Optimal duration-bandwidth localized antisymmetric biorthogonal wavelet filters

The perfect reconstruction cosine/sine-modulated filter banks belonging to the class of modulated filter banks have been studied extensively due to their attractive features (simple structure, analysis and synthesis filters are of equal length, low computational complexity), and consequently, they have received a great interest in audio coding applications. In fact, they are employed in the international speech and audio coding standards and proprietary audio compression algorithms. The oddly and evenly stacked modified discrete cosine transform (MDCT) and the corresponding modified discrete sine transform (MDST), the modulated lapped transform (MLT), the extended lapped transforms (ELTs), and their biorthogonal versions are real-valued cosine/sine-modulated filter banks satisfying the perfect reconstruction property. The modulated complex lapped transform (MCLT) is the complex-valued filter bank whose real part is the MLT or equivalently, the oddly stacked MDCT, and the imaginary part is the oddly stacked MDST. In this chapter, definitions, general properties, and matrix representations of the MDCT/MDST, MLT, ELT, and MCLT filter banks are presented. In order to an analysis/synthesis filter bank be perfect reconstruction, the necessary and sufficient conditions imposed on the analysis and synthesis windowing functions play an important role. Therefore, additionally the windowing procedure and perfect reconstruction (biorthogonal) conditions in the case of identical and (nonidentical) analysis and synthesis windowing functions, design of a windowing function including definitions of commonly windowing functions used in audio coding applications, adaptive switching of transform block sizes and windowing functions, and general perfect reconstruction conditions for the ELT filter bank with multiple overlapping factor both for the orthogonal and biorthogonal cases are derived and/or discussed in detail.

There has been considerable interest in the design of multidimensional (MD) filter banks. MD filter banks find application in subband coding of images and video data. MD filter banks can be designed by cascading one-dimensional (1D) filter banks in the form of a tree structure. In this case, the individual analysis and synthesis filters are separable and the filter bank is called a separable filter bank. MD filter banks with nonseparable filters offer more flexibility and usually provide better performance. Nonetheless, their design is considerably more difficult than separable filter banks. The purpose of this paper is to provide an overview of developments in this field on the design techniques for MD filter banks, mostly two-dimensional (2D) filter banks. In some image coding applications, the 2D two-channel filter banks are of great importance, particularly the filter bank with diamond-shaped filters. We will present several design techniques for the 2D two-channel nonseparable filter banks. As the design of MD filters are not as tractable as that of 1D filters, we seek design techniques that do not involve direct optimization of MD filters. To facilitate this, transformations that turn a separable MD filter bank into a nonseparable one are developed. Also, transformations of 1D filter banks to MD filter banks are investigated. We will review some designs of MD filter banks using transformations. In the context of 1D filter bank design, the cosine modulated filter bank (CMFB) is well-known for its design and implementation efficiency. All the analysis filters are cosine modulated versions of a prototype filter. The design cost of the filter bank is equivalent to that of the prototype and the implementation complexity is comparable to that of the prototype plus a low-complexity matrix. The success with 1D CMFB motivate the generalization to the 2D case. We will construct the 2D CMFB by following a very close analogy of 1D case. It is well-known that the 1D lossless systems can be characterized by state space description. In 1D, the connection between the losslessness of a transfer matrix and the unitariness of the realization matrix is well-established. We will present the developments on the study of 2D lossless systems. As in 1D case, the 2D FIR lossless systems can be characterized in terms of state space realizations. We will review this, and then address the factorizability of 2D FIR lossless systems by using the properties of state space realizations.

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.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This paper presents a generalized novel design of a family of multiplier free orthogonal wavelet filter banks (FBs) with low complexity. In the proposed method, maximum number of zeros at <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z=-1$ </tex-math></inline-formula> is ensured for a desired length of analysis and synthesis filters. A new class of technique is proposed to obtain dyadic filter coefficients based on double shifting orthogonality property with allowable deviation from original filter coefficients. This is achieved by slightly altering perfect reconstruction (PR) condition to obtain dyadic filter coefficient. A general closed form expression is formulated using double shifting orthogonality property to reduce the error between original filter coefficients and proposed dyadic filter coefficients. The designed wavelet FB improved time-frequency product, Sobolev regularity and frequency selectivity (low energy in error) over well known existing wavelet FBs. In addition, a fixed point VLSI architecture is proposed for the designed family of orthogonal wavelet FBs. The suggested architecture is designed and implemented on Kintex7 FPGA board. The proposed family of orthogonal wavelet FBs is multiplier less with significantly reduced adders, shifters and dynamic power dissipation. The performance of the proposed wavelet FB is validated on two different applications namely image compression and medical image retrieval. The proposed FBs are tested on well known datasets such as CLASSIC, EPFL, RAISE, and FiveK for image compression and on three different medical image databases such as NEMA, OASIS, and EXACT09 for image retrieval.

A filter bank is presented, which is based on coupled all-pass filters. The special properties of the filter bank are approximately linear phases of the subbands, perfect reconstruction and a low group delay. The synthesis filter bank is just a simple summation. The filter bank represents a generalization of the well-known lattice wave digital filters (LWDFs). For this reason, most properties of the wave digital filters are maintained, for instance, their excellent stability behavior and an extremely low sensitivity coefficient (see Fettweis, A., Proc. IEEE, vol.74, no.2, p.270-327, 1986). The design of the all-pass filters is performed by an iterative procedure. Finally, the filter bank is illustrated by means of a design example.

