Perfect Reconstruction Filter Banks Research Articles

A design procedure for optimal energy compaction IIR filters

In this brief we consider the energy compaction problem for filters belonging to a restricted class of infinite-impulse response (IIR) filters, having the transfer function h(z)=(H/sub 0/(z/sup 2/)+z/sup -1/ H/sub 1/(z/sup 2/))//spl radic/2, where both polyphase components H/sub 0/(z) and H/sub 1/(z) are stable all pass filters. Assuming rational input spectra, we reduce the infinite series expansion of the compaction gain criterion to a finite sum with terms depending on the parameters of the filter H(z). Simple expressions for computing the gradient of the criterion are derived, and finally a gradient ascent maximization routine is proposed, which provides the IIR filter with optimum parameters. Simulation results illustrate the good compaction gain of the optimal IIR filters, which despite their constrained nature (allpass polyphase components), may well outperform finite-impulse response (FIR) filters with much more parameters. The main envisaged application of our work is building QMF perfect reconstruction filterbanks with maximum coding gain, but the above optimization problem and simple extensions of it can be solved in the same manner and also be used in other filter design applications.

Rate-distortion-optimal subband coding without perfect-reconstruction constraints

We investigate the design of subband coders without the traditional perfect-reconstruction constraint on the filters. The coder uses scalar quantizers, and its filters and bit allocation are designed to optimize a rate-distortion criterion. Using convexity analysis, we show that optimality can be achieved using filterbanks that are the cascade of a (paraunitary) principal component filterbank for the input spectral process and a set of pre and postfilters surrounding each quantizer. Analytical expressions for the pre and postfilters are then derived. An algorithm for computing the globally optimal filters and bit allocation is given. We also develop closed-form solutions for the special case of two-channel coders under an exponential rate-distortion model. Finally, we investigate a constrained-length version of the filter design problem, which is applicable to practical coding scenarios. While the optimal filterbanks are nearly perfect-reconstruction at high rates, we demonstrate an apparently surprising advantage of optimal FIR filterbanks; they significantly outperform optimal perfect-reconstruction FIR filterbanks at all bit rates.

Design of two-channel perfect-reconstruction linear-phase finite-duration impulse response filter banks for subband image coding by the genetic algorithm approach

In this paper, a powerful approach is proposed for the design of two-dimensional two-channel perfect-reconstruction finite-duration impulse response filter banks employing diamond-shaped linear-phase filters. When one of the analysis filter pair is properly designed, the condition of perfect reconstruction is formulated in a spatial-domain constraint equation for designing the other. The method is based on minimizing the integrated square error for the frequency response over the pass band and stop band while imposing the spatial-domain constraint. One numerical design example is given to demonstrate the perfect reconstruction for the proposed system.

Eigenstructure approach for characterization of FIR PR filterbanks with 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.

Design of two-channel perfect-reconstruction linear-phase finite-duration impulse response filter banks for subband image coding by the genetic algorithm approach

In this paper, a powerful approach is proposed for the design of two-dimensional two-channel perfect-reconstruction finite-duration impulse response filter banks employing diamond-shaped linear-phase filters. When one of the analysis filter pair is properly designed, the condition of perfect reconstruction is formulated in a spatial-domain constraint equation for designing the other. The method is based on minimizing the integrated square error for the frequency response over the pass band and stop band while imposing the spatial-domain constraint. One numerical design example is given to demonstrate the perfect reconstruction for the proposed system.

Biorthogonal Butterworth wavelets derived from discrete interpolatory splines

We present a new family of biorthogonal wavelet transforms and a related library of biorthogonal periodic symmetric waveforms. For the construction, we used the interpolatory discrete splines, which enabled us to design a library of perfect reconstruction filterbanks. These filterbanks are related to Butterworth filters. The construction is performed in a "lifting" manner. The difference from the conventional lifting scheme is that all the transforms are implemented in the frequency domain with the use of the fast Fourier transform (FFT). Two ways to choose the control filters are suggested. 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. These filters have a linear-phase property, and the basic waveforms are symmetric. In addition, these filters yield refined frequency resolution.

Warped discrete-Fourier transform: Theory and applications

In this paper, we advance the concept of warped discrete-Fourier transform (WDFT), which is the evaluation of frequency samples of the z-transform of a finite-length sequence at nonuniformly spaced points on the unit circle obtained by a frequency transformation using an allpass warping function. By factorizing the WDFT matrix, we propose an exact computation scheme for finite sequences using less number of operations than a direct computation. We discuss various properties of WDFT and the structure of the factoring matrices. Examples of WDFT for first- and second-order allpass functions is also presented. Applications of WDFT included are spectral analysis, design of tunable FIR filters, and design of perfect reconstruction filterbanks with nonuniformly spaced passbands of filters in the bank. WDFT is efficient to resolve closely spaced sinusoids. Tunable FIR filters may be designed from FIR prototypes using WDFT. In yet another application, warped PR filterbanks are designed using WDFT and are applied for signal compression.

