Perfect Reconstruction Filter Banks Research Articles

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.

M-channel lifting factorization of perfect reconstruction filter banks and reversible M-band wavelet transforms

An intrinsic M-channel lifting factorization of perfect reconstruction filter banks (PRFBs) is presented as an extension of Sweldens' conventional two-channel lifting scheme. Given a polyphase matrix E(z) of a finite-impulse response (FIR) M- channel PRFB with det(E(z))=z/sup -K/, K/spl isin//spl Zopf/, a systematic M-channel lifting factorization is derived based on the Monic Euclidean algorithm. The M-channel lifting structure provides an efficient factorization and implementation; examples include optimizing the factorization for the number of lifting steps, delay elements, and dyadic coefficients. Specialization to paraunitary building blocks enables the design of paraunitary filter banks based on lifting. We show how to achieve reversible, possibly multiplierless, implementations under finite precision, through the unit diagonal scaling property of the Monic Euclidean algorithm. Furthermore, filter-bank regularity of a desired order can be imposed on the lifting structure, and PRFBs with a prescribed admissible scaling filter are conveniently parameterized.

Autoregressive modeling of the Wigner–Ville distribution based on signal decomposition and modified group delay

An autoregressive modeling of the Wigner–Ville distribution (WVD), based on signal decomposition (SD) by a perfect reconstruction filter bank (PRFB) and the modified magnitude group delay function (MMGD), has been proposed. The SD and MMGD, respectively, reduce the existence of crossterms (without any time smoothing) and the Gibb's ripple effect (due to truncation of the WVD kernel, without applying any window), significantly. In view of this, the modeling is not affected by either the crossterms or the Gibb's ripple and the window that would have been used. The proposed method represents actual time–frequency information parsimoniously and compared to the existing WVD modeling methods, its performance is significantly better in terms of both time and frequency resolution (as there is no time and frequency smoothing) and noise immunity/variance and is computationally efficient.

Improved Wigner–Ville distribution performance by signal decomposition and modified group delay

A new approach has been proposed for improving the performance of the Wigner–Ville distribution. This approach is based on signal decomposition and modified magnitude group delay function. Signal decomposition achieved by perfect reconstruction filter bank reduces significantly the existence of crossterms. The Gibbs ripple effect is due to truncation of the Wigner–Ville distribution kernel. The modified magnitude group delay function overcomes this effect without applying any window. Compared to those of Pseudo Wigner–Ville distribution and its versions, the proposed method has significantly improved performance in both time and frequency resolution as there is no time and frequency smoothing. Further, this method obeys better the desirable properties of time–frequency representation and has a better noise immunity.

Two-channel digital and hybrid analog/digital multirate filter banks with very low-complexity analysis or synthesis filters

Multirate filter banks make use of analysis and synthesis filter banks. This paper introduces two-channel digital and hybrid analog/digital multirate filter banks where either the analysis or synthesis filters have a very low complexity. Such filter banks find application, for example, in high-speed analog-to digital converters where it is essential to minimize the complexity of the discrete-time or analog filters. The proposed digital filter banks are approximately perfect reconstruction (PR) filter banks, whereas the hybrid analog/digital filter banks can be chosen to be either approximately PR or approximately perfect magnitude reconstruction filter banks. The design is performed by first optimizing the digital or analog analysis filters and then, with the analysis filters fixed, optimizing the digital synthesis filters. This design procedure makes it possible to obtain analysis filters of very low order and complexity. The overall complexity is also low. Further, the proposed filter banks are, in all cases, very easy to design by making use of well-known and reliable optimization techniques; in particular, as small distortion and aliasing as desired are readily obtained because they are controlled in a linear programming problem. Several design examples are included, illustrating the properties of the proposed filter banks.

New results of phase shifting in the wavelet space

This paper investigates the relationship between even-phase coefficients and odd-phase coefficients in a two-channel perfect reconstruction filter bank. We demonstrate that they are linked to each other by a unique phase-shifting matrix. In the case of multilevel wavelet decomposition, we present an efficient recursive solution to directly perform phase shifting in the wavelet space. Our proposed solution can also be easily generalized into the case of two-dimensional wavelet transform. Direct phase-shifting methods in the wavelet space have potential applications in wavelet-based image/video coding and compressed domain processing.

