A novel center of mass method for estimation of center frequency and spectral edges in CR using filter banks

In this paper, we propose a novel multistage DFT based polyphase filter bank technique using center of mass approach for estimating center frequency, detecting spectral edges and identifying spectral holes in wideband cognitive radio (CR) for efficient utilization of radio frequency spectrum. Spectral holes are identified by measuring energy at the output of individual subband of filter banks. Accuracy of spectral holes detection depends on frequency resolution of subbands and can be increased with an increase in number of DFT points, however, at the expense of computational complexity. In order to reduce complexity our algorithm starts with a coarser spectral resolution in the first stage. If a primary user appears over more than one subband, center frequency can be estimated in the first stage using proposed approach. However, if the primary user appears exclusively within a single subband, center frequency can be estimated at the second stage. For center frequency estimation, we propose a novel center of mass approach to achieve better precision, where mass is related to energy and distance is related to frequency. Exhaustive simulation results show that center frequency estimation using proposed multistage polyphase filter bank based on center of mass reduces computational complexity and has higher precision compared to conventional filter bank methods.     

A new approach to subband adaptive filtering

Subband adaptive filtering has attracted much attention lately. In this paper, we propose a new structure and a new formulation for adapting the filter coefficients. This structure is based on polyphase decomposition of the filter to be adapted and is independent of the type of filter banks used in the subband decomposition. The new formulation yields improved convergence rate when the LMS algorithm is used for coefficient adaptation. As we increase the number of bands in the filter, the convergence rate increases and approaches the rate that can be obtained with a flat input spectrum. The computational complexity of the proposed scheme is nearly the same as that of the fullband approach. Simulation results are included to demonstrate the efficacy of the new approach     

Multirate digital filters, filter banks, polyphase networks, andapplications: a tutorial

The basic concepts and building blocks in multirate digital signal processing (DSP), including the digital polyphase representation, are reviewed. Recent progress, as reported by several authors in this area, is discussed. Several applications are described, including subband coding of waveforms, voice privacy systems, integral and fractional sampling rate conversion (such as in digital audio), digital crossover networks, and multirate coding of narrowband filter coefficients. The M-band quadrature mirror filter (QMF) bank is discussed in considerable detail, including an analysis of various errors and imperfections. Recent techniques for perfect signal reconstruction in such systems are reviewed. The connection between QMF banks and other related topics, such as block digital filtering and periodically time-varying systems, is examined in a pseudo-circulant-matrix framework. Unconventional applications of the polyphase concept are discussed     

Modulated filter banks with arbitrary system delay: efficientimplementations and the time-varying case

We present a new method for the design and implementation of modulated filter banks with perfect reconstruction. It is based on the decomposition of the analysis and synthesis polyphase matrices into a product of two different types of simple matrices, replacing the polyphase filtering part in a modulated filter bank. Special consideration is given to cosine-modulated as well as time-varying filter banks. The new structure provides several advantages. First of all, it allows an easy control of the input-output system delay, which can be chosen in single steps of input sampling rate, independent of the filter length. This property can be used in audio coding applications to reduce pre-echoes. Second, it results in a structure that is nearly twice as efficient as performing the polyphase filtering directly. Perfect reconstruction is a structurally inherent feature of the new formulation, even for nonlinear operations or time-varying coefficients. Hence, the structure is especially suited for the design of time-varying filter banks where both the number of bands as well as the prototype filters can be changed while maintaining perfect reconstruction and critical sampling. Further, a proof of effective completeness is given, and the design of equal magnitude-response analysis and synthesis filter banks is described. Filter design can be performed by nonconstrained optimization of the matrix coefficients according to a given cost function. Design and audio-coding application examples are given to show the performance of the new filter bank     

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     

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.     

