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ISSN 2229-5518

Comparative Analysis of Two-Channel QMF Bank Designed by new Class of Adjustable Window Functions

Jyotsna v. Ogale, Alok Jain

Abstract— This paper proposes an efficient method for the design of Pseudo Quadrature mirror filter (QMF) bank. Parent filter of the filter bank is designed using new class of adjustable window functions. Filter coefficients are optimized by varying the cutoff frequency. A comparative analysis is included to confirm the validity of the proposed work.

Index Terms— Finite impulse response filter, Quadrature mirror filter bank, Optimization, Adjustable window functions, Near perfect reconstruction.

.

1 INTRODUCTION

—————————— ——————————
Quadrature mirror filter banks have received much attention since they are successfully used in the design of multirate sig-
Thus researcher’s main attention lies on minimization of am-

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nal processing system. These filterbanks find wide applica-
tions in many fields of signal processing such as sub-band
coding of speech and image signals [1],[2],[3],[4], speech and
image compression [5],[6]. Because of such wide application,
researchers are giving more attention towards the efficient
design of filterbanks [7],]8],[9],[10],[11],[12],[13],[14],[15],
which was first introduced by Johnston [16]. In QMF bank as
shown in Fig. 1 the input signal x(n) splits into equally spaced
frequency sub-bands using a pair of filter comprising lowpass
plitude distortion which needs optimization of certain param-
eter hence optimization technique have been developed. De-
sign methods [8],[9] developed so far involve minimizing an
error function directly in the frequency domain or time do-
main to achieve the design requirements. In the conventional
QMF design technique [14],[15],[16],[17],[18],[19] to get min-
imum point analytically, the objective function, is evaluated
by discretization, or iterative least squares methods are used
which are based on the linearization of the error function to,
and highpass analysis filters

H 0 ( z) and

H1 ( z) respectively,

modify the objective function. Thus, the performance of the
followed by a two fold decimators to down sample the sub-
band signals. At the receiving end corresponding synthesis

bank has two fold interpolator for both the sub-band signals followed by G0 ( z) and G1 ( z) synthesis filters. The outputs of synthesis filters are then combined to obtain the reconstructed signal y(n). The reconstructed output signal y(n) is distorted due to aliasing ,amplitude and phase distortions. These distor- tions can be completely eliminated in Perfect Reconstruction case ideally but practically it is not possible .Aliasing and phase distortion has been completely eliminated by designing

all the analysis/synthesis FIR linear phase filters by a single lowpass prototype even order, symmetric, linear phase, finite impulse response filter. Amplitude distortion can be mini- mize, but can not be eliminated completely.

————————————————

Jyotsna v. Ogale is currently pursuing doctoral degree program in eletronics and communication engineering in Rajiv Gandhi Technical Uni- versity, India, PH-07592250806. E-mail: jyoti.ogale @yahoo.com

Dr.Alok Jain is currently working as a Prof. and H.O.D. in electronics and

intrumentation engineering in Samrat Ashok Technical Institute (Degree

College of Engineering), Vidisha, Madhya Pradesh, India, PH-

07592250725. E-mail: alokjain6@rediffmail.com

(This information is optional; change it according to your need )

QMF bank designed degrades as the solution obtained is the
minimization of the discretized version of the objective func-
tion rather than the objective function itself, or computational
complexity increased. Various design techniques including
optimization based [20], and non optimization based tech-
niques have been reported in literature for the design of QMF
bank. In optimization based technique, the design problem is
formulated either as multi-objective or single objective nonlin-
ear optimization problem, which is solved by various existing
methods such as least square technique, weighted least square
(WLS) technique [14],[15],[16],[17] and genetic algorithm [21].
Jain and Crochiere [9] have introduced the concept of iterative
algorithm and formulated the design problem in quadratic form in time domain. Thereafter, several new iterative algo- rithms [10],[12],[13],[14],[15],[16],[17],[18],[19],[20],[21] have been developed either in time domain or frequency domain.
In continuation, in this work a single variable linear iterative optimization algorithm proposed in [22] has been used to min- imize reconstruction error near to perfect reconstruction. Kai- ser, D.C, Cosh, Modified Cosh and Exponential Window func- tions [23],[24],[25] are used to design even order symmetric, linear phase, lowpass prototype FIR filter for pseudo QMF bank. Finally comparative evaluations of all the filters design

