Design of quadrature mirror filter bank using Lagrange multiplier method based on fractional derivative constraints

Engineering Science and Technology, an International Journal 18 (2015) 235e243

H O S T E D BY

Contents lists available at ScienceDirect

Engineering Science and Technology, an International Journal journal homepage: http://www.elsevier.com/locate/jestch

Full length article

Design of quadrature mirror filter bank using Lagrange multiplier method based on fractional derivative constraints B. Kuldeep a, A. Kumar a, *, G.K. Singh b a b

Indian Institute of Information Technology Design and Manufacturing, Jabalpur 482011, MP, India Department of Electrical Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, Uttrakhand, India

a r t i c l e i n f o

a b s t r a c t

Article history: Received 21 August 2014 Received in revised form 8 December 2014 Accepted 9 December 2014 Available online 31 January 2015

Fractional calculus has recently been identified as a very important mathematical tool in the field of signal processing. Digital filters designed by fractional derivatives give more accurate frequency response in the prescribed frequency region. Digital filters are most important part of multi-rate filter bank systems. In this paper, an improved method based on fractional derivative constraints is presented for the design of two-channel quadrature mirror filter (QMF) bank. The design problem is formulated as minimization of L2 error of filter bank transfer function in passband, stopband interval and at quadrature frequency, and then Lagrange multiplier method with fractional derivative constraints is applied to solve it. The proposed method is then successfully applied for the design of two-channel QMF bank with higher order filter taps. Performance of the QMF bank design is then examined through study of various parameters such as passband error, stopband error, transition band error, peak reconstruction error (PRE), stopband attenuation (As). It is found that, the good design can be obtained with the change of number and value of fractional derivative constraint coefficients. Copyright © 2015, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/bync-nd/4.0/).

Keywords: Quadrature mirror filter (QMF) Fractional derivative Prototype filter Peak reconstruction error (PRE) Multi-rate

1. Introduction Multi-rate systems, especially multi-rate filter banks are widely used in several applications of signal processing such as: in speech processing [1], image processing and telemedicine [2] etc. Basically, multi-rate filter bank structure is divided into two parts: first is analysis section and second is synthesis section. In analysis part, signal is divided into different sub-bands and some specific processes are applied on the sub-bands. In synthesis section, processed sub-bands are combined and signal is reconstructed. Among all the multi-rate filter banks, two-channel filter bank was first used for sub-band coding of speech signal. Two-channel filter bank is fundamental part of multi-rate filter bank and also called as quadrature mirror filter (QMF) bank. Due to immense application of QMF bank in signal processing, various optimization methods have been developed for the design of optimized QMF bank [3e12]. These optimization methods are

* Corresponding author. E-mail addresses: [email protected] (B. Kuldeep), anilkdee@gmail. com (A. Kumar), [email protected] (G.K. Singh). Peer review under responsibility of Karabuk University.

categorized as: gradient based optimization methods and nature inspired optimization methods. Gradient based optimization methods basically depend upon the gradient of function like field function method [3], weighted least square method (WLS) [4,5], LevenbergeMarquardt algorithm (LM) [6e8], Quasi-Newton method [8] etc. Gradient based method gives stationary solution where gradient is zero. In [3], a novel approach for QMF bank has been proposed using field function method. In [3], modified field function is introduced for obtaining global minima using various extra constraints on prototype filter and filter bank. A new WLS technique for designing FIR filter, based on neural network, has been developed by researchers recently [4]. Another technique based on neural network was proposed for designing QMF bank without any convergence problem [5]. A different approach for the design of two-channel QMF bank using Marquardt optimization method is given in [6]. Design was further modified using LevenbergeMarquardt (LM) method [7]. Recently, authors in [8] have used LevenbergeMarquardt (LM) and Quasi-Newton (QN) optimization method simultaneously for the effective design of twoChannel QMF bank. Recently, nature inspired methods are widely used for obtaining global solution of linear and nonlinear problems because stochastic and meta-heuristic nature. Nature inspired methods are basically

http://dx.doi.org/10.1016/j.jestch.2014.12.005 2215-0986/Copyright © 2015, The Authors. Production and hosting by Elsevier B.V. on behalf of Karabuk University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

