# Design of Two-Channel Quadrature Mirror Filter Banks Using Differential Evolution with Global and Local Neighborhoods.

**ABSTRACT** This paper introduces a novel method named DEGL (Differential Evolution with global and local neighborhoods) regarding the design of two channel quadrature mirror filter with linear phase characteristics. To match the ideal system response characteristics, this improved variant of Differential Evolution technique is employed to optimize the values of the filter bank coefficients. The filter response is optimized in both pass band and stop band. The overall filter bank response consists of objective functions termed as reconstruction error, mean square error in pass band and mean square error in stop band. Effective designing can be performed if the objective function is properly minimized. The proposed algorithm can perform much better than the other existing design methods. Three different design examples are presented here for the illustrations of the benefits provided by the proposed algorithm.

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**ABSTRACT:**Artificial bee colony (ABC) algorithm has been introduced recently for solving optimization problems. The ABC algorithm is based on intelligent foraging behavior of honeybee swarms and has many advantages over earlier swarm intelligence algorithms. In this work, a new method based on ABC algorithm for designing two-channel quadrature mirror filter (QMF) banks with linear phase is presented. To satisfy the perfect reconstruction condition, low-pass prototype filter coefficients are optimized to minimize an objective function. The objective function is formulated as a weighted sum of four terms, pass-band error, and stop-band residual energy of low-pass analysis filter, square error of the overall transfer function at the quadrature frequency and amplitude distortion of the QMF bank. The design results of the proposed method are compared with earlier reported results of particle swarm optimization (PSO), differential-evolution (DE) and conventional optimization algorithms.Swarm and Evolutionary Computation. 12/2014;

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Design of two-channel quadrature mirror filter banks

using Differential Evolution with global and local

neighborhoods

Pradipta Ghosh , Hamim Zafar , Joydeep Banerjee Swagatam Das

Electronics & Tele-Comm. Engineering

Jadavpur University

Kolkata, India

{iampradiptaghosh, hmm.zafar, iamjoydeepbanerjee}@gmail.com

swagatamdas19@yahoo.co.in

Abstract. This paper introduces a novel method named DEGL (Differential

Evolution with global and local neighborhoods) regarding the design of two

channel quadrature mirror filter with linear phase characteristics. To match the

ideal system response characteristics, this improved variant of Differential

Evolution technique is employed to optimize the values of the filter bank

coefficients. The filter response is optimized in both pass band and stop band.

The overall filter bank response consists of objective functions termed as

reconstruction error, mean square error in pass band and mean square error in

stop band. Effective designing can be performed if the objective function is

properly minimized. The proposed algorithm can perform much better than the

other existing design methods. Three different design examples are presented

here for the illustrations of the benefits provided by the proposed algorithm.

Keywords- Filter banks, Quadrature Mirror Filter, Sub-band coding, perfect

reconstruction, DEGL.

1. Introduction

Efficient design of filter banks has become a promising area of research work. An

improved design of filter can have significant effect on different aspects of signal

processing and many fields such as speech coding, scrambling, image processing, and

transmission of several signals through same channel [1]. Among various filter banks,

the two channel QMF bank was first used in Sub-band coding, in which the signal is

divided into several frequency bands and digital encoders of each band of signal can

be used for analysis. QMF also finds application in representation of signals in digital

form for transmission and storage purpose in speech processing, image processing

and its compression, communication systems, power system networks, antenna

systems [2], analog to digital (A/D) converter [3], and design of wavelet base [4].

