Design of Digital Differentiators Using Interior Search ... - Science Direct

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ScienceDirect Procedia Computer Science 57 (2015) 368 – 376

3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015)

Design of Digital Differentiators Using Interior Search Algorithm Manjeet Kumara,‫כ‬, Tarun Kumar Rawata, Aman Jainb, Atul Anshuman Singhb, Aviral Mittalb b

a Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, Dwarka, Delhi-110078, India Undergraduate, Division of Electronics and Communication Engineering, Netaji Subhas Institute of Technology, Dwarka, Delhi-110078, India

Abstract This paper presents a novel metaheuristic algorithm called Interior Search Algorithm (ISA), which is applied for digital differentiator design problem. ISA is based on the principles of aesthetic techniques commonly used in interior design and decoration. ISA has a very quick convergence rate and only one control parameter. The approach presented here has alleviated from the problems of premature convergence, stagnation and revisiting of the same solution over and over again, which is common in other optimization techniques. Statistical and simulation results have been compared with already existing differentiator design methods such as segment rule, genetic algorithm (GA), simulated annealing (SA), pole zero method (PZ) and particle swarm optimization (PSO). The results affirm that the proposed method outperforms its counterparts in terms of absolute magnitude error and phase error. © byby Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2015 2015The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 2015 (ICRTC-2015). (ICRTC-2015) Keywords: Interior Search Algorithm; Metahueristic Algorithm; Digital Differentiator; PSO; GA.

1. Introduction Digital differentiators are used in wide range of applications such as digital image processing1, radar engineering2, biomedical3 and control systems4. They are useful systems for determining the time derivative of a signal. The frequency response of an ideal differentiator is given by * Corresponding author. Tel.: +91 8860010669. E-mail address: [email protected] (M. Kumar)

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of organizing committee of the 3rd International Conference on Recent Trends in Computing 2015 (ICRTC-2015) doi:10.1016/j.procs.2015.07.351

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Manjeet Kumar et al. / Procedia Computer Science 57 (2015) 368 – 376

‫ܪ‬ௗ ሺ߱ሻ ൌ ݆߱

(1)

where ݆ ൌ ξͳ and ߱ is the angular frequency. The digital differentiator design problem essentially comprises of obtaining a class of digital filter that approximates the frequency response of an ideal differentiator as closely as possible. Digital differentiators can be broadly classified into two types, ﬁnite impulse response (FIR), output depends only on present and past inputs, and infinite impulse response (IIR), output depends not only on previous inputs, but also on previous outputs. IIR differentiators have impulse response that is theoretically infinite, and due to their recursive nature, large memory space is required to store previous outputs. Whereas, FIR digital differentiator requires limited memory and has wider range of frequency over which it is stable and has a linear frequency response. Due to these aspects, designing of FIR differentiator is easier and less complex compared to IIR differentiators. Although, in applications where ecient processing of signals is required IIR differentiators are preferred. Many methods of designing digital differentiators already exist in literature. Conventionally, digital differentiators are designed by inverting the transfer functions of integrators with suitable modifications. The integrators are in turn designed using simple linear interpolation between the magnitude responses of diơerent Newton-Cotes integrators such as rectangular, trapezoidal and Simpson integrators5,6,7. A wide band digital differentiator based on wide band third-order trapezoidal integrator designed using Newton-Cotes integration formula has been proposed by Ngo8. Al-Alaoui has proposed segment rule to design digital differentiators based on linear interpolation between diơerent Newton-Cotes integration formulas. Some of the differentiators thus obtained were further optimized using simulated annealing9. Gupta et al. have proposed a class of wide band digital differentiators obtained through linear interpolation of popular digital integration techniques, the SKG (Schneider Kaneshige Groutage), trapezoidal rule, rectangular rule and the optimized 4-segment integrator rule9,10,11,12. Secondorder wide band differentiator obtained through optimization of magnitude response and analysis of pole-zero plot has been presented by Upadhyay13. The differentiator thus obtained has a relative error of less than 2%, making it suitable for real-time application. Another class of differentiator obtained by optimization of the pole-zero locations of existing recursive wide band digital differentiators has been formulated by Upadhyay and Singh 14. The proposed differentiator has a relative error of less than 0.48% over the entire Nyquist band. Pei and Hsu have used fractional delay method to design a first order digital differentiator15. All the above mentioned conventional optimization methods have been found to be eƥcient only in case of unimodal optimization problems. In problems involving multimodal optimization, these methods have demonstrated following shortcomings: (i) requirement of continuous and dierentiable error fitness function, (ii) incapability in searching large search space, (iii) premature convergence or convergence to a local optimal solution, (iv) increased sensitivity to starting points when there is an increase in number of variables and (v) requirement of piecewise linear cost function in case of linear programming method. Hence, it may be concluded that the conventional optimization methods are suitable only for use in unimodal optimization problems. To overcome all the drawbacks of conventional methods, several researchers have proposed many new optimization methods, most of which are based on the natural selection and evolution technique. These methods are formally known as heuristic and metaheuristic evolutionary optimization algorithms. They include, Genetic Algorithm (GA), a search algorithm inspired by Darwin’s ”Survival of the fittest”16, Simulated Annealing (SA), which is based on the thermodynamic eơect17, Particle Swarm Optimization (PSO), which simulates the behavior of bird flocking or fish schooling18, Cat Swarm Optimization (CSO), based on the behaviour of cats for tracking and seeking of an object19, Ant Colony Optimization (ACO), based on the ant food searching behaviour 20, etc. GA has shown the most promising potential to provide answers to the shortcomings of conventional methods. GA is capable of solving multidimensional problem that automatically lead to performance tradeo between design specifications. However, GA exhibits a number of shortcomings21; Because of their slow convergence rate, it shows ineciency in finding global minima, and as a result of this, the solution thus obtained is sub-optimal. Another drawback is the inability to find the optimal solution in computationally tractable time in problems involving large search spaces. Many modifications to the GA have been proposed to overcome the aforementioned diculties21. The standard PSO is a population based stochastic search algorithm. The eciency of PSO lies in the fact that it is simple to implement and requires only a few control parameters. In most of the applications, the standard PSO demonstrates better performance than the conventional methods. However, there are cases where this method results