A general-purpose optimization approach for designing two-channel FIR filterbanks

An efficient general-purpose optimization approach is proposed for designing two-channel finite impulse response (FIR) filterbanks. This technique can be used for optimizing two-channel FIR filterbanks in all alias-free cases proposed in the literature. The generalized problem is to minimize the maximum of the stopband energies of the two analysis filters subject to the given passband and transition band constraints and the given allowable reconstruction error. Therefore, in addition to the perfect-reconstruction filterbanks, nearly perfect-reconstruction banks can be optimized in a controlled manner. The optimization is carried out in two steps. In the first step, for the selected type of the filterbank, a good starting-point filterbank for further optimization is generated using an existing design scheme. The second step involves optimizing the filterbank with the aid of a modified Dutta-Vidyasagar (1977) algorithm. Several examples are included, illustrating the efficiency and the flexibility of the proposed approach.

Oversampled linear-phase perfect reconstruction filterbanks: theory, lattice structure and parameterization

The paper presents the theory, lattice structure, and parameterization for a general class of P-channel oversampled linear-phase perfect reconstruction filterbanks (OLPPRFBs) - systems with sampling factor M (P/spl ges/M) and filter length of L=KM (K/spl ges/1) each. For these OLPPRFBs, the necessary existence conditions on the number of symmetric filters, n/sub /spl beta//, and antisymmetric filters, n/sub /spl alpha//, (i.e., symmetry polarity) are first investigated. VLSI-friendly lattice structures are then developed for two types of OLPPRFBs, type I system (n/sub /spl beta//=n/sub /spl alpha//) and type II system (n/sub /spl beta///spl ne/n/sub /spl alpha//). The completeness and minimality of each type of lattice are also analyzed. Compared with existing work, the proposed lattices are the most general and efficient ones for OLPPRFBs. Besides, through the lattice structures, the sufficiency of the existence conditions is also verified. Next, lifting-based structures are proposed to parameterize a left invertible matrix and all of its left inverses, which leads to unconstrained optimization as well as robust implementation of OLPPRFBs. Finally, several design examples are presented to confirm the validity of the theory and demonstrate the versatility of synthesis filterbanks in the oversampled system.

Filter bank design for subband adaptive microphone arrays

This paper presents a new method for the design of oversampled uniform DFT-filter banks for the special application of subband adaptive beamforming with microphone arrays. Since array applications rely on the fact that different source positions give rise to different signal delays, a beamformer alters the phase information of the signals. This in turn leads to signal degradations when perfect reconstruction filter banks are used for the subband decomposition and reconstruction. The objective of the filter bank design is to minimize the magnitude of all aliasing components individually, such that aliasing distortion is minimized although phase alterations occur in the subbands. The proposed method is evaluated in a car hands-free mobile telephony environment and the results show that the proposed method offers better performance regarding suppression levels of disturbing signals and much less distortion to the source speech.

Theory of paraunitary filter banks over fields of characteristic two

Motivated by our wavelet framework for error-control coding, we proceed to develop an important family of wavelet transforms over finite fields. Paraunitary (PU) filter banks that are realizations of orthogonal wavelets by multirate filters are an important subclass of perfect reconstruction (PR) filter banks. A parameterization of PU filter banks that covers all possible PU systems is very desirable in error-control coding because it provides a framework for optimizing the free parameters to maximize coding performance. This paper undertakes the problem of classifying all PU matrices with entries from a polynomial ring, where the coefficients of the polynomials are taken from finite fields. It constructs Householder transformations that are used as elementary operations for the realization of all unitary matrices. Then, it introduces elementary PU building blocks and a factorization technique that is specialized to obtain a complete realization for all PU filter banks over fields of characteristic two. This is proved for the 2 /spl times/ 2 case, and conjectured for the M /spl times/ M case, where M /spl ges/ 3. Using these elementary building blocks, we can construct all PU filter banks over fields of characteristic two. These filter banks can be used to implement transforms which, in turn, provide a powerful new perspective on the problems of constructing and decoding arbitrary-rate error-correcting codes.

