Subband adaptive filtering with real-valued subband signals foracoustic echo cancellation

Adaptive filtering in subbands was originally proposed to overcome the limitations of conventional least-mean-square (LMS) algorithms. In general, subband adaptive filters offer computational savings, as well as faster convergence over the conventional LMS algorithm. However, improvements to current subband adaptive filters could be further enhanced by a more elegant choice of their design/structure. Classical subband adaptive filters employ DFT-based analysis and synthesis filter banks which results in subband signals that are complex-valued. The authors modify the structure of subband adaptive filters by using single-sideband (SSB) modulated analysis and synthesis filter banks, which result in subband signals that are real-valued. This simplifies the realisation of subband adaptive filters     

On the optimality of nonunitary filter banks in subband coders

This paper investigates the energy compaction capabilities of nonunitary filter banks in subband coding. It is shown that nonunitary filter banks have larger coding gain than unitary filter banks because of the possibility of performing half-whitening in each channel. For long filter unit pulse responses, optimization of subband coding gain for stationary input signals results in a filter bank decomposition, where each channel works as an optimal open-loop DPCM system. We derive a formula giving the optimal filter response for each channel as a function of the input power spectral density (PSD). For shorter filter bank responses, good gain is obtained by suboptimal half-whitening responses, but the impact on the theoretical coding gain is still highly significant. Image coding examples demonstrate that better performance is achieved using nonunitary filter banks when the input images are in correspondence with the signal model.     

Recursive time-varying filter banks for subband image coding

Filter banks, subband/wavelets, and multiresolution decompositions that employ recursive filters have been considered previously and are recognized for their efficiency in partitioning the frequency spectrum. This paper presents an analysis of a new infinite impulse response (IIR) filter bank in which these computationally efficient filters may be changed adaptively in response to the input. The new filter bank framework is presented and discussed in the context of subband image coding. In the absence of quantization errors, exact reconstruction can be achieved. By the proper choice of an adaptation scheme, it is shown that recursive linear time-varying (LTV) filter banks can yield improvement over conventional ones.     

Optimal filter banks for signal reconstruction from noisy subbandcomponents

Conventional design techniques for analysis and synthesis filters in subband processing applications guarantee perfect reconstruction of the original signal from its subband components. The resulting filters, however, lose their optimality when additive noise due, for example, to signal quantization, disturbs the subband sequences. We propose filter design techniques that minimize the reconstruction mean squared error (MSE) taking into account the second order statistics of signals and noise in the case of either stochastic or deterministic signals. A novel recursive, pseudo-adaptive algorithm is proposed for efficient design of these filters. Analysis and derivations are extended to 2-D signals and filters using powerful Kronecker product notation. A prototype application of the proposed ideas in subband coding is presented. Simulations illustrate the superior performance of the proposed filter banks versus conventional perfect reconstruction filters in the presence of additive subband noise     

Adaptive array processing for multipath fading mitigation viaexploitation of filter banks

We propose subband adaptive array processing for mitigation of both intersymbol interference (ISI) and cochannel interference (CCI) in digital mobile communications. Subband adaptive array processing employs filter banks in a front end to an adaptive array receiver. By decomposing the signals into a set of subband signals, the analysis filters enhance the correlation of multipath rays in each subband. This enhancement is blind in the sense that no a priori knowledge of the temporal characteristics or spatial signatures of arriving signals is required. With the increased coherence, the desired signal can be effectively equalized by subsequent spatial processing. Further, the CCI signals and their multipaths can be suppressed with fewer degrees of freedom. The effects of quadrature mirror filter and discrete Fourier transform filter banks on multipath correlation are delineated     

Optimization of filter banks using cyclostationary spectralanalysis

This paper proposes a design method of optimal biorthogonal FIR filter banks that minimize the time-averaged mean squared error (TAMSE) when the high-frequency subband signal is dropped. To study filter banks from a statistical point of view, cyclostationary spectral analysis is used since the output of the filter bank for a wide-sense stationary input is cyclostationary. First, the cyclic spectral density of the output signal is derived, and an expression for the TAMSE is presented. Then, optimal filter banks are given by minimizing the TAMSE with respect to the coefficients of the filters under the biorthogonality condition. By imposing the additional constraints on the coefficients, the optimal biorthogonal linear phase filter bank can be obtained     

基于提升方法的滤波器组因子分解

讨论了FIR滤波器组的分解.2通道完全重构FIR 子波变换分解可为有限步的提升步骤,使用Laurent多项式的辗转相除法给出了这种分解的一个代数方法的证明;证明了二通道子波变换的分解定理不能平行推广到2M通道滤波器组.提出使用M-通道滤波器组构造2M-通道滤波器组,它由多相矩阵的分块化和提升方法实现,这种方法易于构造非线性滤波器组,如整数变换.     

