Frame-Theory-Based Analysis and Design of Oversampled Filter Banks: Direct Computational Method

This paper studies the frames corresponding to oversampled filter banks (FBs). For this class of FB frames, we present a state-space parameterization of perfect reconstruction FB frames and explicit and numerically efficient formulas to compute the tightest frame bounds, to obtain the dual FB frame, and to construct a tight (paraunitary) FB frame from a given untight (nonparaunitary) FB frame. The derivation uses well-developed techniques from modern control theory, which results in the unified formulas for generic infinite-impulse-response (IIR) and finite-impulse-response (FIR) FBs. These formulas involve only algebraic manipulations of real matrices and can be computed efficiently, reliably, and exactly without the approximation required in the existing methods for generic FBs. The results provide a unified framework for frame-theory-based analysis and systematic design of oversampled filter banks     

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 Direct Design of Oversampled Perfect Reconstruction FIR Filter Banks

We address a problem to find optimal synthesis filters of oversampled uniform finite-impulse-response (FIR) filter banks (FBs) yielding perfect reconstruction (PR), when we are given an analysis FB, in the case where all the filters have the same length that is twice a factor of downsampling. We show that in this class of FBs, a synthesis FB that achieves PR can be found in closed form with elementary matrix operations, unlike conventional design methods with numerical optimization. This framework allows filter coefficients to be complex as well as real. Due to the extra degrees of freedom in a synthesis FB provided by oversampling, we can determine optimal coefficients of synthesis filters that meet certain criteria. We introduce in this paper two criteria: variance of additive noise and stopband attenuation. We show theoretical results of optimal synthesis filters that minimize these criteria and design examples of oversampled linear-phase FIR FBs and DFT-modulated FBs. Moreover, we discuss applications to signal reconstruction from incomplete channel data in transmission and inverse transform of windowed discrete Fourier transform with 50% overlapping.     

Two Classes of Cosine-Modulated IIR/IIR and IIR/FIR NPR Filter Banks

This paper introduces two classes of cosine-modulated causal and stable filter banks (FBs) with near perfect reconstruction (NPR) and low implementation complexity. Both classes have the same infinite-length impulse response (IIR) analysis FB but different synthesis FBs utilizing IIR and finite-length impulse response (FIR) filters, respectively. The two classes are preferable for different types of specifications. The IIR/FIR FBs are preferred if small phase errors relative to the magnitude error are desired, and vice versa. The paper provides systematic design procedures so that PR can be approximated as closely as desired. It is demonstrated through several examples that the proposed FB classes, depending on the specification, can have a lower implementation complexity compared to existing FIR and IIR cosine-modulated FBs (CMFBs). The price to pay for the reduced complexity is generally an increased delay. Furthermore, two additional attractive features of the proposed FBs are that they are asymmetric in the sense that one of the analysis and synthesis banks has a lower computational complexity compared to the other, which can be beneficial in some applications, and that the number of distinct coefficients is small, which facilitates the design of FBs with large numbers of channels.     

Stable, causal, perfect reconstruction, IIR uniform DFT filterbanks

A simple design procedure for stable, causal and perfect reconstruction infinite impulse response parallel uniform discrete Fourier transform filter banks (DFT FBs) based on a new polyphase decomposition, the `polyphase-oversampled' FB, is presented. The proposed design results in causal and stable analysis and synthesis filters that are all derived from a single prototype filter, resulting in efficient realisations. A discussion of the FB numerical properties and some design examples are provided     

Design and Factorization of Two-Channel Perfect Reconstruction Filter Banks with Causal-Stable IIR Filters

In this paper, new design and factorization methods of two-channel perfect reconstruction (PR) filter banks (FBs) with casual-stable IIR filters are introduced. The polyphase components of the analysis filters are assumed to have an identical denominator in order to simplify the PR condition. A modified model reduction is employed to derive a nearly PR causal-stable IIR FB as the initial guess to obtain a PR IIR FB from a PR FIR FB. To obtain high quality PR FIR FBs for carrying out model reduction, cosine-rolloff FIR filters are used as the initial guess to a nonlinear optimization software for solving to the PR solution. A factorization based on the lifting scheme is proposed to convert the IIR FB so obtained to a structurally PR system. The arithmetic complexity of this FB, after factorization, can be reduced asymptotically by a factor of two. Multiplier-less IIR FB can be obtained by replacing the lifting coefficients with the canonical signal digitals (CSD) or sum of powers of two (SOPOT) coefficients.     

Framing pyramids

Burt and Adelson (1983) introduced the Laplacian pyramid (LP) as a multiresolution representation for images. We study the LP using the frame theory, and this reveals that the usual reconstruction is suboptimal. We show that the LP with orthogonal filters is a tight frame, and thus, the optimal linear reconstruction using the dual frame operator has a simple structure that is symmetric with the forward transform. In more general cases, we propose an efficient filterbank (FB) for the reconstruction of the LP using projection that leads to a proved improvement over the usual method in the presence of noise. Setting up the LP as an oversampled FB, we offer a complete parameterization of all synthesis FBs that provide perfect reconstruction for the LP. Finally, we consider the situation where the LP scheme is iterated and derive the continuous-domain frames associated with the LP.     