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by above mentioned window functions have been done in
Cosh and Exponential reported in [23],[24],[25] .
terms of minimum stopband attenuation, farend attenuation,
The z-transform of the output signal

y(n) of the two channel

reconstruction error at fixed filter length and at fixed stopband attenuation.
QMF bank, can be written as [18],[19],[20], [26]:

Y (z) = 1 [H (z)G (z) + H (z)G (z)]X (z) + 1 [H (−z)G (z) + H (−z)G (z)]X (−z) (4)

2 0 0 1 1
2 0 0 1 1

Fig. 1. Two-band QMF system
The first term in above equation represents the input/output relation of the overall analysis synthesis filter bank without aliasing and imaging effects. The second term represents the effects of aliasing and imaging. The alias-free two-channel filter bank is obtained by proper combination of the transfer functions of the filters in the analysis and synthesis parts, i.e.,

H 0 ( z), H 1 ( z), G0 ( z) and G1 ( z).

Aliasing can be removed completely by defining the synthesis filters as given below [1, 20,21,26]:

G0 ( z) = 2H1 (− z) and G1 ( z) = −2H 0 (− z)

(5)

By using the relationship H 1 ( z) = H 0 (− z) between the mirror image filters, the expression for the distortion transfer function of the alias free QMF bank can be written as:

T ( z) = H

2 ( z) − H 2 ( z)

(6)

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0 1

Fig. 2. Frequency response of two-band QMF system

It is apparent that the amplitude and phase distortion of the overall QMF bank depend on the performances of the lowpass filter H 0 ( z) . If H 0 ( z) has a linear phase, the phase distortion of the overall filter bank is eliminated. It is shown that for H 0 ( z) being a linear phase FIR filter of the length N, the

2 ANALYSIS OF THE TWO-CHANNEL QMF BANK

overall frequency response
following manner:

T (e jω )

can be expressed in the
The most efficient way for designing and implementing these systems is to first start with a linear phase finite impulse re-

T (e jω ) = e jω ( N −1) H

 0


(e jω )2 (− 1)(N −1) H

(e jω )2 .



(7)

sponse (FIR) prototype filter H ( z) . Let

N −1

Since the filter pair [ H 0

( z) , H1

( z) ] is a halfband filter pair,

n their magnitude responses at the cross over frequency

H 0 ( z) = h(n) z

, with h(n) = h( N n)

(1)

ω = π 2

are equal as shown in Fig. 2. This implies that for
Where,

n =0

h(n) is the impulse response coefficients of a causal

c

odd

values of N, T (e jω )


may have severe amplitude distor-

tions in the vicinity of ωc = π

2 . Therefore, when using the

Nth-order linear phase FIR filter obtained using window tech-

nique as given by :
linear phase FIR filters, the length N should be an even num- ber. With even filter order equation (4) can be reduces to

h(n) = w(n)hi (n)

(2)

T (e jω ) = e jω ( N −1) H

 0


(e jω ) 2 + H

(e jω )2 .



