DESIGN OF ENERGY EFFICIENT FILTER BANKS

Academic Open Internet Journal

ISSN 1311-4360

www.acadjournal.com

Volume 17, 2006

DESIGN OF ENERGY EFFICIENT FILTER BANKS

Ila.Vennila¹, S.Jayaraman², R.Ravi kumar³

^{1.
LECTURER, (S.G), EEE DEPARTMENT, PSG}^COLLEGE^OF^TECHNOLOGY^,^COIMBATORE^,^INDIA^.

^{email:iven@eee.psgtech.ac.in}

^{2.
PROFESSOR and HOD, ECE DEPARTMENT, PSG}^COLLEGE^OF^TECHNOLOGY^,^COIMBATORE^,^INDIA^.

^{3. SENIOR DEVELOPMENT ENGINEER, ASHOK LEYLAND,}^CHENNAI^,^INDIA^.

ABSTRACT

Filter bank has applications in areas like audio and video coding, data communications, etc.; the improvement of Filter bank will have important impacts in these areas. Compact energy filters are efficient. So the design problem of optimum filter bank is considered as the one having minimum energy. Considering the overall performance, the perfect reconstruction condition and regularity criteria are taken into account. The optimization problem is formulated as single objective problem (i.e.) minimization of energy with constraints on regularity criteria and perfect reconstruction conditions. Genetic Algorithm is applied to yield the solution for a two channel QMF FIR filter bank. The performance of the designed filter bank is compared with that of the filter bank obtained by Parks-McClellan (Remez) algorithm.

Key words: Filter bank, energy, perfect reconstruction, optimization, Genetic Algorithm.

1. INTRODUCTION

Filter banks decompose signal spectra into a number of directly adjacent frequency bands and recombine the signal spectra by the use of low-pass, band pass and high pass filters. Decomposition is performed in analysis filter bank and reconstruction in synthesis filter bank. A subband coding (SBC) filter bank consists of an analysis filter bank followed by synthesis filter bank. Quadrature Mirror Filter banks (QMF) are two channel sub band coding (SBC) filter banks with power complementary frequency responses.

A two channel SBC filter bank is shown in figure 1. An analysis filter bank with filters H₀(z) and H₁(z) decomposes the input signal X(z) into the subband signals X₀(z) and X₁(z). This is followed by a synthesis filter bank with filters G₀(z) and G₁(z), which reconstructs the output signal from the subband signals.

Fig.1. Two-channel SBC Filter Bank

The analysis filters H₀(z),H₁(z) and the synthesis filters G₀(z),G₁(z) are designed to satisfy perfect reconstruction and linear phase conditions. A biorthogonal filter bank achieves both the requirements [1] and requires only two filters H₀(z) and H₁(z),to be designed. The synthesis filters are obtained from analysis filters by the relation,

G₀(z)=2 H₁(-z)

G₁(z)=-2H₀(-z) … (1) In addition it is required that the sum of filter lengths be a multiple of 4.

(i.e.) N₀+N₁=4m

where, N₀= length of the filter H₀(z)

N₁= length of the filter H₀(z)

m = a positive integer

2. PROBLEM FORMULATION

The Filter bank is designed to satisfy following criteria.

2.1. Criteria for the Overall Filter Bank

Let n =1, 2… be the coefficients of low-pass filter, and n=1, 2… be those of high-pass filter. Since both filters are symmetric, linear phase is guaranteed. If the delay of the filter bank is , then Perfect Reconstruction can be enforced by a set of equations [2] called the PR conditions stated in Eqn.(2).

… (2)

where, i = 1, 2, …,

, n =0

=0, otherwise.

2.2. Criteria for each Individual Filter

The performance of the designed filters with ideal filters is measured by pass band energy and stop band energy for the given pass band and stop band cut-off frequencies and respectively. When the magnitude responses of the filters and are and respectively, the energy of the filters [3] are stated in Eqn.(3)

… (3)

2.3. Regularity criteria

The regularity criteria are considered for the convergence of the filters [4]. Second order regularity is used for ensuring the low pass filter =0 at and =0 at

= 0.

… (4)

So the design problem is stated as,

Min_, ++ +

Subject to PR conditions

2-order regularity

3. IMPLEMENTATION

The optimization problem has been implemented in MATLAB using Genetic Algorithm toolbox. The real coding is used by taking the filter coefficients as the variables. This reduces the computational complexity, as there is no need for decoding. In order to reduce the search space, the coefficients of the filters designed using Remez algorithm are taken as one of the initial population’s chromosome and other chromosomes are obtained by its random perturbations. In order to preserve the symmetry, only half of the coefficients in both low pass and high pass analysis filters are taken as genes (variables) in the chromosome. As the filters considered are of odd lengths and , the chromosome length becomes . The population size is taken as 20. The number of generations is taken as 100. Stochastic universal sampling is used for selection. Discrete recombination and real mutation are used as crossover and mutation operators. Fitness based reinsertion scheme is used for reinsertion. The generation gap is taken as 0.9. This is to have elitist strategy, which retains 10% of the best individuals of the previous generation. Optimization is performed and the optimal filter coefficients satisfying the given objective are determined. With these optimal analysis low pass and high pass filter coefficients, the synthesis low pass and high pass filter coefficients are found using the relations stated in Eqn.(5)

g₀(n) = 2*(-1)ⁿ * h₁(n)

g₁(n) = -2*(-1)ⁿ * h₀(n) … (5)

Three filter banks of lengths a) (9, 7), b) (13, 7), and c) (13, 11) have been designed. In each of the filter banks, the first and second number represents the length of analysis low pass filter and the length of analysis high pass filter respectively. The cut-off frequencies for the low pass analysis filter are fixed as =0.4p and =0.6p for all the three filter banks. An 8 bit audio signal sampled at a rate of 22.05 kHz is applied as the input and reconstructed by the filter banks designed using GA and Remez algorithm. The PSNR value of the reconstructed signal in dB is calculated by the following formula:

PSNR = 10 log (255² / MSE)

where, MSE is the mean square error of the reconstructed signal.