236

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

motivated by the natural phenomena's such as biological process: reproduction, mutation and interaction and social behavior: flock of birds, schooling of fish, and intelligence of swarm of bee. Nature inspired methods such as genetic algorithms [9], particle swarm optimization (PSO) [10,11] and differential evolution [12] are associated with global solution. In [9], a signed powers-of-two coefficient perfect reconstruction QMF bank using CORDIC genetic algorithm, which is based on coordinate rotation, is designed. In [10], a QMF bank design by PSO with nonlinear unconstraint optimization is given. QMF design was further improved by modified PSO [11]. In [11], modification is done by hybridization of PSO with Scout Bee form of artificial bee colony (ABC) algorithm. In [12], QMF design was further enhanced using adaptive differential evolution algorithm. All these approaches are suitable for low order QMF filter bank design and for higher order, efficient design of QMF bank is not possible due to high ripples that occur in passband and stopband of analysis and synthesis filters. In [13], linear finite impulse response (FIR) filter has been designed by fractional derivative at prescribed frequency point, and it gives smoother passband and stopband region than conventional least square and conventional derivative methods which show ability of fractional derivative towards more confined analysis of two-channel QMF bank and can be used to design higher order two-channel QMF bank. However, as the authors of this paper have been able to ascertain on the basis of detailed literature review, very less [14] attempt has been made for the design of two-channel QMF bank by using fractional derivative. This paper, therefore, presents the design of higher order QMF bank using fractional derivative constraints. Fractional calculus is a branch of mathematics which admits definition of fractional derivative and fractional integral. In fractional calculus, definition of integral and derivative operators are almost same as the conventional definition, only integer order of calculus is changed into fractional order. It means order becomes more generalized in fractional calculus. In term of mathematical framework, fractional derivative is a very old concept, but from application point of view, it is a very novel concept. For function f(x), nth order fractional derivative is defined as

Dnx f ðxÞ ¼

ðdn f ðxÞÞ ðdxn Þ

(1)

Fractional calculus has been identified as very important mathematical tool in many areas of science and engineering such as basic physics [16], numerical methods [17], material science [18], optical engineering [19], earthquake engineering [20], electric network [21], control system [22], fluid mechanics [23], heat mechanics [24], electromagnetic theory [25] and signal processing [26] etc. Fractional calculus has been exploited in many applications of signal processing, such as design of fractional order differentiator for image sharpening, estimation of fractional noise [26] and estimating the fractional order derivative for a given contaminant signal [27]; design of fractional order multi-scale variation modal to preserve textural information of image, to eliminate staircase effect [28] and for image de-noising [29]; design of more flexible RL and RC circuit by fractional order derivative [30]; development of very low sensitive to noise method by using fractional derivative for identification of blur parameters of the motion blurred image [31]. Fractional derivative is also used to update the coefficients according to prescribed adaptive learning algorithm [32]; to design filter for image contrast enhancement [33]; for the design of fixed fractional delay FIR filter [34] and for efficient design of one and two dimensional linear phase FIR filter [13,35]. The most famous definitions of fractional derivative that have been popularized in the world of fractional calculus are:

RiemanneLiouville, M. Caputo, and GrunwaldeLetnikov definitions. In first two definitions, fractional derivative is being evaluated with the use of integration. Because of integration, these two fractional derivatives are more computationally complex as compared to GrunwaldeLetnikov definition. Due to summation form of GrunwaldeLetnikov fractional derivative, this formula became the first choice of computer numerical solver. Because of inherent low complexity, GrunwaldeLetnikov definition of fractional derivative has been used in this paper. Mathematical definition of fractional derivative given by Grunwald-Letnikov is [13]:

Dux f ðxÞ ¼

∞ X ð1Þk Cku ðdu f ðxÞÞ ¼ lim f ðx  kDÞ ðdxu Þ Du D/0

(2)

k¼0

where,Cku is given by

8 1 k¼0 < Gðu þ 1Þ ¼ uðu  1Þðu  2Þ…ðu  k þ 1Þ Cku ¼ Gðk þ 1ÞGðu  k þ 1Þ : k1 1:2:3…k (3) and G(,) is the gamma function. In Section 2, fractional derivative defined by Eq. (2) will be used to design prototype filter for QMF bank. The paper is structured as follows: In this section, brief introduction of two-channel QMF bank and fractional derivative is given. In Section 2, design formulation of QMF is explains. In Section 3, results obtained by the proposed design method are discussed. Finally, conclusions are made in Section 4. 2. Formulation of design problem Two-channel filter bank is fundamental multirate filter bank. Recently, different design formulation and theory of two-channel filter bank were proposed by many researchers [3e12]. Twochannel filter bank divides input signal into two subbands using analysis section and after applying some specific or required operation signal reconstructed using synthesis section of two-channel filter bank. Basic block diagram of two-channel filter bank is shown in Fig. 1. Ideally, reconstructed output should be exact replica of the input signal with some delay. Since, different types of distortion occur in QMF filter bank, exact reconstruction is not possible. Basically, three types of distortion occur in QMF bank: aliasing distortion, phase and amplitude distortion or peak reconstruction error (PRE). Aliasing distortion occurs because of non-idealness (realization of ideal filter is not possible) of filter, which can be eliminated using a suitable design of the synthesis filters. Phase distortion occurs because of nonlinear phase response of the filter, which can be eliminated using linear phase FIR filter. But, peak reconstruction error can never be eliminated completely. In case of a two-channel QMF bank shown in Fig. 1, aliasing distortion is eliminated by [6e8].

