Design of Two-Channel Quadrature Mirror Filter Banks Using Differential Evolution with Global and Local Neighborhoods Pradipta Ghosh, Hamim Zafar, Joydeep Banerjee, and Swagatam Das Electronics & Tele-Comm. Engineering Jadavpur University Kolkata, India {iampradiptaghosh,hmm.zafar,iamjoydeepbanerjee}@gmail.com, [email protected]

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], B.K. Panigrahi et al. (Eds.): SEMCCO 2011, Part I, LNCS 7076, pp. 1–10, 2011. © Springer-Verlag Berlin Heidelberg 2011

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weighted least square (WLS) technique [8-9] have been applied to 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 Y ( z) = 1/ 2[ H 0 ( z)G0 ( z) + H1 ( z )G1 ( z)]X ( z) + 1/ 2[H 0 (− z)G0 ( z) + H1 (− z)G1 ( z)]X (− z)

= T ( z ) X ( z ) + A( z ) X ( − z )

where Y(z) is the reconstructed signal and X(z) is the original signal.

Fig. 1. Two channel QMF BANK

(1)

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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

H 1 ( z ) = H 0 (− z ), G0 ( z ) = H 1 (− z ) and G1 ( z ) = − H 0 ( − z )

(2)

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

Y ( z ) = 1/ 2[ H 0 ( z ) H 0 ( z ) + H 0 (− z) H 0 (− z)]X ( z )

(3)

It implies that the overall design problem of filter bank reduces to determination of the filter taps coefficients of a low pass filter

H 0 (z) , called a prototype filter. Let H 0 (z) be a linear phase finite impulse response (FIR) filter with even length ( N ): H 0 (e jω ) = H 0 (e jω ) e − jω ( N −1) / 2

(4)

If all above mentioned conditions are put together, the overall transfer function of − jω ( N −1) / 2 2 2 (5) QMF bank is reduced to Eqn.(5) T (e jω ) = e H (e jω ) + H (e j (ω −π ) 2

T ( e jω ) =

{

0

0

}

1 − jω ( N −1) / 2 e T ′(e jω ) 2

(6)

Where, T ′(e jω ) = [ H 0 (ω ) 2 − (−1)( N −1) H 0 (π − ω ) 2 ] and N is the no of filters. If

2

2

H 0 (e jω ) + H 0 (e j (ω −π ) = 1 it results in perfect reconstruction; the output signal

is exact replica of the input signal. If it is evaluated at frequency ( ω = 0.5π ), the perfect reconstruction condition reduces to Eqn. (7). H 0 (e j 0.5π ) = 0.707

(7) − jω ( N −1) / 2

It shows that QMF bank has a linear phase delay due to the term e . Then the condition for the perfect reconstruction is to minimize the weighted sum of four terms as shown below:

K = α1 E p + α 2 E s + α 3 Et + α 4 .mor + α 5 E h

(8)

where α 1 − α 5 are the relative weights and E p , E s , E t , mor (measure of ripple) ,

Eh are defined as follows: E p is the mean square error in pass band (MSEP) which describes the energy of reconstruction error between 0 and Ep =

1

π

ω

H

2

0

(0) − H 0 (ω ) dω

ωp , (9)

0

E s is the mean square error in stop band (MSES) which denotes the stop band energy related to LPF between

ω s to π

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Es =

1

π

π

H

(10)

(ω ) dω 2

0

ωs

E t is the square error of overall transfer function at quadrature frequency π / 2 π 1 Et = [ H 0 ( ) − H 0 (0)]2 2 2

(11)

mor = max 10 log 10 T ′(ω ) − min 10 log 10 T ′(ω )

(12)

Measure of ripple ( mor ) ω

ω

Eh is the deviation of T ′(e jω ) from unity at ω = π / 2 π

(13)

E h = T ′( ) − 1 2

The above fitness is considered because we know that condition for perfect reconstruction filter is 2

2

H 0 (e jω ) + H 0 (e j (ω −π ) = 1

(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

r r r r PG = [ X 1,G , X 2,G , X 3,G ,....., X NP ,G ] r

Where each X i ,G

(15)