(8)
where

h (n) is the impulse response of the ideal low pass fil-

i

According to equation (8), the QMF bank with linear phase
ter and is expressed as:
filters has no phase distortion, but the amplitude distortion will always exist except for the trivial first order case. It is an important approximation problem to adjust the coefficients of

h (n) = sin (ωc ( n− 0.5 N ) )

(3)
the

H 0 ( z) that simultaneously provide the selective frequen-

i π ( n− 0.5N )

cy responses of the analysis/synthesis filters and guarantee a
small reconstruction error. A computer-aided optimization
method can be employed, which iteratively adjusts the coeffi-
where,

ωc is cut-off frequency of the ideal low pass filter

cients of H

( z) to achieve

and w(n)

is the window function. The prototype filter is de- 0
signed using well known Kaiser, DC window functions and newly reported window functions such as Cosh , Modified

H (e

jω ) 2

+ H (e

jω ) 2 = 1

(9)

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The optimization algorithm discussed in [22] is applied to vary the cutoff frequency of the lowpass prototype filter. The reconstruction error is selected as the objective function.

The objective function given in equation. (9) has been used to minimize the reconstruction error to near perfect reconstruc-

Specify sampling frequency, number of channel, stop band attenuation, pass band ripple

Initialize pass band and stop band frequen-

iterations are ss the error of present iteration is within the specified tolerance initialized previously or no improvement has been made from the previous value. The proposed tech- nique has been implemented in MATLAB and its flow graph is shown in Fig. 3 given below.

3 CASE STUDY AND PERFORMANCE ANALYSIS

Under the case study we consider two examples, we choose the Johnston’s FIR filters of the length N = 24 and 32 to ex- amine the performances of a linear phase FIR QMF bank.

cies,tol,i_error,step,dir.Calculate cutoff frequency,

We determine first the analysis filters H 0 ( z)

and

H1 ( z) ,

filter length and window coefficients

Design prototype filter and determine re-

and plot their characteristics in frequency domain. Deter-
mine reconstruction error and minimize it by varying cutoff frequency of the prototype filter.

Example 1-

construction error

(error )

With N =24, ωp =0.4π, ωs = 0.6π, function tolerance=1e-6, ini-

tial error = 250 dir = 1, step = 0.05 a QMF bank has been de-

signed by windowed FIR filter.
The significant parameters obtained are listed in TABLE 1.

Yes

Is error tol or

i _ error

= error

TABLE 1: Performance parameters.

Stop

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Window function

Peak reconstruc- tion

error (Ep )

i _ error

= error

Cutoff frequency=cutoff frequen- cy+(step*dir).

Redesign prototype filter using new value of cutoff frequency and deter-

mine reconstruction error error

Is error

> i _ error

No

Step=step/2 dir=-dir

Yes

Example 2-

For N = 32, ωp = 0.4π, ωs = 0.6π, performance parameters are listed in TABLE 2.The corresponding normalized magni-
tude plots of the analysis filters

H 0 ( z) and

H1 ( z) are

tion. In the optimization algorithm cutoff frequency (ωc ) is varied to get the smallest value of reconstruction error. The algorithm adjusts the cutoff frequency (ωc ) by step size in each iteration , calculates the new filter coefficients , computes the reconstruction error, compare it with previous error, accord- ingly step size and search direction has been changed. The
shown in Fig. 4 (a,c,e). Fig. 4 (b,d,f ) shows the peak recon-
struction error of the QMF bank . The significant parameters obtained are: Ep = 0.0052, As = -42.415 dB, Af = -99.53 dB and actual value of stopband attenuation is -86.93 dB for Modi- fied Cosh window based design with multiplicative fac-
tor ρ = 7 .
The simulation results of the proposed method are compared
with the Gradient method [27], General purpose method [24],
Smith Barnwell method [15], Jain Crochiere [9], Chen Lee [8],
Xu-Lu Antoniou [29], Lu-Xu Antoniou [21] and Sahu et al. [17],