Various optimizations based or non optimization based design techniques for QMF

have been found in literature. Recently several efforts have been made for designing

the optimized QMF banks based on linear and non-linear phase objective function

using various evolutionary algorithms. Various methods such as least square

technique [5-7], weighted least square (WLS) technique [8-9] have been applied to

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solve the problem. But due to high degree of nonlinearity and complex optimization

technique, these methods were not suitable for the filter with larger taps. A method

based on eigenvector computation in each iteration is proposed to obtain the optimum

quantized filter weights [10]. An optimization method based on Genetic algorithm

and signed-power-of-two [11] is successfully applied in designing the lattice QMF. In

frequency domain methods reconstruction error is not equiripple [7-9]. Chen and Lee

have proposed an iterative technique [8] that results in equiripple reconstruction error,

and the generalization of this method was carried out in [9] to obtain equiripple

behaviors in stop band. Unfortunately, these techniques are complicated, and are only

applicable to the two-band QMF banks that have low orders. To solve the previous

problems, a two-step approach for the design of two-channel filter banks [12,13] was

developed. But this approach results in nonlinear phase, and is not suitable for the

wideband audio signal. A more robust and powerful tool PSO has also been applied

for the design of the optimum QMF bank with reduced aliasing distortion, phase

aliasing and reconstruction errors [14,15]. The problem with PSO is the premature

convergence due to the presence of local optima.

Amongst all Evolutionary Algorithms (EAs) described in various articles,

Differential Evolution (DE) has emerged as one of the most powerful tools for solving

the real world optimization problems. It has not been applied on the design of the

perfectly reconstructed QMF banks till now. In this context we present here a new

powerful variant of DE called DEGL [20] for the efficient design of two channels

QMF bank. Later in this paper, we will discuss the effectiveness of this algorithm and

compare this with other existing method towards designing of Digital QMF filter. For

proving our point we have presented three different design problems.

The rest of the paper is arranged in the following way: Section 2 contains the

Design Problem, Section 3 gives a brief overview of classical DE, Section 4

introduces a new improved variant of DE termed as DEGL, Section 5 deals with the

design results and comparison of these results with other algorithms and Section 6

concludes this paper.

2. Formulation of Design Problems

For a typical two-channel QMF bank as shown in Fig. 1 the reconstructed output

signal is defined as

)( )]()()()( [ 2 / 1

+

)( )]()()()([ 2 / 1

=

=

)(

11001

z

1

X

00

X

zXzGzHzGzHzXzG

−

zHzGzHzY

−−+−+

(1)

)()()()(

zAzzT

+

whereY(z) is the reconstructed signal and X(z) is the original signal.

Fig. 1: Two channel QMF BANK

Page 3

In eqn.1 the first term T(z) is the desired translation from input to the output, called

distortion transfer function, while second term, A(z)is aliasing distortion because of

change in sampling rate. By setting

)()( ),()(

1001

zHzGzHzH

−=−=

and

)(

1

zG

the aliasing error in (1) can be completely eliminated. Then eqn. (1) becomes

)]()()()( [ 2 / 1)(

0000

zHzHzHzHzY

−−+=

It implies that the overall design problem of filter bank reduces to determination of

)(

0

zH

−−=

(2)

)(

zX

(3)

the filter taps coefficients of a low pass filter

(z)H0

, called a prototype filter. Let

(z) H0

be a linear phase finite impulse response (FIR) filter with even length ( N ):

2/ ) 1

−

(

00

)()(

−

=

Njjj

eeHeH

ωωω

(4)

If all above mentioned conditions are put together, the overall transfer function of

QMF bank is reduced to Eqn. (5)

)(

=

eT

{}

2

(

0

2

0

2/ ) 1

−

(

)()(

2

πωω

j

ω

j

ω

j

−

−

+

j

N

eHeH

e

(5)

)(

2

1

)(

2/ ) 1

−

(

ω

j

ω

j

ω

jN

eTeeT

′

=

−

(6)

Where,

])() 1

−

()([)(

2

0

) 1

−H

(

2

0

ωπω

ω

j

−−=

′

HeT

N

and N is the no of filters.

If

1)()(

2

(

0

2

0

=+

−πωω

jj

eHeH

it results in perfect reconstruction; the output signal is

exact replica of the input signal. If it is evaluated at frequency (

perfect reconstruction condition reduces to Eqn. (7).

