An efficient Quadrature Mirror Filter design and its applications in audio signal processing

An iterative approach for minimax design of multidimensional quadrature mirror filters

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.

Design of quadrature mirror filter bank using polyphase components based on optimal fractional derivative constraints

Scene adaptive coders are constituted by the cascade of a linear transform, scalar quantization, entropy coding and a buffer controlled by a feedback loop for bit rate regulation. The main contribution of this paper is to derive an analytical criterion evaluating the performances of any perfect reconstruction linear transform in the frame of scene adaptive coding. This criterion is, thereafter, used in order to optimize linear multiresolution transforms. The optimization adapts the filters parameters to the codec features and to the statistics of the 2-D sources; so, the authors call these transforms adapted multiresolution transforms (AMTs). The transforms under study are implemented by a cascade of separable perfect-reconstruction (PR) FIR two-band filter banks that can change at each resolution level. Two types of filter banks are envisaged: the PR orthogonal quadrature mirror filter (QMF) bank, which allows to implement the orthogonal AMT and the PR linear-phase filter (LPF) bank, which implements the biorthogonal AMT. They perform the optimization of the filters in their factorized lattice form, taking the finite length of the multipliers into account. Their criterion also allows them to show the performances achieved by these two linear multiresolution transforms compared to other linear (multiresolution) transforms. >

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. >

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>

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. >

This paper presents a new algorithm for the design of a two band alias-free linear phase quadrature mirror filter (QMF) bank. The proposed algorithm is based on the optimum value of passband edge frequency ( p ω ) that is found by approximately satisfying the magnitude response condition of the prototype filter at frequency ( 0.5 ωπ = ). In this algorithm, passband edge frequency is optimized. Performances of this algorithm are evaluated in term of computational time, reconstruction error and number of iterations. The design examples given in this paper clearly illustrate the superiority of this algorithm in terms of the peak reconstruction error, computation time, and number of iterations over Creusere and Mitra algorithm. The proposed algorithm is simple, easy to implement, and linear in nature.

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.

An iterative method for the design of two-band quadrature mirror filter (QMF) banks proposed by Cruesere and Mitra is examined and improved by designing the prototype filters using window method instead of equiripple method. The Kaiser window and Chebyshev window functions are used in the design of prototype filters. The cut-off frequency has been adjusted in this improved method instead of passband frequency which is the case in Cruesere and Mitra algorithm. The paper concludes with design results and comparisons that show that our new improved method takes less computational effort when compared with Cruesere and Mitra method.

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. >

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.

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.

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.