370


in an inferior solution. This is due to the premature convergence and stagnation, that is, the algorithm converges to an unwanted sub-optimal solution and gets stuck. To overcome this, many modifications have been proposed, such as, Craziness PSO (CRPSO)22, and others23,24. Many heuristic evolutionary algorithms, including some of the above mentioned, have been implemented in designing of digital differentiator. Al-Alaoui has proposed a class of wide band digital dierentiators using GA, SA and Fletcher-Powell25. Gupta et. al26 have demonstrated a class of wide band digital differentiator using a modified form of PSO, GA, two variations of PSO27,28,29,30,31 and PSO-GA hybrid techniques32. Recently, a new algorithm called Interior Search Algorithm33 has been proposed. This algorithm takes into account the aesthetic techniques commonly used for interior design and decoration to solve global optimization problems. Like any other metaheuristic algorithm, it works on the principles of diversification and intensification to search for the global best. ISA provides advantages over conventional optimization methods as it requires tuning of only one parameter. It has a faster convergence rate as compared to other algorithms. It also solves the problems of premature and local convergence, which leads to finding the global minimum much more eƥciently. In this paper, ISA has been utilized to obtain a class of digital differentiator. The proposed differentiators are then compared with the differentiators obtained through SA, GA, modified PSO, pole-zero optimization and segment rule. Comparison of diơerent dierentiators has been done graphically on the basis of magnitude responses, magnitude error responses and the phase responses. Further comparisons based on the values of magnitude error and phase error obtained for various dierentiators have also been done. The remaining of the paper is organized as follows: In Section 2, the digital dierentiator design problem is formulated. Section 3 provides a brief review of ISA and its implementation in digital differentiator design. In Section 4, simulation and statistical results with illustrations are presented. Finally, the paper is concluded in Section 5.

Figure 1. Flow chart for Interior Search Algorithm.

371


2. Problem Formulation The input-output relation of an IIR system is governed by the following dierence equation34,35. ெ ‫ݕ‬ሺ݊ሻ ൅ σே ௞ୀ଴ ܽ௞ ‫ݕ‬ሺ݊ െ ݇ሻ ൌ σ௞ୀ଴ ܾ௞ ‫ݔ‬ሺ݊ െ ݇ሻ

(2)

where y(n), x(n),ܾ௞ ,ܽ௞ are the filter’s output, input, numerator and denominator coeƥcients, respectively, and N(≥ M) is the differentiator’s order. Assuming that ܽ଴ = 1, the transfer function of the differentiator can be written as