H1 ðzÞ ¼ H0 ð  zÞ; G0 ðzÞ ¼ H1 ð  zÞ

and

G1 ðzÞ ¼ H0 ð  zÞ (4)

where,H0(z) and H1(z) are analysis filters, while G0(z) and G1(z) are the synthesis filters. From Eq. (4), it can be interpreted that all the filters (H1(z), G0(z) and G1(z)) of QMF bank are derived by prototype filter H0(z). Let prototype filter H0(z) is a linear phase FIR filter has frequency response

  H0 eju ¼ ejuðN1Þ=2 H0 ðuÞ

(5)

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

237

Fig. 1. Quadrature mirror filter bank.

where, H0(u) is amplitude response of the filter. From the analysis of two-channel QMF bank, overall amplitude response T(u) of twochannel QMF bank can be defined as [6e8,15].

0 Z 1@ fðaÞ ¼ p

o n jTðuÞj ¼ jH0 ðuÞj2 þ jH0 ðu  pÞj2

where, Hd(u) is defined using Eqs. (12) and (14) as

(6)

If prototype filter H0(eju) is ideal low pass filter, then its amplitude response can be defined as

H0 ðuÞ ¼ H0 ð0Þ ¼0

0  u  up us  u  p

(7)

where, H0(0) is amplitude response of the ideal filter at zero frequency; up and us are passband and stopband edge frequency. Now using Eqs. (7) and (6) can be rewritten as

jTðuÞj ¼ jH0 ð0Þj2 ¼ jH0 ð0Þj2

0  u  up us  u  p

(8)

For perfect reconstruction, Eq. (8) must satisfied and at u ¼ p/2 Eq. (8) is reduced to

jH0 ðp=2Þj ¼ 0:707H0 ð0Þ

(9)

Eq. (9) is termed as perfect reconstruction condition. Authors in [15] have used the above condition for finding optimum cut-off frequency and optimum passband edge frequency so that amplitude distortion is minimum. In this paper, perfect reconstruction condition is taken as constraint for improve reconstruction called transition band constraint. It is indicated from Eq. (9) that the design of two-channel filter bank is a prototype filter design. Let, H0(z) is a prototype low-pass FIR filter and its amplitude response is given by

H0 ðuÞ ¼

M1 X

  1 u an cos n þ 2

1 ðH0 ðuÞ  Hd ðuÞÞ duA 2

(14)

u2R

8 < H0 ð0Þ 0  u  up Hd ðuÞ ¼ 0 us  u  p : 0:707H0 ð0Þ u ¼ p=2

(15)

and, R is the interested band. Here for simplicity, R is defined as R 2 R1 ∪ R2. where, R1 2 [0,up] ∪ [us,1] and R2 2 {p/2}. Eq. (14) can be rewritten as

fðaÞ ¼

 1 T a Qa  2pT a þ a p

(16)

where, Q ¼ Q1 þ Q2. Q1, Q2, p, a given by

Z Q1 ¼

bðuÞbðuÞT du

(17)

u2R1

Q2 ¼ ðbðuÞ  0:707bð0ÞÞ2

at u ¼ p=2

(18)

Z p¼

Hd ðuÞbðuÞdu

(19)

ðHd ðuÞÞ2 du

(20)

u2R1

Z a¼ u2R1

(10)

where, M is defined as M ¼ N/2 and H0(u) can be rewritten as

To improve the magnitude response of prototype filter H0(u), and reconstruction quality of QMF bank the following constraints are applied:

H0 ðuÞ ¼ aT bðuÞ

H0 ðuÞju¼p=2 ¼ 0:707H0 ð0Þ

n¼0

where; and

iT h a ¼ a0 a1 a2 …aðN=2Þ1

bðuÞ ¼ ½cosðu=2Þ…cosðuðN  1Þ=2ÞT

(11) (12) (13)

