Theory and design of signal-adapted FIR paraunitary filter banks

We study the design of signal-adapted FIR paraunitary filter banks, using energy compaction as the adaptation criterion. We present some important properties that globally optimal solutions to this optimization problem satisfy. In particular, we show that the optimal filters in the first channel of the filter bank are spectral factors of the solution to a linear semi-infinite programming (SIP) problem. The remaining filters are related to the first through a matrix eigenvector decomposition. We discuss uniqueness and sensitivity issues. The SIP problem is solved using a discretization method and a standard simplex algorithm. We also show how regularity constraints may be incorporated into the design problem to obtain globally optimal (in the energy compaction sense) filter banks with specified regularity. We also consider a problem in which the polyphase matrix implementation of the filter bank is constrained to be DCT based. Such constraints may also be incorporated into our optimization algorithm; therefore, we are able to obtain globally optimal filter banks subject to regularity and/or computational complexity constraints. Numerous experiments are presented to illustrate the main features that distinguish adapted and nonadapted filters, as well as the effects of the various constraints. The conjecture that energy compaction and coding gain optimization are equivalent design criteria is shown not to hold for FIR filter banks     

Paraunitary filter banks over finite fields

In real and complex fields, unitary and paraunitary (PU) matrices have found many applications in signal processing. There has been interest in extending these ideas to the case of finite fields. We study the theory of PU filter banks (FBs) in GF(q) with q prime. Various properties of unitary and PU matrices in finite fields are studied. In particular, a number of factorization theorems are given. We show that (i) all unitary matrices in GF(q) are factorizable in terms of Householder-like matrices and permutation matrices, and (ii) the class of first-order PU matrices (the lapped orthogonal transform in finite fields) can always be expressed as a product of degree-one or degree-two building blocks. If q>2, we do not need degree-two building blocks. While many properties of PU matrices in finite fields are similar to those of PU matrices in complex field, there are a number of differences. For example, unlike the conventional PU systems, in finite fields, there are PU systems that are unfactorizable in terms of smaller building blocks. In fact, in the special case of 2×2 systems, all PU matrices that are factorizable in terms of degree-one building blocks are diagonal matrices. We derive results for both the cases of GF(2) and GF(Q) with q>2. Even though they share some similarities, there are many differences between these two cases     

Multidimensional, paraunitary principal component filter banks

In this correspondence, the one-dimensional (1-D) principal component filter banks (PCFB's) derived by Tsatsatsanis and Giannakis (1995) are generalized to higher dimensions. As presented by Tsatsatsanis and Giannakis, PCFB's minimize the mean-squared error (MSE) when only Q out of P subbands are retained. Previously, two-dimensional (2-D) PCFB's were proposed by Tirakis et al. (1995). The work by Tirakis et al. was limited to 2-D signals and separable resampling operators. The formulation presented here is general in that it can easily accommodate signals of arbitrary (yet finite) dimension and nonseparable sampling. A major result presented in this paper is that in addition to minimizing MSE when reconstructing from Q out of p subbands, the PCFB's result in maximizing theoretical coding gain (TCG) thereby performing optimally in a energy compaction sense     

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     

Boundary operation of 2-D nonseparable linear-phase paraunitary filter banks

This paper proposes a boundary operation technique of 2-D nonseparable linear-phase paraunitary filter banks (NS-LPPUFBs) for size limitation. The proposed technique is based on a lattice structure consisting of the 2-D separable block discrete cosine transform and nonseparable support-extension processes. The bases are allowed to be anisotropic with the fixed critically subsampling, overlapping, orthogonal, symmetric, real-valued, and compact-support properties. First, the blockwise implementation is developed so that the basis images can be locally controlled. The local control of basis images is shown to maintain orthogonality. This property leads a basis termination (BT) technique as a boundary operation. The technique overcomes the drawback of NS-LPPUFBs that the popular symmetric extension method is invalid. Through some experimental results of diagonal texture coding, the significance of the BT is verified.     