‫ ܪ‬ሺ‫ ݖ‬ሻ ൌ

షೖ σಾ ೖసబ ௕ೖ ௭ షೖ ଵାσಿ ೖసభ ௔ೖ ௭

(3)

The corresponding frequency response of IIR differentiator becomes: σಾ

‫ ܪ‬ሺ߱ሻ ൌ ଵାσೖసబ ಿ

௕ೖ ௘ షೕೖഘ

ೖసభ ௔ೖ ௘

షೕೖഘ

(4)

The values of the coefficients (ܽ௞ ,ܾ௞ ) of the transfer function will determine the type of filter designed (FIR or IIR). In this paper, a novel fitness function is adopted to achieve small magnitude error which is given as: ଵ

‫ ܬ‬ൌ ‫׬‬଴ ሺȁ‫ܪ‬ௗ ሺ߱ሻȁ െ ȁ‫ ܪ‬ሺ߱ሻȁሻଶ ݀߱

(5)

The cost function is specifically used to minimize the error between the ideal and approximated graphs in the lower and higher frequency ranges. Table 1. Control parameters of ISA Parameter

Value

Population Size

25-50

No. of iteration

50-100

Lower bound (LB)

-2

Upper bound (UB)

2

Alpha (α)

0.2

3. Interior Search Algorithm (ISA) ISA, also known as aesthetic search algorithm, is based on the aesthetic techniques used in interior design and decoration. This algorithm was originally given by Gandomi in 201433. It is a robust and ﬂexible algorithm that can be used to solve wide domains of optimization problems. It is a novel method for solving optimization tasks, has a very quick convergence rate and a capability to handle large search space. This method has been previously used to solve traditional engineering design problem, like welded beam design, pressure vessel design and gear train design33. In this paper ISA has been used to solve the digital differentiator design problem. The flowchart for ISA process is shown in Fig. 1. The implementation steps for digital differentiator design problem using ISA are as follows33. Step 1: Randomly generate a population of elements ‫ ݔ‬௝ between lower bounds (LB) and upper bounds (UB) and ௝ find their fitness values, ݂൫‫ݔ‬௜ ൯.

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Manjeet Kumar et al. / Procedia Computer Science 57 (2015) 368 – 376 ௝

Step 2: Find the fittest element in the ݆௧௛ iteration࢞௚௕ . For this optimization problem, the element with minimum function value is defined as the global best. Step 3: Randomly divide other elements into two groups, composition group and mirror group, by comparing with a tuned parameter α. If ‫ݎ‬௜ İ α, the element goes into the mirror group, otherwise it goes into the composition group. This step is important as a balance needs to be maintained between diversification and intensification. Step 4: In the composition group, change each element randomly within the limited search space (LB and UB). ௝

‫ݔ‬௜ ൌ ࡸ࡮࢐ ൅ ሺࢁ࡮௝ െ ࡸ࡮௝ ሻ‫ݎ‬ଶ

(6)

where ‫ݎ‬ଶ is a random value between 0 and 1. Step 5: For the elements in mirror group, place a mirror randomly between each element and the fittest element. The position of the mirror is evaluated by using the formula given below: ࢐

࢐ିଵ

࢞௠ǡ௜ ൌ ‫ݎ‬ଷ ࢞࢏

࢐

൅ ሺͳ െ ‫ݎ‬ଷ ሻ࢞ࢍ࢈

(7)

where ‫ݎ‬ଷ is a random value between 0 and 1. The location of the image or virtual location of the element depends on the mirror location and can be formulated as: ࢐

࢐

࢐ିଵ

࢞࢏ ൌ ʹ࢞௠ǡ௜ െ ࢞࢏

(8)

Step 6: The location of the global best is slightly changed by using random walk. Random walk works as local search because it searches around global best. ࢐

࢐ିଵ

࢞ࢍ࢈ ൌ ࢞ࢍ࢈ ൅ ߣ࢘࢔

(9)