Now the problem is, how to find the filter co-efficient vector ‘a’ so that magnitude response of the prototype filter H0(u) approximately approaches to desired magnitude response Hd(u). So, in this part of the paper, filter co-efficient vector a is determined by minimizing the following objective function

and

Du H0 ðuÞju¼u0 ¼ 0

(21) (22)

where, u0 is the prescribed frequency point, and u 2 {u1, u2, u3…uL} is fractional order of fractional derivative. Hence, the total numbers of constraints are L þ 1. Constraint given by Eq. (21) is nothing but perfect reconstruction condition given by Eq. (9) [6e8] called transition band constraints such that the magnitude response of prototype filter at u ¼ p/2(H0(p/2)) should be exactly equal to 0.707H0(0) which improves the reconstruction quality of QMF. Constraint given by Eq. (22) is called fractional derivative constraint

238

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

such that fractional derivative of H0(u) at u0 becomes zero, which increases the flatness of H0(u) at u0. The fractional derivative of H0(u) can be computed by

du

 !  PM1 1 n¼0 an cos n þ 2 u

Du H0 ðuÞ ¼

¼

M1 X n¼0

¼

M1 X n¼0

 an

where, vector c(u, u) is given by

6 6 6 6 6 6 6 6 6 cðu; uÞ ¼ 6 6 6 6 6 6 6 6 6 4

: :   pu u ðn þ 1=2Þ cos n þ 1=2 u þ 2

(26)

(27)

3 bT ðp=2Þ T 6 c ðu ; u Þ 7 0 1 7 6 6 cT ðu ; u Þ 7 6 0 2 7 7 C¼6 : 7 6 7 6 : 7 6 5 4 : cT ðu0 ; uL Þ 2

(23)

2

aT cðu0 ; uk Þ ¼ 0

Ca ¼ f

duu     1 u 1 pu uþ ¼ aT cðu; uÞ cos nþ nþ 2 2 2

  pu ð1=2Þu cos 1=2 u þ 2   pu u ð3=2Þ cos 3=2 u þ 2   pu u ð5=2Þ cos 5=2 u þ 2 :

(25)

The above constraints can be rewritten in matrix form as

duu   du cos n þ 12 u an

aT bðp=2Þ ¼ 0:707H0 ð0Þ

3 0:707 6 0 7 7 6 6 0 7 7 6 7 f ¼6 6 : 7 6 : 7 7 6 4 : 5 0 2

(28)

Now, the design problem is reduced to

3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

Minimize fðaÞ ¼

 1 T a Qa  2pT a þ a ; p

subject to Ca ¼ f (29)

(24)

For solving above optimization problem, Lagrange multiplier method is exploited. The optimal solution of this problem is given by

 1 h i CQ 1 p  f ao ¼ Q 1 p  Q 1 C T CQ 1 C T

(30)

This closed form optimized solution ao is used to find optimum prototype filter coefficients h0(n). Eq. (4) shows that all the filters of

Fig. 2. Two-channel QMF bank designed by the proposed method with N ¼ 32, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2, u3 ¼ 3, u4 ¼ 4, u5 ¼ 5, u6 ¼ 6, u7 ¼ 28.8. a Amplitude response of the prototype filter in dB; b amplitude responses of the analysis filters in dB; c reconstruction error in dB.

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

QMF bank are simply derived by the prototype low pass filter H0(z), which has filter coefficients h0(n).

In this section, the proposed method is used for the design of prototype filter for a two-channel QMF bank. Various design examples are appended to show the usefulness and superiority of this method. The effectiveness of this method is evaluated in terms of following powerful parameters: Peak reconstruction error (PRE) in dB:

2 o n   2       PRE ¼ max 10 log H0 eju  þ H0 ejðupÞ 

Zup

at u ¼ us

ðH0 ðuÞ  1Þ2 du

0

(34)

us

ft ¼ ðH0 ðuÞ  0:707H0 ð0ÞÞ2

at u ¼ p=2

(35)

In Section 3.1, performance of the proposed method is presented with distinct design examples. After that, in Section 3.2, comparison of the proposed method with known methods is presented. 3.1. Design example

(32)

Example I: In this example, a two-channel QMF bank is designed with N ¼ 32, up ¼ 0.4p and us ¼ 0.6p. The values of fractional derivative constraints are taken asu1 ¼ 1, u2 ¼ 2.2, u3 ¼ 3, u4 ¼ 4, u5 ¼ 5, u6 ¼ 6, u7 ¼ 28.8, and initial value of filter coefficient is taken ash0(n) ¼ [0, 0, 0, 0……0.707]. The resulting performance parameters obtained are:

(33)

fp ¼ 2:4043  1007 ; fs ¼ 1:0517  1005 ; ft

Passband Error:

1 p

ðH0 ðuÞ  0Þ2 du

(31)

Stopband attenuation:

fp ¼

Zp

Transition band Error [6e8]:

3. Result and discussion

As ¼ 20 log10 jH0 ðuÞj;

1 fs ¼ p

239

¼ 2:0831  1030 ; PRE ¼ 0:0099 dB; As ¼ 30:6055 dB

Stopband Error:

Fig. 3. Two-channel QMF bank designed by the proposed method with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2, u3 ¼ 23.7. a Amplitude response of the prototype filter in dB; b amplitude responses of the analysis filters in dB; c reconstruction error in dB.

240

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

Table 1 Performance parameters in different design examples using the proposed method. Filter taps

u3

ɸp

28 32 40 48 56 64 72 76 82 90 94 100

31.6 24.7 27.1 39.6 44.0 8.30 23.6 9.80 4.80 45.3 12.7 20.5

1.84 5.23 6.10 2.50 7.87 1.21 2.35 4.13 1.59 6.79 8.08 7.69

ɸs            

1007 1008 1009 1009 1011 1010 1011 1011 1011 1012 1012 1012

1.10 2.99 2.32 1.79 1.39 1.10 7.55 2.01 1.59 7.59 7.85 5.57

ɸt            

1005 1006 1007 1008 1009 1010 1012 1012 1013 1013 1015 1014

1.23 1.23 1.23 3.58 1.77 4.45 1.08 1.54 2.01 5.07 6.58 1.90

           

1032 1032 1032 1027 1029 1030 1021 1020 1020 1017 1018 1014

PRE

As

0.0193 0.0141 0.0080 0.0058 0.0051 0.0044 0.0035 0.0020 0.0020 0.0093 0.0093 0.0215

30.45 35.48 45.53 55.65 66.32 77.20 88.56 96.10 109.33 115.33 125.25 108.21

The amplitude response of the prototype filter is shown in Fig. 2(a), while that of the analysis filter is depicted in Fig. 2(b), and the variation in reconstruction error is shown in Fig. 2(c) respectively. Example II: In this example, a two-channel QMF bank is designed with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p. The values of fractional derivative constraints are taken as u1 ¼ 1, u2 ¼ 2.2, u3 ¼ 44. The resulting performance parameters obtained are:

fp ¼ 2:7608  1011 ; fs ¼ 1:3862  1009 ; ft ¼ 1:7799  1029 ; PRE ¼ 0:0051 dB; As ¼ 66:3169 dB Results obtained are depicted in Fig. 3. Several other examples are also designed using the same method with N ¼ 32, up ¼ 0.4p,

us ¼ 0.6p, u1 ¼ 1, u2 ¼ 2.2, and u3 is taken as a variable and obtained results are summarized in Table 1. From simulation results of the designed examples, it can be observed that the proposed method is compatible with higher order filter taps (up to N ¼ 100) QMF bank design. Results clearly show that method gives better performance in terms of PRE, ɸp, ɸs and ɸt. ɸt is appreciably reduced as compared to passband and stopband error. Minimum PRE obtained among different designs is 0.0020 dB with N ¼ 82. Hence, the proposed QMF bank can be used in various applications, where higher order filter bank effectively works such as in echo cancellation, cross talk suppression, and ECG signal processing. Figs. 4e8 presents the variation of PRE, ɸp, ɸs, ɸt and As respectively with respect tou1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable with N ¼ 5 6, up ¼ 0.4p, us ¼ 0.6p. From figures, it is demonstrated that all the performance parameter vary and several maxima and minima occur with the variation of fractional derivative constraint u3. So by varying the value of fractional derivatives constraint, desired value of PRE, ɸp, ɸs, ɸt and As can be achieved. Results in Table 2 show that all the performance parametersPRE, ɸp, ɸs, ɸt and As can be effectively varied with the variation of number of fractional derivatives constraints (Fromu1……u9). As seen in Section 3.1, example II, PRE is 0.0051 for N ¼ 56 andu1 ¼ 1, u2 ¼ 2.2, u3 ¼ 44, but with fractional derivatives constraints, u1 ¼ 1, u2 ¼ 2.2, u3 ¼ 9.1, u4 ¼ 4, u5 ¼ 5, u6 ¼ 6 and u7 ¼ 7, PRE becomes 0.0036. Hence, it can be concluded that, through alteration of value and number of fractional derivatives constraints, PRE can be reduced effectively. The CPU configuration used is Intel (R) Core (TM) i3-2350M @ 2.3 GHz, 2 GB of RAM. Minimum CPU time 0.0156 s (for number of fractional derivative constraints ¼ 1 and N ¼ 32) and maximum CPU time 0.1024 s (for number of fractional derivative constraints ¼ 10 and N ¼ 120).