New method for designing two-channel causal stable IIR perfect reconstruction filter banks and wavelet bases

A new method for designing two-channel causal stable IIR PR filter banks and wavelet bases is proposed. It is based on the struc- ture previously proposed by Phoong et al. (1995). Such a filter bank is parameterized by two functions a(z) and b(z), which can be chosen as an all-pass function to obtain IIR filterbanks with very high stopband attenuation. One of the problems with this choice is that a bump of about 4 dB always exists near the transition band of the analysis and synthesis filters. The stopband attenuation of the high-pass analysis filter is also 10 dB lower than that of the low-pass filter. By choosingb(z) and a(z )a s an all-pass function and a type-II linear-phase finite impulse response (FIR) function, respectively, the bumping can be significantly sup- pressed. In addition, the stopband attenuation of the high-pass filter can be controlled easily. The design problem is formulated as a polynomial approximation problem and is solved efficiently by the Remez exchange algorithm. The extension of this method to the design of a class of IIR wavelet bases is also considered. © 2000 Society of Photo-Optical Instrumenta- tion Engineers. (S0091-3286(00)02909-3)

Design of low-delay FIR half-band filters with arbitrary flatness and its application to filter banks

Half-band filters are a class of important filters among digital filters, and have been widely used in many applications such as filter banks and wavelets. The conventional methods are mainly concerned with FIR half-band filters with exactly linear phase. However, the exactly linear phase filters have a drawback of large group delay when high-order filters are required. In this paper, the design of FIR half-band filters with a lower group delay is considered. In some applications of filter banks and wavelets, a flat magnitude response is required for the half-band filters. Therefore, a new method for designing low-delay FIR half-band filters with arbitrary flatness is proposed. In the proposed method, while taking the specified flatness condition into account, the design problem is formulated by using the complex Remez exchange algorithm in the stopband. Then, a set of filter coefficients can be easily obtained by solving a simple system of linear equations. The optimal solution with an equiripple response in the stopband is obtained by applying an iteration process. Finally, the proposed method is applied to the design of two-channel perfect reconstruction filter banks with low group delay to demonstrate its effectiveness. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(10): 1–9, 2000

An Introduction to Wavelet Transform Tidal Analysis Methods

Continuous wavelet transforms (CWTs) provide an approach to understanding the numerous tidal phenomena that deviate markedly from an assumed statistical stationarity or exact periodicity inherent in traditional tidal methods. Use of wavelets allows determination of the degree of non-stationarity present in time series, such as estuarine and shelf currents, usually treated as stationary. Wavelets also provide a consistent analysis of tidal and non-tidal variance, a feature often essential for dynamical analyses of non-stationary tides. We summarize basic notions of the wavelet transform, also known as a perfect reconstruction filter bank or a multire solution analysis, contrast them with those of harmonic analysis and Fourier transforms, construct a continuous wavelet transform basis with a scale selection especially adapted to tidal problems, describe possibilities for analysis of scalar and vector quantities, define a criterion for knowledge of independence of process between adjoining scales, and illustrate use of wavelet tools with several examples. In contrast to the nearly periodic barotropic tide typical of coastal stations, this paper analyses processes that are in part tidally driven but non-stationary, e.g. baroclinic tidal currents, river tides, continental shelf internal tides, and some kinds of biological activity in the coastal ocean. In all cases, wavelet analysis provides a consistent, linear analysis of tidal and non-tidal variance and reveals features that harmonic analysis on a Fourier transform approach could not elucidate.

A new design method for two-channel perfect reconstruction IIR filter banks

In this paper, a new method for designing perfect reconstruction (PR) two-channel causal stable IIR filter banks is introduced. It is based on a structure previously proposed by Phoong et al. (1995). By using a combination of allpass and linear-phase FIR functions, the bumping problem found in the conventional structural PR filter bank is significantly suppressed. The design problem is formulated as a polynomial approximation problem and is solved effectively using the Remez exchange algorithm. Filter banks with flexible stopband attenuation and system delay can readily be obtained using the proposed algorithm.

Further results on reconstruction methods for processing finite-length signals with perfect reconstruction filter banks

In de Queiroz and Rao (1995), new expressions were developed for the perfect reconstruction (PR) of the boundary regions of a finite-length signal after subband processing using uniform and paraunitary filter banks (PUFBs). The present paper points out an incorrect assertion in the previous paper regarding the sufficiency of its solution and presents the general existence (applicability) conditions for the method. Furthermore, we extend the method to encompass perfect reconstruction filter banks, i.e., not only the paraunitary case.