Linear phase paraunitary filter banks: theory, factorizations anddesigns

M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. The authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. They begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, they develop a minimal factorization for a large class of such systems. This factorization will be proved to be complete for even M. Further, they structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. They then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory     

A new algorithm for linear-phase paraunitary filter banks with pairwise mirror-image frequency responses

Subband filter banks have attracted much attention during the past few years. In this paper, an efficient design algorithm, which leads to linear-phase paraunitary filter banks with pairwise mirror-image frequency responses, is revisited and further studied. New lattice structures are presented to extend the algorithm to the case where the number (M) of channels is odd. Design examples of filter banks with 3 and 5 channels are presented.     

Special paraunitary matrices, Cayley transform, and multidimensional orthogonal filter banks.

We characterize and design multidimensional (MD) orthogonal filter banks using special paraunitary matrices and the Cayley transform. Orthogonal filter banks are represented by paraunitary matrices in the polyphase domain. We define special paraunitary matrices as paraunitary matrices with unit determinant. We show that every paraunitary matrix can be characterized by a special paraunitary matrix and a phase factor. Therefore, the design of paraunitary matrices (and thus of orthogonal filter banks) becomes the design of special paraunitary matrices, which requires a smaller set of nonlinear equations. Moreover, we provide a complete characterization of special paraunitary matrices in the Cayley domain, which converts nonlinear constraints into linear constraints. Our method greatly simplifies the design of MD orthogonal filter banks and leads to complete characterizations of such filter banks.     

CORDIC-lifting factorization of paraunitary filter banks based on the quaternionic multipliers for lossless image coding

Quaternions have offered a new paradigm to the signal processing community: to operate directly in a multidimensional domain. We have recently introduced the quaternionic approach to the design and implementation of paraunitary filter banks: four- and eight-channel linear-phase paraunitary filter banks, including those with pairwise-mirror-image symmetric frequency responses. The hypercomplex number theory is utilized to derive novel lattice structures in which quaternion multipliers replace Givens (planar) rotations. Unlike the conventional algorithms, the proposed computational schemes maintain losslessness regardless of their coefficient quantization. Moreover, the one regularity conditions can be expressed directly in terms of the quaternion lattice coefficients and thus easily satisfied even in finite-precision arithmetic. In this paper, a novel approach to realizing CORDIC-lifting factorization of paraunitary filter banks is presented, which is based on the embedding of the CORDIC algorithm inside the lifting scheme. Lifting allows for making multiplications invertible. The 2D CORDIC engine using sparse iterations and asynchronous pipeline processor architecture based on the embedded CORDIC engine as stage of processor is reported. Also it is necessary to notice, that the quaternion multiplier lifting scheme based on the 2D CORDIC algorithm is the structural decision for the lossless digital signal processing. This approach applies to very practical filter banks, which are essential for image processing, and addresses interesting theoretical questions.     

A simplified lattice factorization for linear-phase paraunitary filter banks with pairwise mirror image frequency responses

In this paper, by extending our previous work on general linear-phase paraunitary filter banks even-channel (LPPUFBs), we develop a new structure for LPPUFBs with the pairwise mirror image (PMI) frequency responses, which is a simplified version of the lattice proposed by Nguyen et al. Our simplification is achieved through trivial matrix manipulations and the cosine-sine (C-S) decomposition of a general orthogonal matrix. The resulting new structure covers the same class of PMI-LPPUFBs as the original lattice, while substantially reducing the number of free parameters involved in the nonlinear optimization. A design example is presented to demonstrate the effectiveness of the new structure.     