Fig. 4. Variation of peak reconstruction error (PRE) with respect to fractional derivative constraint u3 with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable

Fig. 5. Variation of passband error (ɸp) with respect to fractional derivative constraint u3 with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

3.2. Comparison with other methods In Table 3, comparative study of the proposed method with other existing methods [6e8,10e12,15] is carried out. For this, twochannel QMF bank is designed with the same specifications (N ¼ 32, up ¼ 0.4p and us ¼ 0.6p) as used in existing methods. It was found that the performance of proposed method is significantly improved as compared to earlier known techniques in terms of PRE, passband error, stopband error and transition band error. As can be seen, the maximum order of transition band error, passband and stopband error are 1032, 1008 and 1006 respectively. Thus the results show that large amount of reduction occurs in transition

241

Table 2 Performance parameters variation with respect to number of fractional derivatives constraints. U

ɸp

u1 ¼ 1 u1e2 ¼ u1e3 ¼ u1e4 ¼ u1e5 ¼ u1e6 ¼ u1e7 ¼ u1e8 ¼ u1e9 ¼

1.39 1.50 1.31 2.44 2.71 8.13 9.44 4.63 5.14

1e2; 1e3; 1e4; 1e5; 1e6; 1e7; 1e8; 1e9

ɸs         

1010 1010 1010 1009 1009 1009 1009 1008 1008

1.09 1.22 1.25 8.40 9.65 1.66 1.63 6.50 2.06

ɸt         

1009 1009 1009 1010 1010 1009 1009 1009 1008

1.69 1.78 1.69 1.69 1.78 1.51 1.51 1.23 1.69

        

1029 1029 1029 1029 1029 1029 1029 1032 1029

PRE

As

0.0147 0.0088 0.0080 0.0328 0.0267 0.0126 0.0063 0.0374 0.0098

67.16 66.87 66.79 67.44 66.94 67.94 69.14 59.46 55.62

Fig. 6. Variation of stopband error (ɸs) with respect to fractional derivative constraint u3 with N ¼ 56, up ¼ 0.4p and us¼0.6p and u1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable

Fig. 7. Variation of transition band error (ɸt) with respect to fractional derivative constraint u3 with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable

Fig. 8. Variation of stopband attenuation (As) with respect to fractional derivative constraint u3 with N ¼ 56, up ¼ 0.4p and us ¼ 0.6p and u1 ¼ 1, u2 ¼ 2.2 and u3 ¼ variable

242

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

Table 3 Comparison with other method. Type of algorithm

ɸp

Algorithm [6]. Algorithm [7]. Algorithm [8]. Algorithm [10]. Algorithm [11]. Algorithm [12]. Algorithm [15]. Proposed(u3 ¼ 24.7) Proposed(u7 ¼ 28.8)

1.45  7.42  3.29  2.35  5.27  2.48  2.29  5.23 £ 2.40 £

1008 1009 1008 1008 1008 1008 1008 1008 1007

ɸs

ɸt

2.76  1006 1.27  1006 5.30  1005 5.79  1008 1.00  1005 3.37  1006 1.27  1006 2.99 £ 1006 1.05 £ 10e05

9.01  6.63  2.02  e 3.61  e 3.19  1.23 £ 2.08 £

1006 1009 1008 1011 1011 1032 1030

PRE

As

0.0270 0.0102 0.0089 0.0148 0.0126 0.0085 0.0114 0.0141 0.0099

35.20 36.59 34.38 36.87 34.14 36.91 35.60 35.48 30.61

Bold represent the results obtained by proposed method.

Fig. 9. Two-channel QMF bank designed by the proposed method with N ¼ 36, up ¼ 0.4p and us ¼ 0.6pand {u1, u2, u3 and u4} ¼ {13.50, 7.10, 4.44 and 4.19}. a MAD in dB; b MPR in dB; c MSR in dB.