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     

Multirate filter banks with block sampling

Multirate filter banks with block sampling were recently studied by Khansari and Leon-Garcia (1993). In this paper, we want to systematically study multirate filter banks with block sampling by studying general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices, respectively. We show that multirate filter banks with block sampling studied by Khansari and Leon-Garcia are special vector filter banks where the transfer matrices are pseudocirculant. We present some fundamental properties for the basic building blocks, such as Noble identities, interchangeability of down/up sampling, polyphase representations of M-channel vector filter banks, and multirate filter banks with block sampling. We then present necessary and sufficient conditions for the alias-free property, finite impulse response (FIR) systems with FIR inverses, paraunitariness, and lattice structures for paraunitary vector filter banks. We also present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling. As an application of this theory, we present all possible perfect reconstruction delay chain systems with block sampling. We also show some examples that are not paraunitary for conventional multirate filter banks but are paraunitary for multirate filter banks with proper block sampling. In this paper, we also present a connection between vector filter banks and vector transforms studied by Li. Vector filter banks also play important roles in multiwavelet transforms and vector subband coding     

Alias-Free Subband Adaptive Filtering With Critical Sampling

To overcome the limitations of a conventional fullband adaptive filtering, various subband adaptive filtering (SAF) structures have been proposed. Properly designed, an SAF will converge faster at a lower computational cost than a fullband structure. However, its design should consider the following two facts: the interband aliasing introduced by the downsampling process degrades its performance, and the filter bank in the SAF introduces additional computational overhead and system delay. In this paper, to fully exploit the benefits of using an SAF, an almost alias-free SAF structure with critical sampling is proposed. The interband alising is removed from the subband signal by isolating the aliasing using a bandwidth-increased analysis filter. Computer simulations show that the proposed structure converges faster than both an equivalent fullband structure at lower computational complexity and recently proposed SAF structures for a colored input.     

One- and two-level filter-bank convolvers

In a recent paper, it was shown in detail that in the case of orthonormal and biorthogonal filter banks we can convolve two signals by directly convolving the subband signals and combining the results. In this paper, we further generalize the result. We also derive the statistical coding gain for the generalized subband convolver. As an application, we derive a novel low sensitivity structure for FIR filters from the convolution theorem. We define and derive a deterministic coding gain of the subband convolver over direct convolution for a fixed wordlength implementation. This gain serves as a figure of merit for the low sensitivity structure. Several numerical examples are included to demonstrate the usefulness of these ideas. By using the generalized polyphase representation, we show that the subband convolvers, linear periodically time varying systems, and digital block filtering can be viewed in a unified manner. Furthermore, the scheme called IFIR filtering is shown to be a special case of the convolver     

Design of M‐Channel IIR Uniform DFT Filter Banks Using Recursive Digital Filters

In this paper, we propose a method for designing a class of M‐channel, causal, stable, perfect reconstruction, infinite impulse response (IIR), and parallel uniform discrete Fourier transform (DFT) filter banks. It is based on a previously proposed structure by Martinez et al. [1] for IIR digital filter design for sampling rate reduction. The proposed filter bank has a modular structure and is therefore very well suited for VLSI implementation. Moreover, the current structure is more efficient in terms of computational complexity than the most general IIR DFT filter bank, and this results in a reduced computational complexity by more than 50% in both the critically sampled and oversampled cases. In the polyphase oversampled DFT filter bank case, we get flexible stop‐band attenuation, which is also taken care of in the proposed algorithm.     

Filter bank design for a subband adaptive filtering structure with critical sampling

Subband adaptive filtering structures are attractive in applications such as acoustic echo cancellation and channel equalization, due to their properties of decorrelating the input signal and reducing the computational complexity. Recently, a new adaptive filtering structure with critical sampling was proposed. In this paper, we describe an optimization procedure to select the analysis and synthesis filter banks of this new subband structure, so that minimum steady-state mean square error or fastest convergence rate can be achieved. Such filter-bank design method is based on a theoretical analysis of the convergence properties of the adaptation algorithm and uses a nonlinear optimization routine. Computer simulations illustrate the convergence improvements that can be obtained with the filter banks designed by the proposed method.     