Theory and design of two-parallelogram filter banks

It is well known that the analysis and synthesis filters of orthonormal DFT filter banks can not have good frequency selectivity. The reason for this is that each of the analysis and synthesis filters have only one passband. Such frequency stacking (or configuration) in general does not allow alias cancellation when the individual filters have good stopband attenuation. A frequency stacking of this nature is called nonpermissible and should be avoided if good filters are desired. In a usual M-channel filter bank with real-coefficient filters, the analysis and synthesis filters have two passbands. It can be shown that the configuration is permissible in this case. Many designs proposed in the past demonstrate that filter banks with such configurations can have perfect reconstruction and be good filters at the same time. We develop the two-parallelogram filter banks, which is the class of 2-D filter banks in which the supports of the analysis and synthesis filters consist of two parallelograms. The two-parallelogram filter banks are analyzed from a pictorial viewpoint by exploiting the concept of permissibility. Based on this analysis, we construct and design a special type of two-parallelogram filter banks, namely, cosine-modulated filter banks (CMFB). In two-parallelogram CMFB, the analysis and synthesis filters are cosine-modulated versions of a prototype that has a parallelogram support. Necessary and sufficient conditions for perfect reconstruction of two-parallelogram CMFB are derived     

Oversampled filter banks from extended perfect reconstruction filter banks

Oversampled filter banks are currently being proposed for robust transmission applications. In this paper, we completely characterize multidimensional doubly finite-impulse-response (FIR) filter banks, that is, oversampled filter banks whose dual is FIR. Then, we consider the problem of extending perfect reconstruction critically sampled multidimensional filter banks in order to obtain doubly FIR (DFIR) filter banks. As a result, very simple criteria for constructing DFIR filter banks as extensions of orthogonal filter banks are obtained. This paper also analyzes the problem of constructing totally FIR filter banks, i.e., DFIR filter banks that remain DFIR even when some channels are removed. It is shown that any totally FIR filter bank can be implemented as the cascade of a critically sampled DFIR filter bank whose number of channels is equal to the subsampling factor, a redundant finite-dimensional transform, and a suitable set of delays.     

Nonuniform perfect reconstruction filter banks over lattices withapplication to transmultiplexers

This paper presents a multidimensional multirate theory for signals defined over lattices. We extend the notion of the z transform and present linear periodically-shift-varying (LPSV) systems. We use this theory to study transmultiplexers for signals defined over arbitrary lattices (nonuniform or unequal-band case). We give dimensionality conditions for perfect reconstruction and determine the form of the solutions. Finally, we study tree-structured transmultiplexing systems. Such systems permit us to design nonuniform filter banks from uniform filter banks. Furthermore, their multistage implementation allows lower complexity     

THEORY AND DESIGN OF UNIFORM FILTER BANKS USING ALL-PASS FILTERS

In this paper,the theory of uniform filter banks using all-pass tilters is furtherdeveloped.A new structure of two stage filter banks using all-pass filter is proposed,The pre-stage is half-band filter with period,the post-stage is two sets of band-pass filter banks.Thepre-stage filter stop-band just controls the don't-care-band of the post-stage filter banks usingall-pass polyphase,so as to realize a continuous stop-band property Moreover,a method ofsynthesizing filter bank is derived,which eliminates aliasing and amplitude distortions of theanalysis/synthesis system Finally,an example is given.     

Order statistic filter banks

Filter banks play a major role in multirate signal processing where these have been successfully used in a variety of applications. In the past, filter banks have been developed within the framework of linear filters. It is well known, however, that linear filters may have less than satisfactory performance whenever the underlying processes are non-Gaussian. We introduce the nonlinear class of order statistic (OS) filter banks that exploit the spectral characteristics of the input signal as well as its rank-ordering structure. The attained subband signals provide frequency and rank information in a localized time interval. OS filter banks can lead to significant gains over linear filter banks, particularly when the input signals contain abrupt changes and details, as is common with image and video signals. OS filter banks are formed using traditional linear filter banks as fundamental building blocks. It is shown that OS filter banks subsume linear filter banks and that the latter are obtained by simple linear transformations of the former. To illustrate the properties of OS filter banks, we develop simulations showing that the learning characteristics of the LMS algorithm, which are used to optimize the weight taps of OS filters, can be significantly improved by performing the adaptation in the OS subband domain.     