B. Kuldeep et al. / Engineering Science and Technology, an International Journal 18 (2015) 235e243

band error as compared to passband and stopband error. The average percentage differences (PD) for the proposed method and published results [6,7,10,11,15] in term of peak reconstruction error is 28.57%, 2.9%, 33.11%, 21.42% and 13.16%. Percentage difference is calculated by using Eq. (36). It can be seen from Table 3, that transition band error is approximately 1020-1024 times better than the other methods. Stopband error has improved as compared to algorithms [8,11,12]. Compared with the rest of results, passband and stopband errors order are comparable to each other.

PDError ¼

maxjErrorjother  maxjErrorjproposed maxjErrorjother

 100%

(36)

In [3], for N ¼ 36, up ¼ 0.4p and us ¼ 0.6p achieve maximum amplitude distortion (MAD), maximum passband ripple (MPR) and maximum stopband ripple (MSR) are 58.1557 dB, 64.2416 dB and 50.3625 dB respectively for fractional order {u1, u2, u3 and u4} ¼ {13.50, 7.10, 4.44 and 4.19}. Fig. 9(a)e(c) shows, MAD, MPR, and MSR derived by proposed method with same specification given in [3]. It is clear from Fig. 9, MAD, MPR and MSR are 61.84 dB, 65.00 dB and 49.09 dB respectively, which prove that proposed method gives better performance in term of MAD and MPR from method in [3]. 4. Conclusion In this paper, design of a two-channel quadrature mirror filter (QMF) bank through fractional derivative constraints is presented. From the experimental results, it is evident that the proposed design method gives very small reconstruction error as compared to other existing methods. In terms of transition band error, large reduction in error has been observed. Passband and stopband error results are very close to that of other previous work. Proposed method is also gives better performance from [3] in term of MAD and MPR. Variation in all the performance parameters (peak reconstruction error, passband error, stopband error, transition band error, stopband attenuation) with the change in number and value of fractional derivative constraint coefficients makes this design very flexible as compared to other existing methods. QMF bank design with higher filter taps (up to N ¼ 100) shows the superiority of this method. The QMF bank designed by this method can be used in various applications where higher order QMF effectively works such as in echo cancellation, cross talk suppression, and ECG signal processing. References [1] S. Alaya, N. Zoghlami, Z. Lachiri, Speech enhancement based on perceptual filter bank improvement, Int. J. Speech Tech. 17 (3) (2014) 253e258. [2] T.K. Le, Automated method for scoring breast tissue microarray spots using quadrature mirror filters and support vector machines, in: 15th International Conference on Information Fusion, IEEE, July 9e12, 2012, pp. 1868e1875. [3] Y.F. Ho, B.W.K. Ling, L. Benmesbah, T.C.W. Kok, W.C. Siu, K.L. Teo, Two-channel linear phase FIR QMF bank minimax design via global nonconvex optimization programming, IEEE Trans. Signal Process. 58 (8) (2010) 4436e4441. [4] Y.D. Jou, F.K. Chen, WLS design of FIR Nyquist filter based on neural networks, Digital Signal Process. 21 (1) (2011) 17e24. [5] Y.D. Jou, Design of two-channel linear-phase quadrature mirror filter banks based on neural networks, Signal Process. 87 (5) (2007) 1031e1044. [6] O.P. Sahu, M.K. Soni, I.M. Talwar, Marquardt optimization method to design two-channel quadrature mirror filter banks, Digital Signal Process. 16 (6) (2006) 870e879. [7] A. Kumar, G.K. Singh, R.S. Anand, An improved method for the design of quadrature mirror filter banks using the Levenberg-Marquardt optimization, Signal Image Video Process. 7 (2) (2011) 209e220.

