QUADRATURE MIRROR FILTER BANK FOR SUBBAND CODING

Abstract

Substantial progress in the field of multirate filter banks design and their applications have been made since the last two decades. Among the different classes of filter banks, a two-channel filter bank, namely, quadrature mirror filter (QMF) bank has been of great interest in digital signal processing applications during the past decade, ever since their introduction by Croisier et al. These filter banks find applications where the input signal is splitted into a number of consecutive bands in the frequency domain, so that each subband can be independently processed. Typical processing includes under-sampling the subband signals, encoding them, and transmitting over a channel, or merely storing the coded signals. The main aim of doing so is to significantly reduce the number of bits representing the original signal for storing or transferring purposes. Eventually, at some point in the processing, the subband signals are recombined so that the original input signal is properly reconstructed. Originally, they were introduced to reduce or remove the effect of aliasing error in the speech coding. Since then, advances in QMF banks have provided a new generation of subband coders for video signals, digital transmultiplexers used in FDM /TDM conversion, and ECG signal processing. There are many benefits of using multirate systems: quantization noise is localized in subbands; signal-to-quantization noise (SNR) ratio is improved due to independent variation of quantization step size in each band, and bit allocation can be done on the basis of perceptual criteria etc. Hence, the subband coding systems have proven to be powerful tools for data compression schemes such as MP3 and JPEG 2000, denoising, and signal estimation. Ideally, in absence of channel coding, multirate system is a pure delay system, and should not introduce any distortion to the input signal. However, in general it cannot be achieved, and a multirate system suffers from three different types of distortion: aliasing distortion, amplitude distortion and phase distortion. In early stage of research, several methods were developed to diminish these distortions. In order to keep aliasing distortion low, the composing filters have very narrow transition width, which in turn have high orders that make the implementation of these systems complicated. To overcome these drawbacks, the concept of QMF bank was introduced to purge aliasing distortion in the multirate systems. Since then, manysubband coders were devised for speech coding using QMFbank. However, their performances rely on the ability of composing fitters in QMF banks, which provide good isolation between continuous frequency bands. But, the design of QMF banks was thorny due to some constraints such as filter response exhibits an odd symmetric property about the quarter-band frequency. In addition to this, QMF bank design is a multiobjective optimization problem, and the design objectives are not unique and are conflicting, leading to designs with different tradeoff. Depending upon how many channels are used for splitting the input signal, multirate filter banks can be classified in two groups: two-channel and M-channel filter banks. In first group, the input signal is separated into two different subbands; while in second group, it is splitted into M-different subbands. Both the types of filter banks can be further classified as nearly perfect reconstruction (NPR) and perfect reconstruction (PR) filter banks. NPR-QMF banks consist of conventional QMF banks where the analysis and synthesis filters have linear-phase filters. In NPR-QMF bank, aliasing distortion is confiscated using suitable design of synthesis filters, whereas phase distortion is purged with the use of linear-phase finite impulse response (FIR) filters. Amplitude distortion can be minimized using computer aided techniques or equalized by cascading with a filter, and can also be completely eliminated at the expense of phase distortion. It is also possible to remove all the three types of distortions in a two-channel filter bank without restrictive QMF conditions. Such filter banks are termed as exact reconstruction filter banks. In these filter banks, the reconstructed output signal is an exact replica of original signal with some delay. However, to achieve PR filter banks, often one has to sacrifice several good properties such as low stopband attenuation, narrow transition bands and linear-phase, which are desired in many applications such as videos and communication systems in the analysis /synthesis filters. These impairments as well as computational complexity are major issues in the design of filter banks for subband coding systems. Over the past several years, numerous methods have been devised in an attempt to diminish or eliminate above mentioned errors as well as to minimize the overall computation time needed for the implementation of these systems. The first systematic approach was developed in 1980 for designing NPR-QMF bank. Subsequently, many methods have been devised which were based on direct minimization of error function either in frequency domain or in time domain. But, these