Theory and factorization for a class of structurally regular biorthogonal filter banks

Regularity is a fundamental and desirable property of wavelets and perfect reconstruction filter banks (PRFBs). Among others, it dictates the smoothness of the wavelet basis and the rate of decay of the wavelet coefficients. This paper considers how regularity of a desired degree can be structurally imposed onto biorthogonal filter banks (BOFBs) so that they can be designed with exact regularity and fast convergence via unconstrained optimization. The considered design space is a useful class of M-channel causal finite-impulse response (FIR) BOFBs (having anticausal FIR inverses) that are characterized by the dyadic-based structure W(z)=I-UV/sup /spl dagger//+z/sup -1/UV/sup /spl dagger// for which U and V are M/spl times//spl gamma/ parameter matrices satisfying V/sup /spl dagger//U=I/sub /spl gamma//, 1/spl les//spl gamma//spl les/M, for any M/spl ges/2. Structural conditions for regularity are derived, where the Householder transform is found convenient. As a special case, a class of regular linear-phase BOFBs is considered by further imposing linear phase (LP) on the dyadic-based structure. In this way, an alternative and simplified parameterization of the biorthogonal linear-phase filter banks (GLBTs) is obtained, and the general theory of structural regularity is shown to simplify significantly. Regular BOFBs are designed according to the proposed theory and are evaluated using a transform-based image codec. They are found to provide better objective performance and improved perceptual quality of the decompressed images. Specifically, the blocking artifacts are reduced, and texture details are better preserved. For fingerprint images, the proposed biorthogonal transform codec outperforms the FBI scheme by 1-1.6 dB in PSNR.     

余弦调制滤波器组的原型滤波器设计

该文提出了一种近似重构的余弦调制滤波器组的原型滤波器设计方法。该方法将原型滤波器表示成A(z2)B(z)的形式(其中B(z)是最平坦FIR滤波器),通过优化低阶FIR滤波器A(z)的通带边缘频率,间接设计原型滤波器。文中给出的设计例子表明,该方法可获得很高阻带衰减的滤波器组。     

滤波器组的多滤波器联合能量频谱感知算法

针对认知无线电系统,设计了一种基于滤波器组的多滤波器联合能量频谱感知算法.分析了算法的基本原理,给出了算法的流程,并以信噪比为参数说明了算法的优越性.仿真验证算法的检测概率与漏失概率,并对经典滤波器组的能量频谱感知算法的检测概率与漏失概率进行了比较,结果表明:本算法性能优于经典滤波器组能量频谱感知算法的性能.     

Wavelets and recursive filter banks

It is shown that infinite impulse response (IIR) filters lead to more general wavelets of infinite support than finite impulse response (FIR) filters. A complete constructive method that yields all orthogonal two channel filter banks, where the filters have rational transfer functions, is given, and it is shown how these can be used to generate orthonormal wavelet bases. A family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband Butterworth filters. When there is an odd number of zeros at π it is shown that closed forms for the filters are available without need for factorization. A still larger class of orthonormal wavelet bases having the same moment properties and containing the Daubechies and Butterworth filters as the limiting cases is presented. It is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case, and a constructive method is given. It is also shown how compactly supported bases may be orthogonalized, and bases for the spline function spaces are constructed     

FIR principal component filter banks

Two-dimensional (2-D) principal component filter banks (PCFBs) of finite impulse response (FIR) are proposed. For 2-D signals, among all uniform paraunitary FIR analysis/synthesis filter banks, the FIR PCFBs have the most energy compaction and maximize the arithmetic mean to geometric mean ratio (AM/GM ratio) of subband variances, which is the theoretic coding gain (TCC) of the systems under proper assumptions. The theoretic proof and design techniques are provided. Several special cases are discussed. Experimental results show the potential power of the FIR PCFBs