Comparative Analysis of Shift Variance and Cyclostationarity in Multirate Filter Banks

Multirate filter banks introduce periodic time-varying phenomena into their subband signals. The nature of these effects depends on whether the signals are regarded as deterministic or as random signals. We analyze the behavior of deterministic and wide-sense stationary (WSS) random signals in multirate filter banks in a comparative manner. While aliasing in the decimation stage causes subband energy spectra of deterministic signals to become shift-variant, imaging in the interpolation stage causes WSS random signals to become WS cyclostationary (WSCS). We provide criteria to quantify both shift variance and cyclic nonstationarity. For shift variance, these criteria separately assess the shift dependence of energy and of energy spectra. Similarly for nonstationarity, they separately assess the nonstationary behavior of signal power and of power spectra. We show that, under aliasing cancellation and perfect reconstruction constraints of paraunitary and biorthogonal filter banks, these criteria evaluate the behavior of deterministic and WSS random signals in a consistent, dual way. We apply our criteria to paraunitary and biorthogonal filter banks as well as to orthogonal block transforms, and show that, for critical signals such as lines or edges in image data, the biorthogonal 9/7 filters perform best among these     

An integrated framework for adaptive subband image coding

Previous work on filter banks and related expansions has revealed an interesting insight: different filter bank trees can be regarded as different ways of constructing orthonormal bases for linear signal expansion. In particular, fast algorithms for finding best bases in an operational rate-distortion (R/D) sense have been successfully used in image coding. Independently of this work, other research has also explored the design of filter banks that optimize energy compaction for a single signal or a class of signals. In this paper, we integrate these two different but complementary approaches to best-basis design and propose a coding paradigm in which subband filters, tree structure, and quantizers are chosen to optimize the R/D performance. These coder attributes represent side information. They are selected from a codebook designed off-line from training data, using R/D as the design criterion. This approach provides a rational framework in which to explore alternatives to empirical design of filter banks, quantizers, and other coding parameters. The on-line coding algorithm is a relatively simple extension of current R/D-optimal coding algorithms that operate with fixed filter banks and empirically designed quantizer codebooks. In particular, it is shown that selection of the best adapted filter bank from the codebook is computationally elementary     

Optimal signal reconstruction in noisy filter bank systems:multirate Kalman synthesis filtering approach

A multirate Kalman synthesis filter is proposed in this paper to replace the conventional synthesis filters in a noisy filter bank system to achieve optimal reconstruction of the input signal. Based on an equivalent block representation of subband signals, a state-space model is introduced for an M-band filter bank system with subband noises. The composite effect of the input signal, analysis filter bank, decimators, and interpolators is represented by a multirate state-space model. The input signal is embedded in the state vector, and the corrupting noises in subband paths are generally considered as additive noises. Hence, the signal reconstruction problem in the M-band filter bank systems with subband noises becomes a state estimation procedure in the resultant multirate state-space model. The multirate Kalman filtering algorithm is then derived according to the multirate state-space model to achieve optimal signal reconstruction in noisy filter bank systems. Based on the optimal state estimation theory, the proposed multirate Kalman synthesis filter provides the minimum-variance reconstruction of the input signal. Two numerical examples are also included. The simulation results indicate that the performance improvement of signal reconstruction in noisy filter bank systems is remarkable     

两带自适应FIR线性相位双正交滤波器组设计

自适应滤波器组设计是多速率滤波器组理论和应用的一个重要方面。由于其频率响应更好匹配于输入信号的统计特性,这类滤波器组可获得更大的子带编码增益。该文研究了两带自适应FIR线性相位双正交滤波器组的设计问题,给出了设计算法,特别是通过最优IIR双正交滤波器组确定初始点(初始滤波器组)的方法。仿真结果表明,得到的滤波器组的子带编码增益远远超过了最优的IIR正交滤波器组,与已有的设计结果比较,编码增益明显提高。     

Two-dimensional filter bank design for optimal reconstruction usinglimited subband information

In this correspondence, we propose design techniques for analysis and synthesis filters of 2-D perfect reconstruction filter banks (PRFB's) that perform optimal reconstruction when a reduced number of subband signals is used. Based on the minimization of the squared error between the original signal and some low-resolution representation of it, the 2-D filters are optimally adjusted to the statistics of the input images so that most of the signal's energy is concentrated in the first few subband components. This property makes the optimal PRFB's efficient for image compression and pattern representations at lower resolutions for classification purposes. By extending recently introduced ideas from frequency domain principal component analysis to two dimensions, we present results for general 2-D discrete nonstationary and stationary second-order processes, showing that the optimal filters are nonseparable. Particular attention is paid to separable random fields, proving that only the first and last filters of the optimal PRFB are separable in this case. Simulation results that illustrate the theoretical achievements are presented.     

On reconstruction methods for processing finite-length signals withparaunitary filter banks

New expressions are developed for the perfect reconstruction of the boundary regions of a finite-length signal after subband processing. The time-invariant filter bank is required to be uniform and paraunitary, using FIR filters regardless of phase or symmetry. They accommodate a linear boundary extension in the analysis section, and avoid periodic extensions or storage of extended subband signals. The reconstruction methods are based on the formulation of linear systems that are built as a function of the filters     

Reduced-delay quadrature mirror filter structures for subband coding of speech

In subband coding systems of speech, quadrature mirror filter (QMF) banks have been used effectively in a tree-structured form for decomposition and alias-free reconstruction of the speech signal. In the present paper, we derive some new causal and noncausal QMF structures which can reduce group delay. These structures are based on even- as well as odd-length finite impulse response filters.     

