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.     

Eigenstructure approach for characterization of FIR PR filterbankswith order one polyphase

A new approach for the characterization of M-channel finite impulse response (FIR) perfect reconstruction (PR) filterbanks is proposed. By appropriately restricting the eigenstructure of the polyphase matrix of the bank, a complete characterization of order-one polyphase matrices is obtained in which the polynomial part is in a block diagonal form. Nilpotent matrices play a crucial role in the structure. This structure allows imposing restrictions on the order of the inverse of the polyphase matrix and/or analysis-synthesis delay (reconstruction delay). Next, we derive an alternate complete characterization in terms of the degree of the determinant and the McMillan degree of order-one polyphase matrix, which we call the dyadic-based characterization. The characterization of Vaidyanathan and Chen (1995) for matrices with anticausal inverse turns out to be a special case of the proposed characterization. The dyadic-based characterization is more suitable for design without any above-mentioned restriction since it allows better initialization. We finally present design examples with different cost functions     

基于多相滤波的ILS信号的数字正交变换技术

为了尽可能多地用软件来实现仪表着陆系统（ILS）接收机的信号处理,需要对导航信号进行数字化,而基于多相滤波的数字正交变换无需正交本振,结构上也更适用于实时处理系统。介绍了ILS的工作原理及信号体制,研究了基于多相滤波的ILS信号的数字正交变换技术,通过MATLAB仿真验证了设计的可行性。     

On factorization of M-channel paraunitary filterbanks

We systematically investigate the factorization of causal finite impulse response (FIR) paraunitary filterbanks with given filter length. Based on the singular value decomposition of the coefficient matrices of the polyphase representation, a fundamental order-one factorization form is first proposed for general paraunitary systems. Then, we develop a new structure for the design and implementation of paraunitary system based on the decomposition of Hermitian unitary matrices. Within this framework, the linear-phase filterbank and pairwise mirror-image symmetry filterbank are revisited. Their structures are special cases of the proposed general structures. Compared with the existing structures, more efficient ones that only use approximately half the number of free parameters are derived. The proposed structures are complete and minimal. Although the factorization theory with or without constraints is discussed in the framework of M-channel filterbanks, the results can be applied to wavelets and multiwavelet systems and could serve as a general theory for paraunitary systems     

A subband adaptive filter allowing maximally decimation

Conventional subband adaptive filters (ADF) using filter banks have shown degradation in performance because of the nonideal nature of filter banks. For this problem, the authors propose an alias free subband structure for adaptive filtering using polyphase frequency sampling filter (FSF) banks. As a preliminary, they make it clear that the conventional polyphase discrete Fourier transform (DFT) bank is equivalent to a class of the FSF bank. Then, they propose a new class of ADF using the FSF banks based on the frequency sampling theorem. As a result, the proposed technique enables subband adaptive filtering without the degradation effect of both the aliasing and cross terms, even if one chooses critical subsampling     

Theory and design of dyadic subband tree structures with embeddedPDF-optimized quantizers

We develop a methodology for the analysis of signal quantization effects in critically sampled dyadic subband tree structures using a nonlinear gain-plus-additive-noise model for the probability density function (PDF)-optimized quantizer. We constrain the two-band nonquantized and uncompensated structure at each level to be perfect reconstruction (PR). We develop an equivalent uniform filter bank followed by its polyphase structure described by primitive submatrices and compute a rigorously correct mean squared error (MSE) in the frequency domain using cyclostationary concepts in terms of: (1) the allocated quantizer bits; (2) the filter coefficients; (3) an embedded compensation parameter vector. This MSE is then minimized over all three items above. Our optimization method is applied to the specific case of a four-channel dyadic tree with average bit rate constraint. This tree is represented by an eight-channel polyphase equivalent whose interchannel signals are correlated. We show how to represent rigorously the correlation of random noise between channels due to the embedded quantizers. Our design of paraunitary and biorthogonal structures with identical and nonidentical stages is performed, compared, and validated by computer simulation under the assumption of uncorrelated cross band noise. The nonidentical stage biorthogonal filter bank turned out to have the best performance in MSE sense, but the most robust structure is the nonidentical stage paraunitary filter bank     

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     

The generalized lapped pseudo-biorthogonal transform: oversampled linear-phase perfect reconstruction filterbanks with lattice structures

We investigate a lattice structure for a special class of N-channel oversampled linear-phase perfect reconstruction filterbanks with a decimation factor M smaller than N. We deal with systems in which all analysis and synthesis filters have the same finite impulse response (FIR) length and share the same center of symmetry. We provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks (FBs). All filter coefficients are parameterized by rotation angles and positive values. The resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem. We consider next the case where we are given the generalized lapped pseudo-biorthogonal transform (GLPBT) lattice structure with specific parameters, and we a priori know the correlation matrix of noise that is added in the transform domain. In this case, we provide an alternative lattice structure that suppress the noise. We show that the proposed systems with the lattice structure cover a wide range of linear-phase perfect reconstruction FBs. We also introduce a new cost function for oversampled FB design that can be obtained by generalizing the conventional coding gain. Finally, we exhibit several design examples and their properties.     

