Reducing the number of iterations in the WCIP ...

Reducing the number of iterations in the WCIP method in case of space excitation H. Hrizi, N. Sboui Laboratory, Department of Physics, Faculty of Sciences Tunis 2092 El Manar Tunisia

H. Baudrand Electronics Laboratory EN SEEIHT Toulouse France

II. THEORETICAL STUDY Abstract—The WCIP method (Wave Concept Iterative Process) is a numerical method used for modeling electronic HF circuits (High Frequency). In some cases, this method requires a big number of iterations to converge to the optimal result. That’s why it needs much computing time. For this, we use an autoregressive an adaptive filter to reduce the number of iterations calculated by this method. Index Terms—Wave Iterative Method, Convergence, reducing number of iterations, reducing calculation.

I.

INTRODUCTION

The iterative method "WCIP" is a numerical method which allows solving problems of electromagnetism and analysis of electronic circuits High Frequency (HF). This method uses waves instead of electromagnetic field [1-15]. Although this method is absolutely convergent, the number of iterations is relatively high in the case of modeling complex structures requiring a fine mesh. The computational complexity is related to the number of cells describing the circuit. For that, it takes too long to give good results by this method. To avoid this problem, the technique of adaptive filtering is an effective way to ensure fast convergence to the optimal value with minimum error. We use the adaptive LMS algorithm (Least Mean Square), because it is the simplest in terms of computation. In addition, it is the most efficient algorithm in terms of minimization criterion of the mean square error [16-24]. We present a new algorithm based on adaptive filtering techniques and autoregressive. We also show the integration of this filtering algorithm in the method "WCIP" to create a new method that will be called "A-WCIP: Adaptive-WCIP." Finally, we aim to reduce the number of iterations of this method, and therefore we reduce the computation time and improving the speed of convergence of this method.

A. Wave concept iterative algorithm The WCIP method is an integral method based on the wave concept and it is used in solving problems of electromagnetic modeling. It is noted "WCIP" because it treats the waves instead of electromagnetic fields. This method is different from the other integral methods (Method of moments, Galerkin Method ...) because it does not use the scalar product or the matrix inversion. The method defines two operators in spacial and spectral domains. The Fast Fourier Mode Transformation (FMT) and the reverse transformation FMT-1 insure the transition from one area to another. Incident and reflected waves are expressed in function of electromagnetic fields as in the next relation:    Ai =  2 Bi   2 

1 Z 0i 1 Z 0i

−

Z 0i   2 E  i Z 0i  J i 

2  

(1)

The iterative process consists in establishing a recurrent relation between the incident and reflected waves in two different domains as indicated in the following equations:

r

v

r

ˆ A+ B B = Γ  Ω 0 r r ˆB  A = Γ

(2)

The passage from one domain to another is ensured by the Fast Mode Transformation (FMT) and the Fast Opposite Mode Transformation (FMT-1). The iterative process is summarized in Fig. 1. To generalize, the iterative process can be summarized by the following algorithm in Fig.1:

r J1

Begin

r J2

r E1

r E2

Quadripôle

B0

Initial Values

Fig. 2. Electric model

Spatial Domain

∧

Based on the general theory of the quadrupole, we can express the current densities as a function of electric field:

B( x , y ) = ΓΩ A( x , y ) + B 0

 J 1  Y11 Y12   E1   J  = Y Y   E   2   21 22   2 

B ( m ,n ) = FMT ( B( x , y ) )

With: Spectral Domain

α A( m ,n ) = Γ B( m ,n )

A( x , y ) = FMT

−1

[Y ] = 

Y11 Y12   Y21 Y22 

quadrupole. The terms follows: Y11 =

( A( m ,n ) )

J1 E1

Y12 =

No

the admittance matrix of the

Y11 and Y21 of the matrix are defined as

E =0 2

J1 E2

E

1

(5)

and

=0

and

Y21 =

Y22 =

J1 E1

E =0 2

J2 E E2

1

=0

Convergence

Terms of matrix “Y” are obtained by exciting the circuit with the first source and by passing the second source, the elements “Y” are determined by exciting the circuit by the second source, the first source is shorted. From the matrix [Y] we can deduce the distribution matrix [S] given by the following equation:

Yes The end

[s ] = 1 − [ y ] 1 + [y ]

Fig. 1. The iterative algorithm “WCIP”

a- Computing of “Y” and “S” matrices

B. Computational complexity and convergence of the WCIP method

The surface admittance is generally determinated from the next expression:

Yin =

< E0 / J > < E0 / J >

The iterative method is one of the most effective methods in solving problems of electromagnetism and analysis of electronic circuits High Frequency (HF). This method relies on the manipulation of incident and reflected waves instead of electromagnetic field. Although this method is absolutely convergent, the number of iterations is relatively high in the case of modeling complex structures requiring a fine mesh. The computational complexity is strongly related to the number of cells describing the circuit. To this end, this method requires a relatively high computing time to converge to the optimal result. As we note in [3] the convergence is slow and exceeds one thousand iterations.

(3)

The impedance is defined by:

Zin =

1 Yin

(6)

(4)

b- Case of a quadrupole A quadrupole is an electronic device having two ports. To determine the characteristics of a quadrupole, it is assumed that there are two sources defining a transmission line. The electric model of the line is in Fig. 2:

Also as other studies [2, 9, 14, 15] have shown the computational complexity of the method, especially in the case of complex structures that require a large number of pixels. 2

In addition, the great progress in technology especially of high frequency electronics and the emergence of new families of integrated circuits make the computational complexity of calculating very important. More complex circuits require high accuracy at the mesh on the spatial structure. Hence the need for a finer discretisation circuit for maximum information on the distribution of physical quantities. However, though a fine mesh needs more accuracy, it requires high computation time especially if it is a complex structure. This is why the problem of computational complexity in calculation method WCIP becomes a very interesting problem to investigate. The operator in the space field diffraction reflects the relationship between pitches of waves through the plane of the interface circuit. This operator is a key element in the numerical method. In addition it depends on the quality of spatial grid circuit. To do this, we must describe the plan for discretization pixels to achieve a better result. This determines the spatial mesh computational complexity of the method WCIP. Thus, the computational complexity of the iterative method depends on three main parameters: the spatial resolution defined by the mesh size, the number of interfaces and the number of iterations. In addition, the method requires more computing time in the case of modeling a complex electronic structure or containing active ingredients, this is what we will show in the following study. For modeling complex structures or circuits containing active elements, the method "WCIP" takes a long time to converge to the correct result. In addition to the calculation becomes cumbersome if you use a fine mesh. For example, for structures 512x512 cells, it takes 24 minutes for 1000 iterations using the method WCIP. This result is calculated by a machine with an Intel (R) Pentium (R) Dual

1,01 1 [S11]

0,99 0,98 0,97 0,96 0,95 0,94 0

50

100

150

200

250

300

350

400

450

500

Iterations Fig. 4. [S11] parameters at frequency 10GHZ (500 iterations)

[S21]

0,07 0,06 0,05 0,04 0,03 0,02 0,01 0 0

50

100 150 200 250 300 350 400 450 500 Itérations

Fig. 5. [S21] parameters at frequency 10GHZ (500 iterations)

From Fig. 6 and Fig. 7, we note that at frequency 9.8GHZ, the method may not converge with 200 and 1000 iterations which is a big number. For this, the computing numerical complexity becomes slow and the method needs much time to give good results.

Core CPU 2x2.16GHz and 3GB of RAM. a. Study of convergence in the case of spatial excitation The studied circuit is presented in Fig.3. It is a transmission line excited by a source space. The structure has the next characteristics: height h = 6 mm substrate, the permittivity εr=1, the dimensions a=30mm, b=60mm. This structure is constituted by a metal, a dielectric and two sources.