πω

5 . 0

=

), the

707. 0)(

5 . 0

j

0

=

π

eH

(7)

It shows that QMF bank has a linear phase delay due to the term

Then the condition for the perfect reconstruction is to minimize the weighted sum of

four terms as shown below:

EEEK

321

ααα

++=

2/ ) 1

−

(

−

Nj

e

ω

.

htsp

Emor

.

54

αα

++

(8)

where

51

αα −

are the relative weights and

p

E ,

s

E ,

t E ,mor (measure of ripple) ,

h

E are defined as follows:

p

E is the mean square error in pass band (MSEP) which

describes the energy of reconstruction error between 0 and

p

ω ,

ω

d

ω

(

π

ω

∫

0

HHEp

−=

2

00

)) 0 (

1

(9)

s

E is the mean square error in stop band (MSES) which denotes the stop band energy

related to LPF between

s

ω to π

ω

d

ω

(

π

π

∫

ω

HE

s

s

=

2

0

)

1

(10)

t E is the square error of overall transfer function at quadrature frequency

2/

π

2

00

)]0 (

2

1

)

2

([

HHEt

−=

π

(11)

Page 4

Measure of ripple (mor)

)( log 10 min

ω

)( log 10max

ω

1010

ωω

TT mor

′

−

′

=

(12)

h

E is the deviation of

)(

ω

jeT′

from unity at

2/

πω =

1)

2

(

−

′

=

π

TEh

(13)

The above fitness is considered because we know that condition for perfect

reconstruction filter is

2

+

HeH

1)()(

2

(

00

=

−πωω

jj

e

(14)

3. DE with Global and Local Neighborhoods (DEGL)

DE is a simple real-coded evolutionary algorithm [16]. It works through a simple

cycle of stages, which are detailed in [16]. In this section we describe the variant of

DE termed as DEGL.

Suppose we have a DE population

rr

=

r

,......,3 , 2 , 1(

NPi =

r

we define a neighborhood of radius k (where k is a

],.....,,,[

, , 3, 2

)

, 1

G NPGGGG

XXXXP

rr

(15)

Where each of

Gi

X,

is a D dimensional parameter vector.

Now, for every vector

Gi

X,

nonzero integer from 0 to (NP-1)/2, as the neighborhood size must be smaller than the

population size, i.e. 2k + 1 ≤ NP), consisting of vectors

G

,

ki

X

−

r

,. . ,

Gi

X,

r

,…,

Gki

X

,

+

r

.

We assume the vectors to be organized on a ring topology with respect to their

r

and

G

X

, 2

r

.For each member of the population, a local donor vector is created by

indices, such that vectors

GNP

X

,

r

are the two immediate neighbors of vector

G

X, 1

employing the best (fittest) vector in the neighborhood of that member and any two

other vectors chosen from the same neighborhood. The model may be expressed as

rrr

+=

α

).().(

,,,,_,,

GqGpGiGbestnGiGi

XXXXXL

i

rrr

−+−

β

(16)

where the subscript n_besti indicates the best vector in the neighborhood of

k] + i k, - [i q p, ∈

with

iqp

≠≠

. Similarly, the global donor vector is created as

rr

r

+=

α

Gi

X , and

).() .(

, 2

r

, 1

r

,,_,,

GGGiGgbest nGiGi

XXXXXg

rrr

−+−

β

(17)

where the subscript g_best indicates the best vector in the entire population at

generation G and

], 1 [2, 1

NPrr

∈

with

factors. Note that in (16) and (17), the first perturbation term on the right-hand side

(the one multiplied by α ) is an arithmetical recombination operation, while the

second term (the one multiplied by β ) is the differential mutation. Thus in both the

irr

≠≠ 21

.α and β are the scaling

Page 5

global and local mutation models, we basically generate mutated recombinants, not

pure mutants.