A comparison of hardware implementations of the biorthogonal 9/7 DWT: convolution versus lifting

The filter bank approach for computing the discrete wavelet transform (DWT), which we call the convolution method, can employ either a nonpolyphase or polyphase structure. This work compares filter banks with an alternative polyphase structure for calculating the DWT-the lifting method. We look at the traditional lifting structure and a recently proposed "flipping" structure for implementing lifting. All filter bank structures are implemented on an Altera field-programmable gate array. The quantization of the coefficients (for implementation in fixed-point hardware) plays a crucial role in the performance of all structures, affecting both image compression quality and hardware metrics. We design several quantization methods and compare the best design for each approach: the nonpolyphase filter bank, the polyphase filter bank, the lifting and flipping structures. The results indicate that for the same image compression performance, the flipping structure gives the smallest and fastest, low-power hardware.     

Linear-phase FIR interpolation, decimation, and mth-band filters utilizing the farrow structure

This paper introduces novel linear-phase finite-impulse response (FIR) interpolation, decimation, and Mth-band filters utilizing the Farrow structure. In these new overall filters, each polyphase component (except for one term) is realized using the Farrow structure with a distinct fractional delay. The corresponding interpolation/decimation structures can therefore be implemented using only one set of linear-phase FIR subfilters and one set of multipliers that correspond to the distinct fractional delays. The main advantage of the proposed structures is that they are flexible as to the conversion factors, and this also for an arbitrary set of integer factors, including prime numbers. In particular, they can simultaneously implement several converters at a low cost. The proposed filters can be used to generate both general filters and Mth-band filters for interpolation and decimation by the integer factor M. (In this paper, a general filter for interpolation and decimation by M means a filter having a bandwidth of approximately /spl pi//M without the restriction that /spl pi//M be included in the transition band. This is in contrast to an Mth-band filter whose transition band does include /spl pi//M.) In both cases, the overall filter design problem can be posed as a convex problem, the solution of which is globally optimum. Design examples are included in the paper illustrating the properties and potentials of the proposed filters.     

Results on the factorization of multidimensional matrices for paraunitary filterbanks over the complex field

This paper undertakes the study of multidimensional finite impulse response (FIR) filterbanks. One way to design a filterbank is to factorize its polyphase matrices in terms of elementary building blocks that are fully parameterized. Factorization of one-dimensional (1-D) paraunitary (PU) filterbanks has been successfully accomplished, but its generalization to the multidimensional case has been an open problem. In this paper, a complete factorization for multichannel, two-dimensional (2-D), FIR PU filterbanks is presented. This factorization is based on considering a two-variable FIR PU matrix as a polynomial in one variable whose coefficients are matrices with entries from the ring of polynomials in the other variable. This representation allows the polyphase matrix to be treated as a one-variable matrix polynomial. To perform the factorization, the definition of paraunitariness is generalized to the ring of polynomials. In addition, a new degree-one building block in the ring setting is defined. This results in a building block that generates all two-variable FIR PU matrices. A similar approach is taken for PU matrices with higher dimensions. However, only a first-level factorization is always possible in such cases. Further factorization depends on the structure of the factors obtained in the first level.     

On the perfect reconstruction problem in N-band multirate maximallydecimated FIR filter banks

The problem of finding N-K filters of an N-band maximally decimated FIR analysis filter bank, given K filters, so that FIR perfect reconstruction can be achieved, is considered. The perfect reconstruction condition is expressed as a requirement of unimodularity of the polyphase analysis filter matrix. Based on the theory of Euclidean division for matrix polynomials, the conditions the given transfer functions must satisfy are given, and a complete parameterization of the solution is obtained. This approach provides an interesting alternative to the method of the complementary filter in the case of N>2,K相似文献   

Improved structures of maximally decimated directional filter Banks for spatial image analysis

This paper introduces an improved structure for directional filter banks (DFBs) that preserves the visual information in the subband domain. The new structure achieves this outcome while preserving both the efficient polyphase implementation and the exact reconstruction property. The paper outlines a step-by-step framework in which to examine the DFB, and within this framework discusses how, through the insertion of post-sampling matrices, visual distortions can be removed. In addition to the efficient tree structure, attention is given to the form and design of efficient linear phase filters. Most notably, linear phase IIR prototype filters are presented, together with the design details. These filters can enable the DFB to have more than a three-fold improvement in complexity reduction over quadrature mirror filters (QMFs).     