4. RESULTS AND DISCUSSION

The performance the filter bank designed using the Genetic algorithm approach has been compared with that of the filter bank designed with the same specifications using Remez algorithm. The stop band energy, pass band energy of the analysis filters h₀(n) and h₁(n) and hence the total energy, regularity criteria, PR conditions and PSNR values are calculated for the optimized filter bank and Remez filter bank and the values for the three filter banks are tabulated in Table1-Table3. The magnitude responses of both the methods for the analysis low pass and high pass filters are given in Fig.2-Fig.4.

Table 1. Results for (9, 7) Filter Bank

S.No	Parameter	Remez algorithm	Genetic algorithm
1		0.4835	0.3775
2		0.0027	0.0001
3		0.0027	0.0004
4		0.4837	0.4184
5	Total Energy	0.9726	0.7724
6	Regularity criterion-0	0.1130	0.1540
7	Regularity criterion-1	0.1129	0.0155
8	PR conditions y (1) y (2) y (3) y (4)	0.0143 0.0749 0.0231 0.9746	0.0069 0.0411 0.0270 0.9573
9	PSNR	64.83	65.40

From Table 1, the following observations have been made:

· The total energy of the filter bank is reduced by GA.

· Better satisfaction of regularity criteria 0 and 1, by GA filter bank than by Remez filter bank.

· The values of PR conditions are better satisfied by GA filter bank than by Remez filter bank.

· The PSNR value is about 0.57 dB higher for GA filter bank.

(a) Low pass filter (b) High pass filter

Fig. 2. Magnitude Responses of Analysis filters (9,7)

From Fig.2, it is observed that the pass band of the low pass filter is widened by GA and its stop band attenuation is slightly more than that of Remez in second side lobe. In high pass filter, throughout the stop band, GA has higher attenuation than Remez, and its pass band behaviour is almost identical with Remez.

Table 2. Results for (13, 7) Filter Bank

S. No	Parameter	Remez algorithm	Genetic algorithm
1		0.4016	0.3259
2		0.0004	0.0028
3		0.0027	0.0000
4		0.4837	0.4066
5	Total Energy	0.8884	0.7353
6	Regularity criterion-0	0.0508	0.0205
7	Regularity criterion-1	0.1129	0.0132
8	PR conditions y (0) y (1) y (2) y (3) y (4)	0.0064 0.0278 0.0493 0.0254 0.9679	0.0051 0.0310 0.0179 0.0296 0.9413
9	PSNR	65.36	66.36

From Table 2, the following observations have been made:

· The total energy of the filter bank is reduced by GA.

· Better satisfaction of regularity criteria 0 and 1, by GA filter bank than by Remez filter bank.

· The values of PR conditions are better satisfied by GA filter bank than by Remez filter bank except for y(1) and y(3).

· The PSNR value is 1dB higher for GA filter bank.

(a) Low pass filter (b) High pass filter

Fig. 3. Magnitude Responses of Analysis filters (13,7)

From Fig.3, it is observed that, there is an improvement of stop band attenuation by GA

over Remez both in low pass and high pass filters.

Table 3. Results for (13, 11) Filter Bank

S. No	Parameter	Remez algorithm	Genetic algorithm
1		0.4016	0.3264
2		0.0004	0.0041
3		0.0004	0.0014
4		0.4016	0.3236
5	Total Energy	0.8040	0.6555
6	Regularity criterion-0	0.0508	0.0010
7	Regularity criterion-1	0.0502	0.0741
S. No	Parameter	Remez algorithm	Genetic algorithm
8	PR conditions y (0) y (1) y (2) y (3) y(4) y(5)	0.0029 0.0098 0.0420 0.0237 0.0309 0.9687	0.0044 0.0119 0.0588 0.0074 0.0455 0.9615
9	PSNR	65.79	66.16

From Table 3, the following observations have been made:

· The total energy of the filter bank is reduced by GA.

· Regularity criterion-0 for GA filter bank is better but the criterion-1 is better satisfied by Remez filter bank only.

· The values of PR conditions are better satisfied by Remez filter bank than by GA filter bank except for y(3) and y(5).

· The PSNR value is about 0.37 dB higher for GA filter bank.

(a) Low pass filter (b) High pass filter

Fig. 4. Magnitude Responses of Analysis filters (13,11)

From Fig.4, it is observed that in low pass filter, a very high improvement of the stop band attenuation is realized by GA over Remez Algorithm but in high pass filter, the performance of GA is poor in the stop band and the pass band behaviour is almost identical with Remez.

5. CONCLUSION

Three filter banks have been designed by GA and Remez exchange algorithms for energy minimization. In all the cases, minimum energy is obtained by GA filter bank only. For (9,7) filter bank, all the constraints are better satisfied by GA. For (13,7) and (13,11) filter banks, GA satisfies all except two PR conditions. The PSNR of the reconstructed signal is higher for all the filter banks designed by GA. Hence it is concluded that, for energy minimization problem, under the given constraints, GA filter banks perform better than Remez by giving minimum energy and higher PSNR.