Optimization of linear multiresolution transforms for scene adaptive coding

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.

The authors have experimented with a variety of analysis/synthesis systems in the presence of quantization. The systems were block discrete Fourier transform (DFT), block DFT with trapezoidal windowing, block discrete cosine transform, Hanning windowed short-time Fourier transform (STFT) oversampled by a factor of two, Kaiser windowed STFT oversampled by a factor of four, quadrature mirror filter (QMF) banks, perfect-reconstruction QMF banks, pseudo-QMF banks, and Princen-Bradley filter banks. All systems were evaluated for speech compression; a subset was evaluated for image compression. It is shown that for speech the more sophisticated systems (e.g., QMF) provide the best objective performance, but comparable subjective performance can be obtained at a fraction of the complexity using simpler systems (e.g. pseudo-QMF and Princen-Bradley filter banks). For images, a tree-structured QMF bank did not perform well for 16*16 frequency bands when compared to block-oriented analysis/synthesis systems. >

An oversampled linear phase (LP) perfect reconstruction (PR) filter bank (FB) with a lattice structure involving a noise robust building block is presented. The structure is based on the generalized lapped pseudo-biorthogonal transform (GLPBT), which represents a large class of lapped transforms. Given an analysis bank of the GLPBT and a correlation matrix of noise, the proposed synthesis bank reduces the noise component and attains PR. Numerical results show that this class of filter bank is effective in transmission of oversampled data over noisy channels.

In this paper, we present a new structure for linear-phase filter banks without redundancy, which have unequal-length filters at each subband. First, we extend the simplified lattice structure of the linear-phase filter bank to the unequal-length linear-phase paraunitary filter bank. In general, the unequal-length linear-phase paraunitary filter bank has equal-length filters at both the analysis and the synthesis bank because the synthesis bank is a transposed version of the analysis one, while the biorthogonal one does not. So, we discuss the conditions that the linear-phase biorthogonal filter bank has unequal-length filters at 1) the analysis bank, 2) the synthesis bank, and 3) both the analysis and the synthesis banks. Finally, several design and image coding examples are shown.

An iterative approach for minimax design of multidimensional quadrature mirror filters

A review is presented first of the evolution of transmultiplexers since about 1966, in the context of a long progression of theoretical advances and developments leading to recent proposals to fundamentally improve OFDM type systems using principles of perfect reconstruction filter (PRF) banks. The equivalence of transmultiplexers to OFDM type multi-user systems is discussed. The desirable goals for performance and implementation of transmultiplexers or multiband, multiuser communication systems that are addressed and met in this paper using filter bank trees are set down. Then modifications and extensions are presented of the designs and architectures of wavelet packet based synthesis and analysis pairs of filter bank trees (Sablatash and Lodge in Digital Signal Process 13: 58–92, 2003) that can be used as transmultiplexers. These exhibit a number of advantages over the previous designs and address three shortcomings of the designs used to illustrate basic principles in Sablatash and Lodge (Digital Signal Process 13:58–92, 2003). The first of these is the asymmetry of the magnitude frequency responses of the multiplexer channels, which is addressed using a symmetric design for a lowpass and highpass quadrature mirror filter (QMF) pair described herein. The second is the problem of minimizing the total delay of the signal in passing through the analysis and synthesis filter banks. This is addressed using an architecture involving DFT polyphase synthesis filter banks to replace the wideband VSB filters at the roots of the two identical synthesis filter bank trees, but results in the multiplexer having fewer levels. In this way a tradeoff is effected of lower delay and complexity with fewer levels of bandwidth on demand. At the receiver matching DFT polyphase analysis filters and the other matching analysis filters are implemented. The third shortcoming is the difficulty in designing a synchronization scheme if the filters in the synthesis and analysis filter banks have non-linear phase. This is addressed by designing linear phase filters that do not affect the ISI to any significant degree for communication purposes, although exact perfect reconstruction is lost, but greatly ease and improve the design of the synchronization scheme. Relationships of this paper and its advantages over recent research studies and IEEE 802.22 standards proposals using PR filter banks for multi-user systems to greatly improve on OFDM systems are discussed.