243

[8] A. Kumar, S.M. Rafi, G.K. Singh, A hybrid method for designing linear-phase quadrature mirror filter bank, Digital Signal Process. 22 (3) (2012) 453e462. [9] S.Y. Park, N.I. Cho, Design of signed powers of two coefficient perfect reconstruction QMF bank using cordic algorithms, IEEE Trans. Circuits Syst. 53 (6) (2006) 1254e1265. [10] J. Upendar, C.P. Gupta, G.K. Singh, Design of two-channel quadrature mirror filter bank using particle swarm optimization, Digital Signal Process. 2 (20) (2010) 304e313. [11] S.M. Rafi, A. Kumar, G.K. Singh, An improved particle swarm optimization method for multirate filter bank design, J. Franklin Inst. 350 (4) (2013) 757e769. [12] P. Ghosh, S. Das, H. Zafar, Adaptive-differential-evolution-based design of two-channel quadrature mirror filter Banks for sub-band coding and data transmission, IEEE Trans. Syst Man Cyber. 42 (6) (2012) 1613e1623. [13] C.C. Tseng, S.L. Lee, Design of linear phase FIR filters using fractional derivative constraints, Signal Process. 92 (5) (2012) 1317e1327. [14] B. Kuldeep, V.K. Singh, A. Kumar, G.K. Singh, Design of two-channel filter bank using nature inspired optimization based fractional derivative constraints, ISA Trans. (2014). http://dx.doi.org/10.1016/j.isatra.2014.06.005i. [15] A. Kumar, G.K. Singh, R.S. Anand, An improved method for designing quadrature mirror filter banks via unconstrained optimization, J. Math. Modell. Algorithms 9 (1) (2010) 99e111. [16] L. Longjin, F.Y. Ren, W.Y. Qiu, The application of fractional derivatives in stochastic models driven by fractional Brownian motion, Phys. A Stat. Mech. Appl. 389 (21) (2010) 4809e4818. [17] Q. Yang, F. Liu, I. Turner, Numerical methods for fractional partial differential equations with Riesz space fractional derivatives, Applied Math. Modell. 34 (1) (2010) 200e218. [18] A.A. Jarbouh, Modelisation of the rheological behavior of viscoelastic materials using the fractional derivatives and transfer technique, Energy Procedia 19 (2012) 212e225. s, Design of an ultra-broadband all-optical fractional [19] C.C. Laborde, M.V. Andre differentiator with a long-period fiber grating, Opt. Quant. Electron. 42 (9e10) (2011) 571e576. [20] P.D. Spanos, G.I. Evangelatos, Response of a non-linear system with restoring forces governed by fractional derivativesdTime domain simulation and statistical linearization solution, Soil Dyn. Earthquake Engg. 30 (9) (2010) 811e821. [21] A.G. Radwan, M.E. Fouda, Optimization of fractional-order RLC filters, Circuits Syst. Signal Process. (2013). [22] F. Padula, A. Visioli, Optimal tuning rules for proportional-integral derivative and fractional order proportional-integral derivative controllers for integral and unstable processes, IET Control Theory Appl. 6 (6) (2012) 776e786. [23] A. Heibig, L.I. Palade, Well posedness of a linearized fractional derivative fluid model, J. Math. Analysis Appl. 380 (1) (2011) 188e203. [24] T. Blaszczyk, J. Leszczynski, E. Szymanek, Numerical solution of composite left and right fractional caputo derivative models for granular heat flow, Mech. Research Commun. 48 (2013) 42e45. [25] I.T. Rekanos, Semi-local approximation of fractional derivatives in FDTD modeling of ColeeCole Media:A questionable approach, IEEE Antennas Wireless Propag. Lett. 11 (2012) 740e743. [26] C.C. Tseng, S.L. Lee, Design of fractional order digital differentiator using radial basis function, IEEE Trans. Circuits Syst. 57 (7) (2010) 1708e1718. [27] D. Chen, Y.Q. Chen, D. Xue, Robust fractional order differentiator, in: 24th Chinese Conference on Control and Decision, IEEE, May 23e25, 2012, pp. 1037e1042. [28] J. Zhang, Z. Wei, L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vis. 43 (1) (2012) 39e49. [29] X. Hu, Y. Li, A new variational model for image de-noising based on fractionalorder derivative, in: International Conference on Systems and Informatics, IEEE, May 19e20, 2012, pp. 1820e1824. [30] A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits, Circuits Syst. Signal Process. 31 (6) (2012) 1901e1915. [31] Y. Ye, X. Pan, J. Wang, Identification of blur parameters of motion blurred image using fractional-order derivative, in: 11th International Conference on Information Science, Signal Processing and Their Applications, IEEE, July 2e5, 2012, pp. 539e544. [32] S.G. Osgouei, M. Geravanchizadeh, Speech enhancement using convex combination of fractional least-mean-squares algorithm, in: 5th International Symposium on Telecommunications, IEEE, Dec. 4e6, 2010, pp. 869e872. [33] S. Khanna, V. Chandrasekaran, Fractional derivative filter for image contrast enhancement with order prediction, in: International Conference on Image Processing, IET, July 3-4, 2012, pp. 1e6. [34] C.C. Tseng, S.L. Lee, Designs of fixed-fractional-delay filters using fractionalderivative constraints, IEEE Trans. Circuits Syst. 59 (10) (2010) 683e687. [35] C.C. Tseng, S.L. Lee, Designs of two-dimensional linear phase FIR filters using fractional derivative constraints, Signal Process. 93 (5) (2013) 1141e1151.