Combining the discrete wavelet transform and mixed-domain filtering

A novel filtering method is proposed that combines the discrete orthogonal wavelet transform (DWT) with the mixed-domain (mixed-D) filtering method. The method uses the DWT to pre- and postprocess those dimensions of the signal that are transformed to the discrete-frequency domain by mixed-D filtering. Using the DWT in this manner provides a controlled mechanism to partition the spectrum of the input signal into subband signals, which then may be selectively filtered during the linear difference equation (LDE) step of the mixed-D algorithm. It is shown that, when the DWT is computed using filters with ideal high- and lowpass frequency responses, the LDE filters used in the mixed-D filtering stage are unchanged by the introduction of the DWT (although the frequency tuple associated with each LDE filter is altered). This indicates that the mixed-D filtering scheme can be easily used in subband coding systems. Results are given for the filtering of a three-dimensional (3-D) linear trajectory signal, representing a common application in video processing.     

调制混合滤波器组的宽带模拟信号采样研究

对于超宽带模拟信号,很难用单个模拟数字转换器（ADC）直接进行采样。该文提出了一种新的并行调制混合滤波器组结构用于实现超宽带模拟信号的采样,首先,将每一路宽带模拟输入信号进行余弦调制,并用相同的低通模拟滤波器均匀分割输入信号的带宽;然后,采用相同的ADC将子带信号数字化;各路子带信号通过上采样器后用数字综合滤波器综合得到原宽带模拟输入信号的数字重构。综合滤波器采用总体最小二乘准则下的特征值滤波器设计方法得到。本文所提出的系统结构不需要使用高速的采样保持电路,降低了系统实现的难度,并且设计的系统具有与其它混合滤波器组相近的重构性能。仿真结果表明了本方法的有效性。      

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.     

Residual echo signal in critically sampled subband acoustic echocancellers based on IIR and FIR filter banks

The residual echo signal characteristics of critically sampled subband acoustic echo cancellers are analyzed. For finite impulse response (FIR) filter banks, the residual echo signal usually has a relatively broad spectral nature around the subband edges. The residual echo signal of power symmetric infinite impulse response (PS-IIR) filter banks, on the other hand, has very narrowband spectral components around the subband edges. These components can be efficiently removed with PS-IIR notch filters that integrate neatly into the filter banks without introducing perceptually noticeable degradation to the near-end speech. This solution has very low computational complexity and does not impinge on the system performance. Simulation studies with recordings from the cockpit of a car, based on a fast QR least-squares adaptive algorithm, demonstrate the potential of this approach for a practical AEC system     

Synthesis filter bank optimization in two-dimensional separablesubband coding systems

Subband coding is a popular and well established technique used in visual communications, such as image and video transmission. In the absence of quantization and transmission errors, the analysis and synthesis filters in a subband coding scheme can be designed to obtain perfect reconstruction of the input signal, but this is no longer the optimal solution in the presence of quantization of the subband coefficients. We presuppose the use of a two-dimensional (2-D) separable subband scheme and we address the problem of designing, for a given analysis filter bank and assuming uniform quantization of the subband coefficients, the set of row and column synthesis filters that minimize the mean squared reconstruction error at the output of the subband system. Since the corresponding optimization problem is inherently nonlinear, we propose a suboptimal solution that extends a one-dimensional (l-D) optimal filter design procedure, already presented in the literature, to a 2-D separable synthesis filter bank. The separable 2-D extension is not trivial, since the processing in one direction, e.g., the rows, alters the statistics of the signals for the design of the filters in the other direction, e.g., the columns. To further simplify the filter design, we propose to model the input image as a 2-D separable Markov process plus an additive white component. Several design examples using both synthetic signals and real world images are presented, showing that the filters designed using the proposed technique can give a significant gain with respect to the perfect reconstruction solution, especially when the dither technique is used for quantization. The simulation results also show that the proposed image model can be conveniently used in the synthesis filter design procedure.     

The design of 2-D nonseparable directional perfect reconstruction filter banks

In this paper, we develop a directional 2-D nonseparable filter bank that can perfectly reconstruct the downsampled subband signals. The filter bank represents two powerful image and video processing tools: directional subband decomposition and perfect reconstruction. The directional filter banks consist of (1) the input signal and the subband signals modulation, (2) diamond shape prefilter, and (3) four different parallelogram shape prefilters. This paper addresses the design and implementation of a two-band filter bank that is proved to be able to provide perfect reconstruction of the downsampled subband signals. Finally, we use a conventional 1-D half-band filter as a prototype and then apply the McClellan transform for the specific 2-D diamond shape and parallelogram shape subfilters. This method is extremely simple in designing the analysis/synthesis subfilters for the filter bank.