[S11]

1,01 1

0,99 0,98 0,97 0,96 0,95 0,94 0

20

40

60

80

100 120 140 160 180 200 Itérations

Fig. 3. Line transmission structure

Fig. 6. [S11] Parameters at frequency 9.8GHZ (200 iterations)

The Fig. 4 and Fig. 5 show that the WCIP method begins the convergence from 500 iterations, for this it needs a big number of iterations to achieve the convergence to the optimal results.

3

frequencies. This provides the stability and convergence of our system irrespective of the conditions of the input signal [16].

1,01 1

Starting

[S11]

0,99 0,98 0,97

Initial values

0,96 0,95

« 1……..Nmin » Iterations «WCIP»

0,94 0

100 200 300 400 500 600 700 800 900 1000 Iterationss

Fig. 7. [S11] Parameters at frequency 9.8GHZ (1000 iterations)

« Nmin…….Nmax » Iterations «A-WCIP»

Finally, we note that the computational complexity of calculating in the iterative method is important when we have a spatial excitation of circuits. Thus, it is necessary to improve this method advantage by other techniques to ensure faster convergence to good results and to reduce the complexity and computation time.

No

Converge

C. The New iterative algorithm “A-WCIP”

Yes

The new algorithm is rated "A-WCIP" (Adaptive WCIP) because it is based on improving the method WCIP by adaptive filtering technique which ensures fast convergence to the optimal result [16]. As shown in Fig. 8, the idea is to add to the classical algorithm WCIP a new algorithm describing a filter AR (autoregressive) and an adaptive filter based on the LMS algorithm.

The end Fig. 9. The new iterative process “A-WCIP”

Consequently, the new algorithm with "A-WCIP", we improve the convergence of the classical iterative WCIP which calculates only a limited number of iterations equal to "Nmin". The number "Nmax" iterations required to achieve convergence can be achieved by the new algorithm "AWCIP", which is a very fast algorithm. Thus, a significant gain in computation time will be realized by this new approach.

III. RESULTS AND SIMULATIONS Fig. 8. The new functional bloc

In the previous theoretical study, the input signal will be appointed in the next section by the coefficients S11 or S21 of the diffraction matrix. To model the electronic structure, we use the new iterative adaptive algorithm "A-WCIP." To validate the results found by the new approach, we compare with the results of the iterative basis "WCIP."

Thus, the algorithm of the new method will be noted as "A-WCIP" (Adaptive Wave Concept Iterative Process) as in Fig. 9. We introduce an input sequence having a length equal to "Nmin", iterations of this sequence are calculated by the conventional algorithm WCIP. The new algorithm "A-WCIP" predicted the outcome of the remaining iterations until "Nmax" iterations where convergence to the optimal value is reached. The best of our packaging system is first made by a good selection of the optimal order "m" filter "AR". Then, it is made by a good selection of step adaptation of the LMS algorithm. We conclude that the packaging of this system is mainly based on the nature of the input signal. This is an important point of our approach, because it ensures that the system adapts to all types of input signals which vary from one frequency to another, especially when we test a wide range of

A. Case of a Space excitation We test the new technique "A-WCIP" in the case of a circuit with spatial excitation. We study the structure shown in Fig. 3 is a transmission line excited by an excitation source space. We present in Fig. 10, 11 and 12, the convergence curves of "S" parameters in function of iterations. These parameters are calculated by two methods: the traditional and the new method (“WCIP” and "A-WCIP") at different frequencies. In the case of spatial excitation, we note that the method WCIP requires a large number of iterations that can 4

reach 2000 iterations to converge to the correct result. In the method "A-WCIP" WCIP algorithm is used to calculate the first iteration ("Nmin" iterations) and other iterations will be calculated by the new filtering technique. The number "Nmax" represents the maximum number of iterations to achieve convergence. We find the following results from the new technique "A-WCIP" converges to good results and that in comparison with the method "WCIP" basis. S11 : WCIP

Finally, we can say that we have reduced the number of iterations calculated by the iterative method from 2000 to 200 or 400 iterations in the case of a circuit with spatial excitation. Thus, we reduce greatly the computation time and the computational complexity of this method. If you change the frequency study, we find that the method "A-WCIP" always ensures a fast convergence towards the optimal result. This demonstrates the robustness of this new approach. CONCLUSION