approaches were not suitable for the longer filter due to high degree of nonlinearity. Furthermore, in most of the algorithms, convergence to optimum solution depends on the initial guess value and sometimes, these approaches do not find the global optimum solution due to several local minima. To overcome this problem, an iterative technique based on linearization of the cost function has been developed which provides certain convergence and permits the design of equiripple filters that satisfy the flatness constraint required for good amplitude reconstruction. Later on, a number of iterative algorithms have been proposed for designing QMF banks based on this algorithm. Although, a computational load is reduced in these algorithms, the design problem is solved by nonlinear optimization. Recently, several new algorithms have been developed for improving the performance of design in term of the reconstruction error, computational time, and number of iterations. Though, almost all methods developed so far give better performance in term of the reconstruction error, but converse to optimum solution in larger number of iterations, and more computational time is required. In view of the above, there is strong motivation to develop a new algorithm that can minimize the reconstruction error, number of iterations as well as computational time in case of longer filter. Initially the research effort was directed towards the design of a two-channel QMF bank. Later on, it was extended to multi-channel QMF banks. Among the three different basic approaches of M-channel filter banks, the cosine-modulated (CM) filter banks have emerged as an attractive choice of filter banks with respect to implementation cost and design saving. To date, several algorithms have been proposed for designing CM filter banks. But these methods require more iterations and large computation time. Thus, similar to two-channel QMF banks, an efficient algorithm is still awaited to propose which can minimize the reconstruction error, aliasing error, computational time, and number of iterations. This dissertation, therefore, presents an efficient and iterative algorithm for the design of a linear-phase QMF bank. The proposed algorithm is based on the optimum cutoff frequency(coc) and perfect reconstruction condition at frequency {(0=kI2). Algorithm employs different window functions for designing the prototype filter for a two-channel QMF bank. A unique method has been developed for optimizing Q)c which leads to improved performance of filter banks relative to other existing algorithms. It is observed that the reconstruction error can be further minimized with the use of window shape parameter. Several design examples have been considered to illustrate the advantageous features of the proposed algorithm over other exiting methods. Finally, the methodology has been extended for designing M-channel cosine-modulated filter banks.

In M-channel filter bank, the amplitude distortion depends on degree of overlapping between analysis filters. The roll-off factor (RF) decides overlapping between the adjacent channels. If 0 < RF < 1, then each subband is overlapped by its adjacent subbands, and if 1< RF < 2, then each subband is overlapped by four subbands. Hence, the roll-off factor controls the performance of filter banks. In digital filter design, three most popular approximation criteria used are the least squares (L2), the minimax (LJ, and the peak constrained least square (PCLS) design criteria. The L-, design criterion does not provide any provision for minimization of the maximum approximation error, while the L^ design criterion has large error energy when compared to that of L2 design criterion. Therefore, the filter banks designed using of equiripple filter yields high L2 error in the stopband region. Whereas the filter banks designed using window functions result in reduced stopband L2 error. However, passband ripple turns out to be approximately equal to the stopband ripple. The PCLS design criterion generalizes both the design criteria. By keeping above facts in mind, a new technique for designing CM filter banks was devised in 2008. But, it is not so efficient and work for limited value of the roll-off factor. In the above context for the purpose of this work, an improved iterative method is developed for M-channel CM filter banks with prescribed stopband attenuation, passband ripple, and channel overlapping by incorporating above mentioned facts in the extended algorithm. The method employs the weighted constrained least squares (CLS) technique for the design of prototype filter for CM filter banks. A comparative study reveals that the proposed method is superior in terms of the reconstruction error and aliasing error, and is more flexible than other existing methods. Similar to the cut-off frequency, a suitable value of the passband edge frequency (co )can reduce the amplitude distortion. There are very few references available in which CO has been optimized for designing a prototype filter for the two-channel QMF banks and also for M-channel CM filter banks. In most of the references available so far, the complex objective functions and optimization techniques have been used that require large computation time and more number of iterations. Therefore, a unique iterative method has been developed which casts the design problem as a linear minimization of the filter coefficients such that their value at co = nl2 is 0.707. Another method has also been developed to solve the design problem in which instead of coc, the passband edge frequency