where λ is a scale factor which is set according to size of the search space. Here, λ is set to be 0.01 × (UB − LB). Step 7: Calculate the fitness values of new locations of the element and images. If the new value is better, replace it, otherwise keep the initial value. Step 8: Repeat steps 2 to 7 until the stopping criteria (maximum number of iterations) is met. The global minimum fitness value and its corresponding element values are used as coeƥcients in optimal differentiator design. 4. Simulation Results To obtain the proposed differentiators, MATLAB simulation has been extensively performed. The best chosen parameters used for ISA optimization algorithm are reported in Table 1. All the MATLAB simulations have been performed in MATLAB 7.12 version on Intel Core(TM), 3.20 GHz with 4 GB RAM. The best optimal coeƥcients for the designed digital differentiator of second, third and fourth orders have been calculated using ISA, GA, SA, PSO and PZ and are reported in Table 2. In order to show the eƥciency of the proposed differentiator design method, several other differentiators obtained through other methods have been tabulated in Tables 2 and 3, to facilitate comparison. The second, third and fourth order differentiators proposed in this paper have been designed for the normalized frequency range of 0 İ ω İ 1. Table 3 presents a comparison of total absolute magnitude error and maximum phase error of diơerent reported differentiators. They have also been compared graphically in Figs. 2−4. These comparisons have been done on the basis of obtained magnitude response curve, magnitude error curve and phase curve. From Table 3 it can be seen that the proposed differentiators have total absolute magnitude error of 1.609128, 1.323152 and 1.228652 and maximum phase error of 89.9095, 89.9110 and 99.6842, in case of second, third and fourth order differentiators respectively. On the basis of above discussion, it can be concluded that the proposed differentiators are superior to other differentiators tabulated in Table 3. In Figs. 2(a), 3(a), 4(a) it can be observed that the proposed differentiators closely approximates the ideal magnitude response over the entire frequency range.

373


From Figs. 2(b), 3(b), 4(b) it is evident that the deviation of magnitude from ideality varies least in case of proposed differentiators over a wide range of frequency. The phase deviations for diơerent order digital differentiators are depicted in Figs. 2(c), 3(c), 4(c). In view of the above facts, it can be concluded that ISA performs better than the above reported literature.

Table 2. Optimal coefficient of digital differentiator of different algorithms Reference

Method

Order

Numerator coefficients

Denominator coefficients

Al-Alaoui9

3- Segment

3

0.01903 -0.029052

1.0000 0.1846

1.123 -1.181

-0.001748 0.03484

4-Segment

4

1.09677 0.0433

1.0000 1.19769 1.21566

-0.015308 -1.11198

1.197690 0.21580

1.1543 -0.4464

1.0000 0.7945

-0.7079

0.0832

25

Alaoui and Baydoun

GA GA

2 3

1.1533 -0.4432

-0.7060

1.0000 0.7981 0.0884

-0.0041

0 1.0000 0.9050 0.1713

GA

4

1.1553 -0.3170 -0.7560

SA

2

1.1538 -0.5408 -0.613

1.0000 0.7121 0.0607

SA

3

1.1555 -0.358 -0.714

1.0000 0.8662 0.1612

-0.0833

0.0028

SA

4

1.1540 0.2290 -0.8794

1.0000 1.3788 0.623

-0.4486 -0.0549

0.1059 0.0059

0.9269 -0.4628

1.0000 0.9883

-0.0817 Alaoui and Baydoun25

26

Gupta et al.

PSO

13

D.K Upadhyay

Present Study

2

-0.0006

0.0066 0

-0.99996

0.1595 1.0000 1.0000 0.2759

PSO

3

0.3237 1.0000 -0.7133 -0.6124

0.1595

PSO

4

0.1806 1.0000 -0.7133

1.0000 0.55207 -0.1578

0.4568 0.2611

-0.0708 -0.01

1.161 -0.79856

1.0000 0.6844

-0.55728

0.0676

1.144341 -0.574021

1.0000 0.696004

PZ ISA

2 2

-0.599883

0.063291

ISA

3

1.155408 -0.359594

1.0000 0.865227

-713884 -0.079311

0.158882 0.003219

ISA

4

1.156659 0.637118

1.0000 1.730208

-0.904285 -0.751576

1.016742 0.222981

-0.136173

0.011178

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Manjeet Kumar et al. / Procedia Computer Science 57 (2015) 368 – 376 Table 3. Comparison of performance criteria attained by already existing literature Reference