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。      

A Class of Multimode Transmultiplexers Based on the Farrow Structure

This paper introduces multimode transmultiplexers (TMUXs) in which the Farrow structure realizes the polyphase components of general lowpass interpolation/decimation filters. As various lowpass filters are obtained by one set of common Farrow subfilters, only one offline filter design enables us to cover different integer sampling rate conversion (SRC) ratios. A model of general rational SRC is also constructed where the same fixed subfilters perform rational SRC. These two SRC schemes are then used to construct multimode TMUXs. Efficient implementation structures are introduced and different filter design techniques such as minimax and least-squares (LS) are discussed. By means of simulation results, it is shown that the performance of the transmultiplexer (TMUX) depends on the ripples of the filters. With the error vector magnitude (EVM) as the performance metric, the LS method has a superiority over the minimax approach.     

On orthonormal wavelets and paraunitary filter banks

The known result that a binary-tree-structured filter bank with the same paraunitary polyphase matrix on all levels generates an orthonormal basis is generalized to binary trees having different paraunitary matrices on each level. A converse result that every orthonormal wavelet basis can be generated by a tree-structured filter bank having paraunitary polyphase matrices is then proved. The concept of orthonormal bases is extended to generalized (nonbinary) tree structures, and it is seen that a close relationship exists between orthonormality and paraunitariness. It is proved that a generalized tree structure with paraunitary polyphase matrices produces an orthonormal basis. Since not all phases can be generated by tree-structured filter banks, it is proved that if an orthonormal basis can be generated using a tree structure, it can be generated specifically by a paraunitary tree     

Restoring Coefficient Symmetry in Polyphase Implementation of Linear-Phase FIR Filters

Polyphase implementation of FIR filters effectively reduces the multiplication rate and data storage in a multirate system. However, the coefficients of the polyphase components are no longer symmetric even though the overall filter has a symmetric (or anti-symmetric) impulse response. In this paper, we introduce a new technique that recasts pairs of the original polyphase components as sums or differences of auxiliary pairs of symmetric and anti-symmetric impulse response filters. The coefficient symmetry of these auxiliary polyphase components can be fully exploited to reduce arithmetic complexity without undue complications. Our new technique makes use of the fact that the impulse responses of the non-symmetric polyphase components exist in time-reversed pairs which can be synthesized from pairs of symmetric and anti-symmetric impulse response filters. This results in a factor-of-two reduction in the number of multipliers required to implement the polyphase components.     

Reducing the Multiplier-Complexity of Massively Parallel Polyphase 2D IIR Broadband Beam Filters

The superior broadband performance of 2D IIR frequency-planar beam filters, relative to conventional 2D FIR true-time-delay beamforming, has recently been reported using computational electromagnetics and real-time emulations on an antenna test range, resulting in significant improvements of bit-error-rates (BERs) in the presence of broadband interference. Further, massively parallel systolic VLSI circuit polyphase architectures have also been reported (Madanayake et al. in Int. J. Circuit Theory Appl. 2010) for the case of the direct-form signal flow graph (SFG) architecture, operating at a maximum throughput of M-(antenna)-frames-per-clock-cycle (MFPCC). The superior broadband performance of 2D IIR frequency-planar beam filters is extended here from the direct-form signal flow graph (SFG) architecture (Madanayake et al. in Int. J. Circuit Theory Appl. 2010) to the novel differential-form SFG architecture in order to reduce overall complexity. The proposed method employs a differential-form polyphase 2D IIR frequency-planar beam SFG, and a corresponding circuit architecture, to implement the required input-output 2D space-time difference equation. The resultant digital hardware has the significant advantage of much-reduced multiplier complexity, relative to the direct-form structure. For example, when look-ahead pipelining is not employed and for polyphase architectures having two, three, and four phases, the corresponding reductions in multiplier complexity are 20%, 28.6% and 33.3%, respectively. A proof-of-concept prototype circuit is designed and implemented on a Xilinx Sx35 FPGA device for the two-phase case, operating at a frame-rate of 132 million linear frames per second on the uniform linear array (ULA), corresponding to 2-frames-per-clock-cycle at a circuit clock frequency of 66 MHz. The circuit is optimized for low critical path delays (CPDs) using look-ahead pipelining of order three. For ultra-wideband (UWB) radio-frequency (RF) implementations, in such fields as radio astronomy, radar and wireless communications, custom VLSI versions of the proposed circuits are required.     

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.