Theory of rate-distortion-optimal, constrained filterbanks-application to IIR and FIR biorthogonal designs

We design filterbanks that are best matched to input signal statistics in M-channel subband coders, using a rate-distortion criterion. Previous research has shown that unconstrained-length, paraunitary filterbanks optimized under various energy compaction criteria are principal-component filterbanks that satisfy two fundamental properties: total decorrelation and spectral majorization. In this paper, we first demonstrate that the two properties above are not specific to the paraunitary case but are satisfied for a much broader class of design constraints. Our results apply to a broad class of rate-distortion criteria, including the conventional coding gain criterion as a special case. A consequence of these properties is that optimal perfect-reconstruction (PR) filterbanks take the form of the cascade of principal-component filterbanks and a bank of pre- and post-conditioning filters. The proof uses variational techniques and is applicable to a variety of constrained design problems. In the second part of this paper, we apply the theory above to practical filterbank design problems. We give analytical expressions for optimal IIR biorthogonal filterbanks; our analysis validates a conjecture by several researchers. We then derive the asymptotic limit of optimal FIR biorthogonal filterbanks as filter length tends to infinity. The performance loss due to FIR constraints is quantified theoretically and experimentally. The optimal filters are quite different from traditional filters. Finally, a sensitivity analysis is presented.

Wavelet families of increasing order in arbitrary dimensions

We build discrete-time compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The associated scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, in-place calculation, and integer-to-integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. While we concentrate on the discrete-time case, some discussion of convergence and stability issues together with examples is given.

Time-domain design and lattice structure of FIR paraunitary filter banks with linear phase

In this paper, we present a novel way to design biorthogonal and paraunitary linear phase filter banks. The square error of the time-domain constraints on the perfect reconstruction filter bank is expressed in a quadratic form with respect to the filter coefficients. This cost function is minimized by solving a set of linear equations iteratively. This means one can design biorthogonal filter banks without using nonlinear programming techniques. With some modifications, the method can be extended to the design of paraunitary filter banks. In addition, the lattice structure of paraunitary filter banks with filter length KM+ β is also derived. Finally, some design examples are presented to validate the proposed method.

Optimal subband coders of quantized multidimensional signals

A new general method is presented for the analysis of the effects of quantization and transmission noise in subband filter-bank multidimensional signal and image coders. Using general statistical noise characterization, the filter banks are obtained, which are optimal in the sense of minimizing the variance of the difference of the reconstructed signal from the original. It is shown that these are not identical to those achieving perfect reconstruction in lossless coding. A general method is described for the optimal selection of the synthesis filters if the analysis filters and quantizers are given. Assuming the use of these optimal synthesis filters, a method is then developed for the determination of the optimal analysis filters and the optimal bit-allocation to the subbands. The effects of quantization on perfect reconstruction filter-banks are analyzed next, and an explicit expression is determined for the quantizer noise-to-signal ratio for such banks. Given arbitrary quantizers, the analysis filters which are optimal in the sense of minimizing the output noise power are obtained. Finally, the jointly optimal selection is determined for the analysis filters and the bit allocations of the subbands. It is seen that this selection is identical to that determined for the theoretically optimal filter-banks.

Linear-phase perfect reconstruction filter bank: lattice structure, design, and application in image coding

A lattice structure for an M-channel linear-phase perfect reconstruction filter bank (LPPRFB) based on the singular value decomposition (SVD) is introduced. The lattice can be proven to use a minimal number of delay elements and to completely span a large class of LPPRFBs: all analysis and synthesis filters have the same FIR length, sharing the same center of symmetry. The lattice also structurally enforces both linear-phase and perfect reconstruction properties, is capable of providing fast and efficient implementation, and avoids the costly matrix inversion problem in the optimization process. From a block transform perspective, the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform (GLBT) with an arbitrary number of channels M and arbitrarily large overlap. The relaxation of the orthogonal constraint allows the GLBT to have significantly different analysis and synthesis basis functions, which can then be tailored appropriately to fit a particular application. Several design examples are presented along with a high-performance GLBT-based progressive image coder to demonstrate the potential of the new transforms.

On the McClellan transformation of 1-D even-length linear phase filterbanks

It was shown by Wei et al. (1997) that by applying the extended McClellan transformation on one-dimensional (1-D) even length linear phase perfect reconstruction (PR) filterbanks (FBs), the resulting multidimensional (M-D) FB does not have the PR property. The two types of extended McClellan transformation that were considered in Wei et al. were originally proposed by Mersereau et al. (1976) and Karam (1996), respectively, for the design of single filters. We shall show in this letter that by using a different type of transformation, PR can indeed be preserved in the M-D FB. A new class of transformation that achieves PR is presented. Furthermore, by choosing an appropriate transformation the M-D filters will have linear phase.

Two-dimensional perfect reconstruction FIR filter banks with triangular supports

We present a theory and design of two-dimensional (2-D) perfect reconstruction (PR) filter banks (FBs) (PRFBs) in which the supports of the analysis and synthesis filters consist of two triangulars. The two-triangular FB can be realized by designing an appropriate 2-D complex prototype whose passband support is a triangle that is a half of a parallelepiped-shaped passband support defined by the sampling matrix. Then a complex prototype filter is modulated by the DFT, and each analysis filter is derived by taking the real part of the modulated output. We show that the two-triangular FB satisfies the condition of permissibility. A necessary and sufficient condition for 2-D PRFBs is derived. Moreover, we present a design method of the 2-D PRFB that minimizes the cost function consisting of the frequency constraint and PR condition. Finally, a design example is presented to confirm the validity of the proposed method.

Nonuniform perfect reconstruction filter banks over lattices with application 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.