M-channel linear phase perfect reconstruction filter bank with rational coefficients

This paper introduces a general class of M-channel linear phase perfect reconstruction filter banks (FBs) with rational coefficients. A subset of the presented solutions has dyadic coefficients, leading to multiplierless implementations suitable for low-power mobile computing. All of these FBs are constructed from a lattice structure that is VLSI-friendly, employs the minimum number of delay elements, and robustly enforces both linear phase and perfect reconstruction property. The lattice coefficients are parameterized as a series of zero-order lifting steps, providing fast, efficient, in-place computation of the subband coefficients. Despite the tight rational or integer constraint, image coding experiments show that these novel FBs are very competitive with current popular transforms such as the 8/spl times/8 discrete cosine transform and the wavelet transform with 9/7-tap biorthogonal irrational-coefficient filters.

Design of 2D filter banks based on 1D lattice structure

AbstractThis paper proposes a design procedure for the 2D linear‐phase para‐unitary filter bank, based on the 1D lattice structure. As the first step, the perfect reconstruction conditions of 1D and 2D cases are compared, and it is shown that the 2D filter bank satisfying the perfect reconstruction condition can be realized by the 1D lattice structure with a constraint. It is known, on the other hand, that the 1D perfect reconstruction filter bank can be converted to the lattice structure, using only half of the filter coefficients, and a simple optimization procedure can derive the whole filter coefficients. Based on those investigations, it is shown that the 2D filter bank can be designed by directly designing only half of the channels of the 2D filter bank in the time domain. © 2002 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 85(12): 38–46, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10005

Multidimensional causal, stable, perfect reconstruction filter banks

Following our earlier one-dimensional (1-D) work, we show that it is possible to have a large family of biorthogonal perfect reconstruction multidimensional (n-D) subband coding filter banks, which are causal and IIR both at the analysis and at the synthesis ends. While the need for perfect reconstruction IIR filter banks are more apparent for multidimensional signals than for 1-D signals, lack of design techniques have made it impossible to use such schemes. We argue that such filter banks, including their multichannel counterparts, should not only exist in a large variety, but we also indicate a number of tools potentially useful in their design. Specifically, a complete parameterization of such filters leading to design methods in 2-D are given. Parameterizations in frequency domain terms as well as in terms of state space descriptions of filters are provided. These latter techniques have potential for better numerical implementation. The development is valid for two-band as well as for multiband subband coding schemes. Examples of bi-orthogonal version of continuous multidimensional wavelets generated by these iterated filter banks, that have not been previously constructed, are worked out.

Modulated, perfect reconstruction filterbanks with integer coefficients

We present design methods for perfect reconstruction (PR) integer-modulated filterbanks, including biorthogonal (low-delay) filterbanks. Both the prototype filter and the modulation sequences are composed of integers, thus allowing efficient hardware implementations and fast computation. To derive such filterbanks, we first start with the PR conditions known for cosine modulation and extend them to more general, integer modulation schemes. For the design of biorthogonal PR integer prototypes, a lifting strategy is introduced. To find suitable integer modulation schemes, new algebraic methods are presented. We show solutions where the PR conditions on the prototype filters and the modulation matrices are entirely decoupled and where some simple coupling is introduced. Both even and odd numbers of channels are considered. Design examples are presented for both cases.

Allpass delay chain-based IIR PR filterbank and its application to multiple description subband coding

A class of infinite impulse response (IIR) perfect reconstruction (PR) filterbank is obtained with an allpass delay chain and finite impulse response (FIR) matrices at the analysis side. Such a formulation leads to a filterbank that can be optimized to any desired response. For a first-order allpass, the synthesis bank becomes FIR. Design examples showing warping of the frequency scale and filters with unequal passband widths are presented. The latter part of the work deals with designing a filterbank for a multiple description coding scenario. This is achieved in two parts: by obtaining the MMSE synthesis bank for a given analysis bank and channel state and by optimizing the analysis response to minimize the average distortion in the presence of channel erasures and quantization. The proposed IIR filterbank, as well as the well-known FIR paraunitary filterbank, are optimized for various channels, and their performances for the two-channel case have been compared with the derived optimal performance.