This paper investigates the existence of a large subclass of M-channel linear-phase perfect-reconstruction FIR filter banks-systems with analysis and synthesis filters of length L/sub i/=K/sub i/M+/spl beta/, where /spl beta/ is an arbitrary integer, 0/spl les//spl beta/<M, and K/sub i/ is any positive integer. For this subclass of systems, and for any arbitrary channel M, we first derive the necessary conditions for the existence of linear-phase perfect-reconstruction filter banks. Next, we turn our attention to a large class of linear-phase paraunitary filter banks, those with even M and even /spl beta/, and develop a simple lattice structure for this case. Finally, several design examples are presented to confirm the validity of the theory.

The design of two channel linear phase quadrature mirror filter (QMF) banks is considered with constraints on the peaks of the reconstruction error and that of the subband filter ripple sizes. We further considered the design specification without explicitly specified transition band for the subband filters. An iterative algorithm is proposed to design QMF banks that can incorporate with various design constraints. We have suggested the possible design tradeoff between transition band bandwidth and the ripple size of the subband subband filter and/or the reconstruction error. Various design examples are presented to illustrate the versatility of the digital QMF banks obtained by the proposed design method. The algorithm is proved to converge and was found to converge efficiently in all the simulations.

The design process is reduced to designing the filter bank to satisfy a set of conditions and it is known that the reconstructed signal always possesses some alias and distortion. It is the present objective to find the necessary and sufficient conditions on the filters so that the resulting QMF (quadrature mirror filter) bank cancels most alias components. Having found these conditions, the author suggests an algorithm to design the PAC (partial alias cancellation) QMF bank. Examples are given to demonstrate the theory.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An investigation into the performance of a multirate sub-band adaptive beamforming array is carried out. Quadrature mirror filter (QMF) banks are used here to partition the frequency band, rather than the conventional Fast Fourier transform (FFT) algorithm. The MSE performances for the sub-band adaptive beamformer with (i) different filter order of the QMF banks but fixed number of sub-bands and (ii) fixed length of the QMF bank but different number of sub-bands are analysed. Both cases show reduction in MSE as either the QMF order is increased or the number of bands is increased. The selection of the QMF bank and the number of bands are left to the user, depending upon the computational complexity and implementation cost. Two examples are chosen from the above analysis and further investigations are carried out on them. The behaviour of the different structures, in terms of eigenvalue performance, radiation characteristics and mean square errors (MSE) are studied and analysed. The filter orders used are 16 (for eight sub-bands) and 32 (for four sub-bands) in length. The MSE are almost identical for both cases.

This paper proposes an efficient method for the design of two-channel, quadrature mirror filter (QMF) bank using polyphase components. The design problem is formulated to minimize an objective function, which is a linear combination of pass-band error and stop-band residual energy of the low-pass analysis filter of the filter bank. Unlike the other techniques which involves linear and non linear optimization, our formulation reduces the design equation to an eigenvector problem. An objective function that minimizes the stop-band energy and square error of the distortion transfer function uses the polyphase components and optimization is done using the power method. Simulation results of the algorithm shows the improved performance of the proposed method. Polyphase decomposition results dramatic computational efficiency and reduces number of iterations.

This paper describes a two channel linear phase quadrature mirror filter bank using Kekre's matrix. It has been derived mathematically that the rows of the Kekre's second order matrix satisfies the conditions for perfect reconstruction. The filters obtained from the Kekre's matrix satisfies the aliasing cancellation conditions and the distortion transfer function is a all pass filter with linear phase response. It is shown that when a sampled speech signal is passed through the filter the signal experiences a one sample delay without any change in amplitude.

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>

The relationship between block digital filtering and quadrature mirror filter (QMF) banks is explored. Necessary and sufficient conditions for alias cancellation in QMF banks are expressed in terms of an associated matrix, derived from the polyphase components of the analysis and synthesis filters. These conditions, called the pseudocirculant conditions, make it possible to unite QMF banks with the framework of block digital filtering directly. Absence of amplitude distortion in an alias-free QMF bank translates into the 'losslessness' property of the pseudocirculant matrix involved. >

A scheme for implementing a perfect reconstruction QMF (quadrature mirror filter) bank is considered. The scheme is based on a uniform DFT (discrete Fourier transform) L-band analysis filter bank realized using a polyphase decomposition of a lowpass L-th band filter (composed of stable allpass subfilters) followed by an L-band synthesis filter bank (realized using the transpose of the analysis filter bank with the allpass filters replaced by their inverse filters). In the subband coding of finite-length sequences, the instability problem of the inverse filters can be avoided by using the original stable allpass filter in each branch of the synthesis bank with time-reversed input and output. Conditions for perfect reconstruction are derived and illustrated with examples. >