transmultiplexer (TMUX) structure capable of generating a large set of user-bandwidths and center frequencies. The structure utilizes fixed integer sampling rate conversion (SRC) blocks, Farrow-based variable interpolation and decimation structures, and variable frequency shifters. A main advantage of this TMUX is that it needs only one filter design beforehand. Specifically, the filters in the fixed integer SRC blocks as well as the subfilters of the Farrow structure are designed only once. Then, all possible combinations of bandwidths and center frequencies are obtained by properly adjusting the variable delay parameter of the Farrow-based filters and the variable parameters of the frequency shifters. The paper includes examples for demonstration. It also shows that, using the rational SRC equivalent of the Farrow-based filters, the TMUX can be described in terms of conventional multirate building blocks which may be useful in further analysis of the overall system.