is optimized. This method employs the constrained equiripple FIR filter design technique for designing a prototype filter for filter banks. The design examples given clearly illustrate the superiority of the developed algorithm in terms of peak reconstruction error, computational time, and number of iterations over the other existing algorithms. Furthermore, it is simple, linear in nature, and easy to implement. Finally, it has been extended for designing Mchannel CM filter banks. The robustness of the extended algorithm to CM filter banks has been shown by simulation results. Some applications, which are carried out in real-time or quasi-real time require a filter bank with low computational time. For example, for high quality reconstruction of the sound signal, a filter bank with high stopband attenuation, small channel overlap, and efficient resolution switching is required in perceptual audio coding. In ECG signal processing, especially for heart beat detection, and for image processing, the filter banks with fast switching resolution, and adjustable stopband attenuation is required. For such applications, iterative algorithm developed so far are not much suitable, the closed form method is more preferred. Literature available so far on two-channel linear-phase QMF banks, reveal that there is still need for a computationally efficient technique, which shall not use any optimization technique for designing a two-channel linear-phase QMF bank, and also for cosine-modulated filter bank. In the above context, a simple and efficient closed form method is developed for designing a two-channel linear-phase QMF bank with prescribed stopband attenuation and channel overlap. The developed methodology is based on optimum passband edge frequency, which is evaluated by empirical formulas instead of using time consuming single or multivariable optimization algorithm. This technique makes the use of window techniques to design prototype filter for QMF banks. Later on, it has been modified to exploit the novelty of the spline function in transition band of the ideal filter. Finally, the modified technique has been extended for designing M-channel cosine-modulated filter banks with prescribed stopband attenuation and channel overlap. When compared to other existing techniques, it was found that the developed closed form techniques perform better in terms of computation time and amplitude distortion. Like iterative and weighted least squares techniques, several other techniques using constrained or unconstrained optimizations have been developed for designing a twochannel linear-phase QMF bank. A detailed survey reveals that in the most of the techniques reported earlier, the design problem has been formulated as a linear or nonlinear combination of the reconstruction error and stop band residual energy or passband energy, and the filter response in transition band has not been taken into account simultaneously. Only one reference is available in which design problem is formulated using the above considerations. In this, three different weights have been chosen based on trial and error method, which is very time-consuming and also works for limited values. Thus, the literature survey reflects the need for the development of a technique, considering filter responses in transition band as well as in passband and stopband for the efficient design of QMF banks. The methodologyproposed in this work casts the filter bank design problem as a low pass prototype filter design problem whose responses into passband and stopband are ideal, and their filter coefficients value at quadrature frequency is 0.707. Originally, the design problemis constructed as an unconstrained optimization that minimizes the weighted sum of error of transfer function of the filter bank at quadrature frequency, stopband energy and the passband error of a prototype filter. A new method is developed for the design of a low pass prototype filter for QMF banks. For solving the given unconstrained optimization problem, Quasi-Newton technique is exploited. Numerical examples and comparisons with several existing methods evidently show the performances and effectiveness of this method. The design problem is further constructed as a constrained optimization problem in two different ways which are solved by two different optimization techniques: sequential quadratic programming (SQP) optimization technique and marquardt optimization technique. Performances of these methods are evaluated in terms of reconstruction error, errors in passband, stopband and transition band. The proposed techniques have been applied for subband coding of speech, electrocardiogram (ECG), and ultrasound image signals to examine the performance. The fidelity parameters obtained clearly illustrate that good designs can be obtained with these techniques.