Now we combine the local and global donor vectors using a scalar weight

) 1 ,0(

∈ω

to form the actual donor vector of the proposed algorithm

r

ω

+=

GiGiGi

LgV

,,,

)..1 (.

rr

ω

−

(18)

Clearly, if

in (18) reduces to that of DE/target to-best/1. Hence the latter may be considered as a

special case of this more general strategy involving both global and local

neighborhood of each vector synergistically. From now on, we shall refer to this

version as DEGL (DE with global and local neighborhoods). The rest of the algorithm

is exactly similar to DE/rand/1/bin. DEGL uses a binomial crossover scheme.

3.1.

Control Parameters in DEGL

DEGL introduces four new parameters. They are:α ,β ,ω and the neighborhood

radius k. In order to lessen the number of parameters further, we take

The most important parameter in DEGL is perhaps the weight factorω , which

controls the balance between the exploration and exploitation capabilities. Small

values of ω (close to 0) in (11) favor the local neighborhood component, thereby

resulting in better exploration. There are three different schemes for the selection and

adaptation of ω to gain intuition regarding DEGL performance. They are Increasing

weight Factor, Random Weight Factor, Self-Adaptive Weight Factor respectively. But

we have used only Random Weight Factor for this design problem. So we will

describe only the incorporated method in the following paragraphs.

3.2. Random Weight Factor:

In this scheme the weight factor of each vector is made to vary as a uniformly

1

=ω

and in addition

F

== βα

, the donor vector generation scheme

F

== βα

.

distributed random number in (0, 1) i.e.

) 1 , 0 (

,

rand

Gi

≈

ω

. Such a choice may

decrease the convergence speed (by introducing more diversity). But the minimum

value is 0.15.

3.3. Advantage of Random Weight Factor:

This scheme had empirically proved to be the best scheme among all three schemes

defined in original DEGL article for this kind of design problem. The most important

advantage in this scheme lies on the process of crossover. Due to varying weight

factor the no of possible different vector increases. So the searching is much wider

than using other two schemes.

Page 6

4. Design Problems

4.1.

Parameter initializations:

For the design purpose we set the searching upper bound = 0.5 and searching lower

bound = -0.5; Function bound Constraint for DEGL is set to be 0. The initial

population size is 100. The no of generations for DEGL is set equal to 500. Next we

had to set the values of the constant terms i.e.

51 αα −

in Eqn. 8. For all the examples,

the relative weights of fitness function are determined based on trial and error method

using concepts of QMF filter. The values of the constants are as follows.

, 95.

α

, 07.

α

, 07.

α

, 10

4

=

α

5

α

4.2.

Problem Examples:

4.2.1.

Two-channel QMF bank for N = 22,

1=

2=

3=

4

−

. 10

1

−

=

πω

4 . 0

=

p

,

πω

6 . 0

=

s

, with 11

filter coefficient,

100

hh →

ω

p

.

For filter length N = 22,

normalized amplitude response for H0, H1 filters of analysis bank and amplitude of

distortion function

)(ω

T′

are plotted in Figs. 2(a) and 2(b), respectively. From Fig.

2(c), this represents attenuation characteristic of low-pass filter H0. Fig. 2(d)

represents the reconstruction error of QMF bank. Table 1 provides the filter

characteristics.

(a)

(c)

Fig. 2. The frequency response of Example 2. (a) Normalized amplitude response of analysis

bank. (b) Amplitude of distortion function. (c) Low-pass filter attenuation characteristics in dB.

(d) Reconstruction error in dB, for N =22

π

4 . 0

=

, edge frequency of stop-band

πω

6 . 0

=

s

. The

(b)

(d)

Page 7

=

0 h

-0.00161

=

1 h

-0.00475

=

2 h

0.01330

=

3 h

0.00104

=

4 h

-0.02797

=

5 h

h

0.00940

=

6 h

0.05150

=

7 h

-0.03441

=

8h

-0.10013

=

9 h

0.12490

=

10

0.46861

4.2.2.