Method

Order

Al-Alaoui9

3- Segment 4-Segment 25

Alaoui and Baydoun

Alaoui and Baydoun25

Gupta et al.26

Total Absolute Magnitude Error

Maximum Phase Error

3

5.503161

269.739995

4

13.040399

172.601481

GA

2

2.084897

89.900552

GA

3

1.858099

89.902310

GA

4

1.318132

89.784931

SA

2

1.596657

89.909167

SA

3

1.511931

89.912323

SA

4

1.381094

89.601356

PSO

2

36.687205

89.998382

PSO

3

91.464599

89.715870

PSO

4

3.371737

179.770729

D.K Upadhyay13

PZ

2

4.793251

89.910229

Present Study

ISA

2

1.609128

89.909579

ISA

3

1.323152

89.911009

ISA

4

1.228652

99.684216

5. Conclusions The paper proposes a novel class of digital differentiators which have been approximated using the ISA algorithm. The ISA algorithm is simple as it has only one tuning parameter and thus has a faster convergence rate. The algorithm was simulated for the design of second, third and fourth order differentiators and the optimal coeƥcients of the transfer functions were derived. The magnitude response of the proposed differentiators accurately approximates the ideal differentiator over the entire Nyquist frequency range. Hence, the proposed differentiators are either comparable or better than the existing differentiators. Therefore, the proposed digital differentiators are viable alternatives over the existing digital differentiators. This work can be further extended for the design of digital integrator, two-dimensional differentiator, Hilbert transformer and many more attractive real time signal processing applications.

Magnitude

2.5

Absolute Magnitude Error

Ideal ISA SA GA PSO P-Z

3

2

1.5 2.965 2.96

0.05

100

0.045

80

0.04

60

0.035

40

0.03

20 Phase (deg)

3.5

ISA SA PSO GA P-Z

0.025

0.02

0 47

-20

46

Ideal ISA SA PSO GA P-Z

2.955

1

2.95

0.015

-40

0.01

-60

0.005

-80

0

-100

2.945 2.94 2.935 2.93

0.5

2.925 2.92 2.915 0.922

0

0

0.1

0.2

0.3

0.924

0.926

0.4 0.5 0.6 Normalized Frequency

(a)

0.928

0.93

0.7

0.932

0.934

0.8

45

44

43

42

41 0.635

0.64

0.645

0.65

0.655

0.66

0.665

0.936

0.9

1

0

0.1

0.2

0.3


(b)

0.7

0.8

0.9

1

0

0.1

0.2

0.3


0.7

0.8

0.9

1

(c)

Figure 2. Comparison of second order digital differentiator using ISA and other reported work (a) magnitude response (b) absolute magnitude error (c) phase response.

375


3.5

0.12 Ideal ISA SA PSO GA 3 Segment

2

1.5 2.96

2.955

1

0.08

80 60 40 20

Phase (deg)

Magnitude Response

2.5

0.1


3

100 ISA SA PSO GA 3 Segment

0.06

0 37.8

-20

37.78 37.76

0.04

Ideal ISA SA PSO GA 3 Segment

-40

2.95

-60

2.945

0.02

0.5

2.94

-80

2.935 0.935

0 0

0.1

0.2

0.3

0.936

0.937

0.938

0.939

0.4 0.5 0.6 0.7 Normalized Frequency

0.94

0.941

0.8

0.9

0

1

0

0.1

0.2

0.3


(a)

0.8

0.9

-100

1

37.74

0

0.1

0.2

37.72 37.7 37.68 37.66 37.64 37.62 0.6961 0.6962 0.6963 0.6964 0.6965 0.6966 0.6967 0.6968 0.6969

0.3


(b)

0.8

0.9

1

(c)

Figure 3. Comparison of third order digital differentiator using ISA and other reported work (a) magnitude response (b) absolute magnitude error (c) phase response.

3

Magnitude

2.5


Ideal ISA PSO GA SA 4 segment

2

1.5 2.885 2.88 2.875

1

2.87 2.865

0.05

100

0.045

80

0.04

60

0.035

40

ISA SA PSO GA 4 segment

0.03 0.025 0.02

Phase (deg)

3.5

20 0 55.18

-20

0.015

-40

0.01

-60

55.16

Ideal ISA PSO GA SA 4 segment

2.86 2.855 2.85

0.5

2.845

-80

2.84

0.005

2.835 0.9

0 0

0.1

0.2

0.3

0.905


(a)

0.91

0.8

0.915

0.9

1

0

-100

0

0.1

0.2

0.3


(b)

0.8

0.9

1

0

0.1

0.2

0.3

55.14 55.12 55.1 55.08 55.06 55.04 55.02 55 0.4904 0.4905 0.4906 0.4907 0.4908 0.4909 0.491 0.4911 0.4912 0.4913


0.8

0.9

1

(c)

Figure 4. Comparison of fourth order digital differentiator using ISA and other reported work (a) magnitude response (b) absolute magnitude error (c) phase response.

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