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for N = 32, and are summarized in TABLE 2. The results show that Modified Cosh window based design performs better than Gradient, General purpose, Smith Barnwell, Jain Cro- chiere, Chen-Lee, Xu-Lu-Antoniou, Lu-Xu-Antoniou and Sahu methods in terms of peak reconstruction error Ep and far end stop band attenuation Af. It gives much better minimum stop band attenuation than [8],[9],[17],[21],[22],[27]. The obtained actual value of stop band attenuation as shown in Fig. 4(c) is highest amongst all the proposed windows. According to the results obtained, some observation about filter characteristics is also made. Variation of peak reconstruction error with filter length at constant As is shown in Fig. 5. It is observed that Modified Cosh window based design shows superior perfor- mance. Similarly from Fig. 6 it is observed that peak recon- struction error decreases as stopband attenuation increases for Kaiser, DC, Cosh and Exponential windows. For Modified Cosh window peak reconstruction error increases with in- creasing values of stopband attenuation. At most it is observed

20

0

-20

-40

-60

-80

-100

-120

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(c) Frequency response plot (Modified Cosh)

-3

that by introducing a third parameter (p) in the window func-
tion a better window function can be obtained for FIR filter
design where higher main-lobe width and smaller ripple ratio
is important. It also leads to better side-lobe roll off ratio. Thus
it gives a better response over cosh window, Kaiser window, DC and Exponential window

6 x 10

4

2

IJSE0 R

20 -2

0 -4

-20

-40

-60

-6

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(d) Reconstruction error plot (Modified Cosh)

-80

-100

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(a) Frequency response plot (Cosh)

0.06

0.04

0.02

20

10

-10

-30

-50

-70

0

-0.02

-0.04

-0.06

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(b) Reconstruction error plot (Cosh)

-90

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(e) Frequency response plot (Exponential)

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0.1

0.05

0

TABLE 2: Performance comparison of proposed QMF bank with previous work.

-0.05

Work N A s

(dB)

Ep

(dB)

A f

(dB)

Phase response

-0.1

0 0.1 0.2 0.3 0.4 0.5

Normalized frequency

4(f) Reconstruction error plot (Exponential)

Gradient method [27] 32 -33.6 0.009 - Nonlinear General purpose [28] 32 -49.2 0.016 - Linear Smith–Barnwell [15] 32 -39.0 0.019 Nonlinear

Jain-Crochiere[9] 32 -33.0 0.015 -54 Linear

0.025

0.02

Chen-Lee[8] 32 -34.0 0.016 -52 Linear

Xu-Lu-Antoniou[29] 32 -35.0 0.031 -54 Linear Lu-Xu-Antoniou[21] 32 -35.0 0.015 -- Linear Sahu et al.[17] 32 -33.913 0.0269 -- Linear

0.015

Kaiser

DC

Kaiser

32 -53.90

32 -53.90

0.01048

0.04244

-53.9

-61.04

Linear

Linear

IJSDC ER

0.01

0.005

0

Cosh

Modified Cosh

Exponential

Cosh

Proposed

Modified

cosh

Exponential

32 -50.50

32 -42.41

32 -44.5

0.0532

0.0052

0.0879

-74.01

-99.53

-63.45

Linear

Linear

Linear

10 20 30 40 50 60 70

Filter length

Fig. 5 Filter length versus Peak reconstruction error plot

0.015

0.014

0.012

0.01

Kaiser DC Cosh

Modified Cosh

Exponential

0.008

0.006

0.004

0.002

0

50 55 60 65 70 75 80 85 90 95 100

Stopband attenuation (dB)

Fig. 6 Stopband attenuation versus Peak reconstruction error plot

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4 CONCLUSION

New variable windows have been proposed for designing the low-pass prototype filters for QMF banks. Linear iterative op- timization algorithm has been use to optimize the filter coeffi- cients to get minimum value of reconstruction error by vary- ing the filter cutoff frequency. The Modified Cosh window based design showed optimum performance in terms of re- construction error and far end attenuation at the cost of in- creased arithmetic complexity. Better far end rejection feature helps to reduce the aliasing energy leak into a sub band from that of the signal in the other sub band.

5 ACKNOWLEDGEMENT

The corresponding author wish to thank Dr. Alok Jain of the same institute for his valuable and constructive suggestions.

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