On the completeness of the lattice factorization for linear-phase perfect reconstruction filter banks

In this letter, we re-examine the completeness of the lattice factorization for M-channel linear-phase perfect reconstruction filter bank (LPPRFB) with filters of the same length L=KM as discussed by Tran et al. (see IEEE Trans. Signal Processing, vol.48, p.133-47, Jan. 2000). We point out that the assertion of completeness is incorrect. Examples are presented to show that the proposed lattice structure of Tran et al. is not complete when K>2. In addition, we verify that the lattice structure is complete only when K/spl les/2.

Butterworth wavelet transforms derived from discrete interpolatory splines: recursive implementation

In the paper we present a new family of biorthogonal wavelet transforms and the related library of biorthogonal symmetric waveforms. For the construction we used the interpolatory discrete splines which enabled us to design a library of perfect reconstruction filter banks. These filter banks are related to Butterworth filters. The construction is performed in a “lifting” manner. The difference from the conventional lifting scheme is that the transforms of a signal are performed via recursive filtering with the use of IIR filters. These filters have linear phase property and the basic waveforms are symmetric. The filters allow fast cascade or parallel implementation. We present explicit formulas for construction of wavelets with arbitrary number of vanishing moments. In addition, these filters yield perfect frequency resolution. The proposed scheme is based on interpolation and, as such, it involves only samples of signals and it does not require any use of quadrature formulas.

DFT modulated filter bank design for oversampled subband systems

Filter banks are widely used in digital signal processing, often integrated in a multirate scheme, to reduce the implementation cost and to improve algorithmic performance. DFT modulated filter banks are commonly used to design oversampled subband schemes. Oversampled subband schemes are attractive as they trade off between complexity gain and aliasing distortion. Ideally, the overall effect of a subband system is that of a pure delay, i.e. the subband system is perfectly reconstructing. In this paper we present design techniques for perfect and nearly perfect reconstruction oversampled DFT modulated filter banks. One of the disadvantages of a perfect reconstruction filter bank design is that the stopband attenuation of the filters is typically small, which could have a negative impact on intermediate subband operations such as subband adaptive filtering. Nearly perfect reconstruction filter banks can be designed to overcome this problem, leading to a better performance of the intermediate algorithms.

Two-dimensional separable filters for optimal reconstruction of JPEG-coded images

Transform coding is a technique used worldwide for image coding, and JPEG has become the most common tool for image compression. In a JPEG decoder, the quantized transform coefficient blocks are usually processed using the inverse discrete cosine transform (DCT) in order to reconstruct an approximation of the original image. The direct and inverse DCT pair can be arranged in the form of a perfect reconstruction filter bank, and it can be shown that, in the presence of quantization of the transform coefficients, the perfect reconstruction synthesis is not the best choice. In this paper, we propose a procedure for the design of separable 2-D synthesis filters that minimize the reconstruction error power for transform coders. The procedure is used to design a family of filters which are used in the decoder instead of the inverse DCT. The appropriate reconstruction filters are selected on the basis of the standard quantization information provided in the JPEG bit stream. We show that the proposed decoding method gives some gain with respect to the usual decoder in most cases, Moreover, it only makes use of the standard information provided by a JPEG bit stream.

A simplified lattice factorization for linear-phase perfect reconstruction filter bank

We propose a simplified version of lattice factorization for linear-phase perfect reconstruction filter bank (LPPRFB) derived by T.D. Tran et al. (see IEEE Trans. Signal Processing. vol.48, no.1, p.133-47, Jan. 2000). The proposed new lattice structure spans the same class of LPPRFB, while substantially reducing free parameters in nonlinear optimization and saving computation cost in hardware implementation. To further address the importance of our proposed structure, we generalize our factorization to multidimensional LPPRFB (MD-LPPRFB), and show its effectiveness.