Two-channel QMF bank for N = 32,

filter coefficient,

0

h →

πω

4 . 0

=

p

,

πω

6 . 0

=

s

, with 16

16

h

.

For filter length N = 32,

setting the initial values and parameters, the DEGL algorithm is run to obtain the

optimal filter coefficients. The normalized amplitude response for H0, H1 filters of

analysis bank and amplitude of distortion function

and 4(b), respectively. From Fig. 4(c), this represents attenuation characteristic of

low-pass filter H0. Fig. 4(d) represents the reconstruction error of QMF bank. Table 2

presents a comparison of proposed DEGL method with design based on CLPSO and

DE algorithms. Table 1 provides the filter characteristics.

(a)

(c)

Fig. 4. The frequency response of Example 3. (a) Normalized amplitude response of analysis

bank. (b) Amplitude of distortion function. (c) Low-pass filter attenuation characteristics in dB.

(d) Reconstruction error in dB, for N =32

πω

4 . 0

=

p

, edge frequency of stop-band

πω

6 . 0

=

s

. After

)(ω

T′

are plotted in Figs. 4(a)

(b)

(d)

Page 8

The optimized 16-filter coefficients for the analysis bank low-pass filter are given

below.

=

0.0012

=

-0.0024

=

-0.0016

0 h

1h

2 h

=

3 h

0.0057

=

4 h

0.0013

=

5 h

h

-0.0115

=

6 h

0.0008

=

7 h

h

0.0201

=

8h

h

-0.0058

=

9 h

h

-0.0330

=

10

0.0167

=

11

h

0.0539

=

12

-0.0417

=

13

-0.0997

=

14

0.1306

=

15

h

0.4651

Tab1e 1. Filter’s performance measuring quantities for N=22

SBEA (stop-band edge attenuation )

SBLFA ( stop-band first lobe attenuation )

MSEP( mean square error in pass band )

MSES ( mean square error in stop band )

N=22

20.8669 dB

31.1792 dB

× 8.95

× 1.23

0.0186

N=32

34.7309 dB

44.6815 dB

10× 1.58

× 3.15

0.0083

07

10

10

=

07

=

40

=

60

10

=

mor (measure of ripple)

Table 2. Performance comparison of proposed DEGL method with other algorithms

Filter constants and parameters Name of the algorithm

mor MSEP

CL- PSO[19] 0.0256

10× 1.84

DE[16] 0.0123

10× 9.72

DEGL 0.0083

10×1.58

MSES SBEA

23.6382

29.6892

SBLFA

31.9996

39.8694

04

=

03

10

10×

10×

×1.12

6.96

3.15

=

06

=

04

=

07

=

60

=

34.7309 44.6815

5. Discussion of Results

We have described here two design problems with no of filter coefficients equal to 11

and 16. One comparison table is also given for the 2nd design problem. In Table 2 the

results for N=32 are compared with the results of same design problem using CL-PSO

[19] and DE [16] algorithms. The result shows that DEGL based design method is

much superior to those two methods. The results clearly show that this method leads

to filter banks with improved performance in terms of peak reconstruction error, mean

square error in stop band, pass band. The measure of ripple, which is an important

parameter in signal processing, is also considerably lower than DE and CLPSO based

methods for all the problems. Almost all the problems satisfy Eqn. 13 which is the

condition for perfect reconstruction filter which is one of the achievements of this

algorithm. The corresponding values of MSEP and MSES are also much lower in this

method than the other two methods as shown in table 4. Also DEGL based method is

much better in terms SBEA and SBLFA.

6. Conclusions

In this paper, a DEGL Algorithm based technique is used for the design of QMF bank

Simulation The result shows that design of filter using DEGL is very effective and

efficient for QMF filter design for any number of filter coefficients. We could also

Page 9

use other improved DE algorithms and many other evolutionary algorithms for design

purpose. We can also use multi objective algorithms. Our further works will be

focused on the improvement of the result obtained using some new design scheme

and further optimization techniques.

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Technical Report TR-95-012, ICSI,