Discrete Zak transforms, polyphase transforms, and applications

We consider three different versions of the Zak (1967) transform (ZT) for discrete-time signals, namely, the discrete-time ZT, the polyphase transform, and a cyclic discrete ZT. In particular, we show that the extension of the discrete-time ZT to the complex z-plane results in the polyphase transform, an important and well-known concept in multirate signal processing and filter bank theory. We discuss fundamental properties, relations, and transform pairs of the three discrete ZT versions, and we summarize applications of these transforms. In particular, the discrete-time ZT and the cyclic discrete ZT are important for discrete-time Gabor (1946) expansion (Weyl-Heisenberg frame) theory since they diagonalize the Weyl-Heisenberg frame operator for critical sampling and integer oversampling. The polyphase representation plays a fundamental role in the theory of filter banks, especially DFT filter banks. Simulation results are presented to demonstrate the application of the discrete ZT to the efficient calculation of dual Gabor windows, tight Gabor windows, and frame bounds     

分布式子带自适应滤波算法

为了解决分布式最小均方算法在输入信号相关性较高时收敛速度较慢、分布式仿射投影算法计算复杂度较高等问题,本文提出了两种分布式子带自适应滤波算法,即递增式和扩散式子带自适应滤波算法.分布式子带自适应滤波算法将节点信号进行子带分割来降低信号的相关性,从而加快收敛速度.由于用于子带分割的滤波器组中包含了抽取单元,所以分布式子带自适应滤波算法和对应的分布式最小均方算法的计算复杂度相近.仿真结果表明,与分布式最小均方算法相比,分布式子带自适应滤波算法具有更好的收敛性能.     

Least squares approximation of perfect reconstruction filter banks

Designing good causal filters for perfect reconstruction (PR) filter banks is a challenging task due to the unusual nature of the design constraints. We present a new least squares (LS) design methodology for approximating PRFBs that avoids most of these difficult constraints. The designer first selects a set of subband analysis filters from an almost unrestricted class of rational filters. Then, given some desired reconstruction delay, this design procedure produces the causal and rational synthesis filters that result in the best least squares approximation to a PRFB. This technique is built on a multi-input multi-output (MIMO) system model for filter banks derived from the filter bank polyphase representation. Using this model, we frame the LS approximation problem for PRFBs as a causal LS equalization problem for MIMO systems. We derive the causal LS solution to this design problem and present an algorithm for computing this solution. The resulting algorithm includes a MIMO spectral factorization that accounts for most of the complexity and computational cost for this design technique. Finally, we consider some design examples and evaluate their performance     

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.     

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     

Architectural Synthesis of Computational Engines for Subband Adaptive Filtering

Architectural synthesis of low-power computational engines (hardware accelerators) for a subband-based adaptive filtering algorithm is presented. The full-band least mean square (LMS) adaptive filtering algorithm, widely used in various applications, is confronted by two problems, viz., slow convergence when the input correlation matrix is ill-conditioned, and increased computational complexity for applications involving use of large adaptive filter orders. Both of these problems can be overcome by the use of a subband-based normalized LMS (NLMS) adaptive filtering algorithm. Since this algorithm is not amenable to pipelining, delayed coefficient adaptation in the NLMS updation is used, which provides the required delays for pipelining. However, the convergence speed of this subband-based delayed NLMS (DNLMS) algorithm degrades with increase in the adaptation delay. We first present a pipelined subband DNLMS adaptive filtering architecture with minimal adaptation delay for any given sampling rate. The architecture is synthesized by using a number of function preserving transformations on the signal flow graph (SFG) representation of the subband DNLMS algorithm. With the use of carry-save arithmetic, the pipelined architecture can support high sampling rates limited only by the delay of two full adders and a 2-to-1 multiplexer. We then extend this synthesis methodology to synthesize a pipelined subband DNLMS architecture whose power dissipation meets a specified budget. This low-power architecture exploits the parallelism in the subband DNLMS algorithm to meet the required computational throughput. The architecture exhibits a novel tradeoff between algorithmic performance (convergence speed) and power dissipation. Finally, we incorporate configurability for filter order, sample period, power reduction factor, number of subbands and decimation/interpolation factor in the low-power architecture, thus resulting in a low-power subband computational engine for adaptive filtering.     

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     

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