(i = 1,2,3,......, NP ) a D dimensional parameter vector. Now, for

r every vector X i ,G we define a neighborhood of radius k (where k is a nonzero integer from 0 to (NP-1)/2, as the neighborhood size must be smaller than the population size, r r r i.e. 2k + 1 ≤ NP), consisting of vectors X i − k ,G ,. . , X i ,G ,…, X i + k ,G . We assume the vectors to be organized on a ring topology with respect to their indices, such that vecr r r tors X NP ,G and X 2 ,G are the two immediate neighbors of vector X 1,G .For each member of the population, a local donor vector is created by 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

r r r r r r Li ,G = X i ,G + α .( X n _ besti ,G − X i ,G ) + β .( X p ,G − X q ,G ) (16)

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where the subscript n_besti indicates the best vector in the neighborhood of X i ,G and p, q ∈[i - k, i + k] with p ≠ q ≠ i . Similarly, the global donor vector is created as

r r r r r r g i ,G = X i ,G + α .( X n _ gbest ,G − X i ,G ) + β .( X r1,G − X r 2,G )

(17)

where the subscript g_best indicates the best vector in the entire population at generation G and r 1 , r 2 ∈ [ 1 , NP ] with r1 ≠ r 2 ≠ i . α and β are the scaling 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 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 ω ∈ ( 0 ,1 ) to form the actual donor vector of the proposed algorithm

r r r Vi ,G = ω . g i ,G + (1 − ω ).. Li ,G

(18)

Clearly, if ω = 1 and in addition α = β = F , the donor vector generation scheme 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 α = β = F . The most important parameter in DEGL is perhaps the weight factor ω , 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 distributed random number in (0, 1) i.e. ω i ,G ≈ rand (0,1) . Such a choice may decrease the convergence speed (by introducing more diversity). But the minimum value is 0.15.

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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.

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. α1 − α 5 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. α1 = .95, α 2 = .07, α 3 = .07, α 4 = 10 −4 , α 5 = 10 −1. 4.2 Problem Examples 4.2.1 Two-Channel QMF Bank for N = 22, ω p = 0.4π , ω s = 0.6π , with 11 Filter Coefficient, h0 → h10 . For filter length N = 22,

ω p = 0.4π

, edge frequency of stop-band

ω s = 0.6π . The

normalized amplitude response for H0, H1 filters of analysis bank and amplitude of distortion function T ′(ω ) 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. h0 = -0.00161 h1 = -0.00475 h2 = 0.01330 h3 = 0.00104 h4 = -0.02797

h5 = 0.00940 h6 = 0.05150 h10 = 0.46861. 4.2.2

h7 = -0.03441 h8 = -0.10013 h9 = 0.12490

Two-Channel QMF Bank for N = 32, ω p = 0.4π , ω s = 0.6π , with 16 Filter Coefficient, h0 → h16 .

For filter length N = 32, ω p = 0.4π , edge frequency of stop-band ω s = 0.6π . After 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 T ′(ω ) are plotted in Figs. 4(a)

Design of Two-Channel Quadrature Mirror Filter Banks

(a)

(b)

(c)

(d)

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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.

(a)

(b)

(c)

(d)

Fig. 3. 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.

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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. The optimized 16-filter coefficients for the analysis bank low-pass filter are given below. h0 = 0.0012; h1 = -0.0024; h2 = -0.0016; h3 = 0.0057; h4 = 0.0013; h5 = -0.0115

h6 = 0.0008; h7 = 0.0201; h8 = -0.0058; h9 = -0.0330; h10 = 0.0167; h11 = 0.0539 h12 = -0.0417; h13 = -0.0997; h14 = 0.1306 ; h15 = 0.4651 Tab1e 1. Filter’s performance measuring quantities

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 ) mor (measure of ripple)

N=22 20.8669 dB 31.1792 dB 8.95 × 10 = 07

N=32 34.7309 dB 44.6815 dB 1.58 × 10 = 07

1.23 × 10= 04 0.0186

3.15 × 10= 06 0.0083

Table 2. Performance comparison of proposed DEGL method with other algorithms Name of the algorithm CL- PSO[19] DE[16] DEGL

mor 0.0256 0.0123 0.0083

Filter constants and parameters MSEP MSES SBEA 23.6382 1.12 ×10 = 03 1.84 ×10 = 04 9.72 × 10 = 06 6.96 × 10 = 04 29.6892 = 07 1.58 × 10 3.15 × 10= 06 34.7309

SBLFA 31.9996 39.8694 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 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.

Design of Two-Channel Quadrature Mirror Filter Banks

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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 use other improved DE algorithms and many other evolutionary algorithms for design purpose. We can also use multi objective algorithms and multimodal optimization algorithms [21]. 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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