Orthonormal FBs With Rational Sampling Factors and Oversampled DFT-Modulated FBs: A Connection and Filter Design

Methods widely used to design filters for uniformly sampled filter banks (FBs) are not applicable for FBs with rational sampling factors and oversampled discrete Fourier transform (DFT)-modulated FBs. In this paper, we show that the filter design problem (with regularity factors/vanishing moments) for these two types of FBs is the same. Following this, we propose two finite-impulse-response (FIR) filter design methods for these FBs. The first method describes a parameterization of FBs with a single regularity factor/vanishing moment. The second method, which can be used to design FBs with an arbitrary number of regularity factors/vanishing moments, uses results from frame theory. We also describe how to modify this method so as to obtain linear phase filters. Finally, we discuss and provide a motivation for iterated DFT-modulated FBs.     

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     

Bound Ratio Minimization of Filter Bank Frames

This paper investigates and solves the problem of frame bound ratio minimization for oversampled perfect reconstruction (PR) filter banks (FBs). For a given analysis PRFB, a finite dimensional convex optimization algorithm is derived to redesign the subband gain of each channel. The redesign minimizes the frame bound ratio of the FB while maintaining its original properties and performance. The obtained solution is precise without involving frequency domain approximation and can be applied to many practical problems in signal processing. The optimal solution is applied to subband noise suppression and tree structured FB gain optimization, resulting in deeper insights and novel solutions to these two general classes of problems and considerable performance improvement. Effectiveness of the optimal solution is demonstrated by extensive numerical examples.     

Variable-length lapped transforms with a combination of multiple synthesis filter banks for image coding.

A class of lapped transforms for image coding, which are characterized by variable-length synthesis filters, is introduced. In this class, the synthesis filter bank (FB) is first defined with an arbitrary combination of finite impulse response synthesis filters of perfect reconstruction FBs. An analysis FB is then obtained using direct matrix inversion or iterative implementation of Neumann series expansion. Moreover, to improve compression, we introduce a unitary transform that follows the analysis FB. This class enables a greater freedom of design than previously presented variable-length lapped transforms. We illustrate several design examples and present experimental results for image coding, which indicate that the proposed transforms are promising and comparable with conventional subband transforms including wavelets.     

Multidimensional orthogonal filter bank characterization and design using the Cayley transform.

We present a complete characterization and design of orthogonal infinite impulse response (IIR) and finite impulse response (FIR) filter banks in any dimension using the Cayley transform (CT). Traditional design methods for one-dimensional orthogonal filter banks cannot be extended to higher dimensions directly due to the lack of a multidimensional (MD) spectral factorization theorem. In the polyphase domain, orthogonal filter banks are equivalent to paraunitary matrices and lead to solving a set of nonlinear equations. The CT establishes a one-to-one mapping between paraunitary matrices and para-skew-Hermitian matrices. In contrast to the paraunitary condition, the para-skew-Hermitian condition amounts to linear constraints on the matrix entries which are much easier to solve. Based on this characterization, we propose efficient methods to design MD orthogonal filter banks and present new design results for both IIR and FIR cases.     

New design and realization techniques for a class of perfect reconstruction two-channel FIR filterbanks and wavelets bases

This paper proposes two new methods for designing a class of two-channel perfect reconstruction (PR) finite impulse response (FIR) filterbanks (FBs) and wavelets with K-regularity of high order and studies its multiplier-less implementation. It is based on the two-channel structural PR FB proposed by Phoong et al (1995). The basic principle is to represent the K-regularity condition as a set of linear equality constraints in the design variables so that the least square and minimax design problems can be solved, respectively, as a quadratic programming problem with linear equality constraints (QPLC) and a semidefinite programming (SDP) problem. We also demonstrate that it is always possible to realize such FBs with sum-of-powers-of-two (SOPOT) coefficients while preserving the regularity constraints using Bernstein polynomials. However, this implementation usually requires long coefficient wordlength and another direct-form implementation, which can realize multiplier-less wavelets with K-regularity condition up to fifth order, is proposed. Several design examples are given to demonstrate the effectiveness of the proposed methods.     

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 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     

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     

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.     

On simplified order-one factorizations of paraunitary filterbanks

This paper revisits the order-one factorization of causal finite impulse response (FIR) paraunitary filterbanks (PU FBs). The basic form of the factorization was proposed by Vaidyanathan et al. in 1987, which is a cascade of general unitary matrices separated by diagonal delay matrices with arbitrary number of delay elements. Recently, Gao et al. have proved the completeness of this factorization and developed a more efficient structure that only uses approximately half number of free parameters. In this paper, by briefly analyzing Gao et al.'s derivation, we first point out that Gao et al.'s factorization contains redundant free parameters. Two simplified structures of Vaidyanathan's factorization are then developed, i.e., a post-filtering-based structure and a prefiltering-based structure. Our simplification relies on consecutive removal of extra degrees of freedom in adjacent stages, which is accomplished through the C-S decomposition of a general unitary matrix. Since the conventional C-S decomposition leads to a redundant representation, a new C-S decomposition is developed to minimize the number of free parameters by further incorporating the Givens rotation factorization. The proposed structures can maintain the completeness and the minimality of the original lattice. Compared with Gao et al.'s factorization, our derivations are much simpler, while the resulting structures contain fewer free parameters and less implementation cost. Besides, these new factorizations indicate that for a PU FB with a given filter length, the symmetric-delay factorization offers the largest degrees of design freedom. Several design examples are presented to confirm the validity of the theory.     

Impulse sampled FIR interpolation with SC active-delayed blockpolyphase structures

The authors propose new switched-capacitor (SC) active-delayed block (ADB) polyphase structures for analogue finite impulse response (FIR) interpolation. Their system functions are not distorted by the input sample-and-hold filtering effect at the lower input sampling rate. Both canonic and non-canonic ADB polyphase architectures which are superior in terms of a lower speed requirement for OAs, and fewer OAs, respectively, will be proposed for SC FIR interpolators