A new approach to the design of M-channel pseudo quadrature mirror filter (QMF) banks is presented. The analysis and synthesis filters are cosine-modulated versions of the prototype filters. The prototype filter is constrained to be a linear-phase spectral-factor of a 2M/sup th/ band filter. As a result, the reconstructed signal has no amplitude nor phase distortions. The only error at the output is aliasing, which is comparable to the stopband attenuation of the prototype filter. Using this approach, it is possible to design a pseudo-QMF bank where the stopband attenuation of the analysis (and thus synthesis) filters is on the order of 100 dB. Consequently, the resulting reconstruction error is also on the order of -100 dB. An example is included to demonstrate the theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Describes a perfect-reconstruction two-channel QMF (quadrature mirror filter) bank, in which the analysis and synthesis filters have linear phases. First, a technique for designing filter banks is presented. Next, the results of two examples demonstrate that the method can lead to good filters although its procedure is very simple; also, this yields filters without multipliers such as the SSKF (symmetric short kernel filter) bank. >

There are some applications that require oversampled filter banks, for example, adaptive signal processing using multirate filter banks. Little study was done, however, for such filter banks compared with maximally decimated ones. In this paper, we propose a design method for oversampled filter banks which satisfy the perfect reconstruction and linear phase properties. Moreover, we consider the design method for both orthogonal and biorthogonal ones. When the decimation ratio is not a divisor of the number of channels, we find it impossible for analysis and synthesis filters to have the same frequency responses by considering the mechanism of aliasing cancellation. In other words, biorthogonal filter banks can be designed under an arbitrary decimation ratio, while it is restricted to a divisor of the number of channels in the orthogonal case. We first derive the conditions for perfect reconstruction and linear phase for both biorthogonal and orthogonal ones. Then, we present the design method for both cases. Finally, some design examples are shown. © 1997 Scripta Technica, Inc. Electron Comm Jpn Pt 3, 80(10): 78–86, 1997

We present a flexible, efficient design technique for the prototype filter of an oversampled near perfect reconstruction (NPR) generalized discrete Fourier transform (GDFT) filter bank. Such filter banks have several desirable properties for subband processing systems that are sensitive to aliasing; e.g., subband adaptive filters. Our design criteria for the prototype filter are explicit bounds on the aliased components in the subbands, the aliased components in the output, and the distortion induced by the filter bank. It is shown that the design of an optimal prototype filter can be transformed into a convex optimization problem that can be efficiently solved. Our design technique provides an efficient and effective tool for exploring many of the inherent trade-offs in the design of the prototype filter, including the trade-off between aliasing in the subbands and the distortion induced by the filter bank. In our examples we calculate several of these trade-offs and demonstrate that our method can generate filters with significantly better performance than filters obtained using current design methods.

A proficient approach to the design of Quadrature Mirror Filter (QMF) Bank is studied and an algorithm is offered for straightforward accomplishment of QMF bank with near perfect reconstruction. Objective function is built upon perfect reconstruction condition. The cutoff frequency is repetitively varied to adjust the filter coefficients using the anticipated algorithm and reconstruction error is minimized. Using the optimal value of cutoff frequency the prototype filter is designed and other filters of the bank is considered from that single prototype filter. Kaiser and Dolph-Chebyshev window functions are used to investigate the betterment of the design method. The performance of the projected design method are evaluated in terms of computation time required i.e. CPU time, Perfect Reconstruction Error (PRE), Magnitude Square Error (MSE) and Number of Iterations (NOI). The proposed algorithm is simple, easy to implement, and linearly converges to optimal solution. Finally, the sub-band decomposition of an audio signal is done using the filter bank.

Tree-structured IIR/FIR uniform-band and octave-band filter banks with very low-complexity analysis or synthesis filters

A new approach is presented for the design of two-channel perfect-reconstruction (PR) FIR filter banks employing linear-phase analysis and synthesis filters. The approach guarantees the designed filter bank to be PR through a set of implicit mathematical relationships. Furthermore, the approach formulates the design as an unconstrained quadratic programming problem to simultaneously optimize the frequency responses of the analysis and synthesis filters. The criterion for optimality is least-squares. The design technique is extendable to M-channel (M/spl ges/2), and this extension is currently under development.

This paper introduces a class of hybrid analog/digital filter banks with approximately perfect magnitude reconstruction. The filter bank consists of analog analysis and digital synthesis filter banks. The analog analysis filters are formed as a sum and difference of two allpass subfilters, respectively, resulting in filters with low orders and few free parameters, which is advantageous from implementation and design points of view. The digital synthesis filters are odd-order linear-phase FIR filters with symmetric and anti-symmetric impulse responses, respectively. The filter design is performed by first optimizing the analog analysis filters. Then, with the analog filters fixed, optimum digital synthesis filters, in the minimax sense, are obtained with the aid of linear programming.

This paper investigates the signal reconstruction in the IIR multirate filter banks subject to subband noises. It uses the technique of linearizing change of variables to overcome the difficulty of nonlinear matrix inequality constraints associated with the mixed H2/H-infinity optimization of IIR multirate filter banks, and generalizes the authors previous results on FIR filter banks to the design of mixed H2/H-infinity optimal IIR synthesis filter banks. A design example is presented to demonstrate the significant performance improvement of the designed multirate filter banks in the presence of subband noises. The limitation of the obtained design method is also pointed out.

This paper presents the theory and design of a class of perfect-reconstruction (PR) recombination nonuniform cosine-modulated filter-banks (CMFBs) when the numbers of channels in the two filter-banks (FBs) are not coprime to each other. Since the analysis and synthesis filters are linear periodic time varying, a new method is proposed to minimize the additional aliasing components. A detailed design example is given to demonstrate the usefulness of the proposed method.