During the last two decades, there has been substantial progress in multirate digital filters and filter banks. This includes the design of quadrature mirror filters (QMF). A twochannel QMF bank is extensively used in many signal processing fields such as subband coding of speech signal, image processing, antenna systems, design of wavelet bases, and biomedical engineering and in digital audio industry. Therefore, new efficient design techniques are being proposed by several authors in this area. This paper presents an overview of analysis and design techniques of the twochannel QMF bank. Application in the area of subband coding and future research trends are also discussed.
The concept of quadrature mirror filter (QMF) bank was first introduced by Croisier et al. [
In comparison to earlier band pass filter based subband coding systems, the QMF bank based systems have many advantages as given next.
Aliasing distortion is eliminated in QMF bank based subband coding systems; therefore, the transition width of the filters is not much important. Lower order filters with wider transition band can be used [
Computation complexity is reduced in case of subband coding system based on QMF banks [
Lower bit rates are possible, without degrading the quality of decoded speech signals.
QMF based subband coders [
Twochannel filter banks can be classified into three types: quadrature mirror filter banks, orthogonal filter banks, and biorthogonal filter banks [
QMF filter sections may be cascaded in a tree structure to generate multichannel filter banks [
A typical twochannel QMF bank shown in Figure
Twochannel quadrature mirror filter bank.
The design techniques for QMF bank can be classified as optimizationbased or nonoptimization based. Various optimization based techniques [
Several methods [
The organization of the paper is as follows. Section
The twochannel QMF bank structure is known as critically sampled filter bank as decimation, and interpolation factors are equal to number of bands. The frequency responses of the analysis filters
Frequency response of the analysis filters
By using inputoutput relationship of decimator and interpolator, we can write
To obtain the perfect reconstruction QMF bank, PHD and AMD should also be eliminated; that can be possible if the reconstructed signal
From (
QMF bank can be implemented efficiently by using polyphase decomposition [
Polyphase implementation of decimation filter.
For complete implementation of twochannel QMF bank using polyphase framework, a total of only about
As we have already discussed in Section
In the design of QMF bank, the coefficients of the filters
The general methods for the design of QMF bank can be as follows:
elimination of PHD completely and optimization of the filter coefficients to minimize AMD in an alias free system;
elimination of AMD completely and optimization of the filter coefficients to minimize PHD in an alias free system;
elimination of the three distortions ALD, AMD, and PHD simultaneously.
As discussed in the previous section, (
Figure
(a) Responses of
Systematic computer aided optimization techniques [
For the design of linear phase FIR lowpass analysis filter
Jain and Crochiere [
Chen and Lee [
A general purpose approach was proposed by Bregovic and Saramaki [
Sahu et al. [
Due to nonlinearity and nonconvexity of the objective functions, conventional numerical/mathematical methods may find difficulty to achieve an optimal design. In most cases, they are not able to find global optimum solution. Nowadays researchers also started using of natureinspired optimization techniques to design QMF bank. Popular swarm intelligence approach, known as particle swarm optimization (PSO), has also been applied in [
Differential evolution (DE) is one of the most powerful evolutionary algorithm (EAs) and has been used for various signal processing applications. Ghosh et al. [
Comparison of various nearly PR algorithms for design of twochannel QMF bank based on significant parameters for
Methods  PRE (dB) 





JainCrochiere [ 
0.015 


33.00  44.25 
ChenLee [ 
0.016 


34.00  44.40 
Gradient method [ 
0.016 


33.60  35.00 
Sahu [ 
0.027 


33.93  44.25 
Upender et al. [ 
0.015 


36.87  44.75 
LuXuAntoniou [ 
0.015 


35.00  44.30 
XuLuAntoniou [ 
0.031 


35.00  43.60 
Steepest Descent [ 
0.050 


34.56  40.00 
Kumar et al. [ 
0.010 


36.59  43.50 
Ghosh [ 
0.0085 


36.91  44.88 
Bergovic [ 
0.009  0.155 

49.20  61.00 
Algorithm in [ 
0.023 


35.40  44.10 
It is possible to completely eliminate amplitude distortion, rather than just minimize it. In this method, again analysis filters and synthesis filters are related as in (
Digital Butterworth, Chebyshev and Elliptic filters are special cases [
Enkanayake and Premaratne et al. [
We can obtain a perfect reconstruction QMF bank by eliminating all three distortions, namely, ALD, AMD, and PHD, simultaneously. The reconstructed signal is therefore just a time delayed version of the input signal
By substituting in (
In some PR systems, it is desirable that the analysis filters are constrained to have linear phase. To obtain FIR linear phase PR QMF banks, it is necessary to give up [
Comparison of three types of twochannel QMF banks.
Feature  NPR 
Perfectreconstruction system  NPR 

Phase response  Linear  Linear/nonlinear  Nonlinear 
Aliasing  Canceled  Canceled  Canceled 
Phase distortion  Eliminated  Eliminated  Minimized 
Amplitude distortion  Minimized  Eliminated  Eliminated 
Overall group delay 


Complicated 
Relation between analysis filters 

Not explicit. 

Power complementarity  Approximately holds  Does not hold  Approximately holds 
Subband coding of signals is an effective method to achieve bandwidth compression when the signal energy is dominantly concentrated in a particular region of frequency. In subband coding, the signal is subdivided into several frequency bands and each band is digitally encoded separately. Vetterli [
Single level decomposition.
The individual subbands can be processed according the required applications. XX subband can be further decomposed for multilevel decomposition. The reconstruction of the full band signal is done using interpolators and synthesis filters.
Following fidelity assessment parameters can be used to analyze the satisfactory reconstruction of original image [
Mean square error (MSE):
Peak signal to noise ratio (PSNR):
Twochannel QMF was first used in subband coding and then find applications in various signal processing fields. Since 1976, various design techniques for QMF banks have been proposed by researchers. Due to different applications, the QMF banks are growing field in digital signal processing. The research areas in QMF banks are design of twochannel nearly PR QMF banks, PR QMF banks, designing Mchannel QMF banks, and applications of QMF banks. There are few points that may need to be further researched.
While designing QMF banks, the performance and effectiveness of a method are measured in terms of peak reconstruction error (PRE), mean square error in pass band, and stop band regions, error in transition band, stop band attenuation, group delay, phase response and computational complexity. There is a need for suitable algorithms for finding optimum solutions for these significant parameters.
If the characteristics of prototype filter are assumed to be ideal in its pass band and stop band regions, consequently, reconstruction error lies only in transition band. Hence, a suitable objective function is required to minimize this error. In earlier proposed design techniques, suitable modifications can be made in such a way that the reconstruction error and computational complexity will reduce by using new algorithms.
Classical optimization methods may fail to achieve optimal design as they do not guarantee for convergence on the global optimum. Therefore, genetic algorithms (based on law of nature) and populationbased optimization algorithms [
Hybrid optimization methods may be used to design QMF banks for improved performance. The combination of classical optimization method and genetic algorithm can be used.
QMF banks are used extensively in designing efficient subband coders for speech, image, and video signals. Tree structure QMF bank can be used in wireless communication as interference canceler [
In this paper, a comprehensive overview of twochannel QMF banks has been presented. Different design techniques for nearly perfect reconstruction and perfect reconstruction QMF banks have been discussed. New algorithms are being proposed for design of QMF bank with improved performance in terms of reconstruction error and computational time. Advances in QMF banks provide efficient subband coders for speech, image, and video signal. Application of twochannel QMF bank for subband coding of image signal is also presented. Some future research trends in designing of QMF banks and its applications have also been discussed which may be helpful for the researchers.