S11 : A-WCIP

1,01

Along this paper, we presented the technique to improve the iterative method. We used a new algorithm based on adaptive filtering technique to accelerate the convergence to the correct result. The convergence speed of the classical iterative method has been improved. In the new method "AWCIP", the iterative algorithm is used to calculate only a minimum number "Nmin" of iterations, we reduce the iterations calculated by the iterative method from 1000 to 25 iterations. The remaining iterations after "Nmin" are processed by the new filtering algorithm to achieve convergence to the optimal value after "Nmax = 1000" iterations. Thus, we have a very rapid convergence in comparison with the conventional method in which the algorithm computes all 1000 iterations. Finally, a very significant reduction of computational time was obtained. And beyond, we ensure fast convergence with an average error limited where we provide a significant improvement for the iterative method in waves, especially in terms of computation time.

1 0,99 0,98 [S] 0,97 0,96 0,95 0,94 200

400

600

800

1000

1200

1400

1600

1800

2000

Iterations

Fig. 10. Variation of S11 in fonction of iterations (Nmax=2000, Nmin=400, Frequency=10GHZ)

S21 : WCIP

S21 : A-WCIP

0,07 0,06 [S]

0,05

REFERENCES

0,04 0,03

[1] N. Sboui, A. Salouha, L. Latrach, A. Gharsallah, A. Gharbi and H. Baudrand, “Efficient Analysis of Switchable FSS Structure using the WCIP Method”, ACES JOURNAL, VOL. 27, NO. 3, MARCH 2012. [2] L. Latrach, “Contribution à l’Etude des Circuits Electroniques Hautes Fréquences Actifs et Passifs implantés sur substrats diélectriques homogènes et inhomogènes par une Méthode Itérative en Ondes’’, Thèse de doctorat, FST, Tunis, 2010. [3] F. Mejri and T. Aguili, “NEW FORMULATION OF THE ITERATIVE METHOD: APPLICATION TO A MICROWAVES ATTENUATOR”, Progress In Electromagnetics Research B, Vol. 22, 321-340, 2010. [4] H. Baudrand, R.S. N’Gongo. "Applications of wave concept iterative procedure", Recent Res. Devel. Microwave Theory and Tech Vol 1 pp. 187–197, 1999. [5] A. Gharsallah, A. Gharbi and H. Baudrand, ‘Efficient analysis of multiport passive circuits using iterative technique,’ Electromagnetics, vol. 21, pp. 73-84, 2001. [6] S. Akatimagool, D. Bajon, H. Baudrand. "Analysis of Multi-Layer Integrated Inductors with Wave Concept Iterative Procedure (WCIP)", IEEE MTT-S 2001, vol-3 pp 1941-1944, May 2001. [7] T. I. E. Brahim, H. Zairi, A. Gharsallah, A. Gharbi, H. Baudrand, "Analysis of Dual-Mode Microstrip Patch Filter using Multiscale Wave Concept Iterative Process," ACES Journal, vol.25 no.9 pp. 801-808, September 2010.

0,02 0,01 0 200

400

600

800

1000 1200 1400 1600 1800 2000 Iterations

Fig. 11. Variation of S21 in fonction of iterations (Nmax=2000, Nmin=400, Frequency=10GHZ)

S11 : WCIP

S11 : A-WCIP

1,01 1 0,99 0,98

[S]

0,97 0,96 0,95 0,94 0,93

100

200

300

400

500

600

700

800

900

1000

Iterations

Fig. 12. Variation of S11 in function of iterations (Nmax=1000, Nmin=200, Frequency=7GHZ)

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[8] A. Gharsallah, R. Garcia, A. Gharbi, H. Baudrand, "Wave Concept Iterative Process Merges with Modal Fast Fourier Transformation to Analyze Microstrip Filters," ACES Journal, vol.16 no.1 pp. 61-67, March 2001. [9] N. Sboui, A. Gharsallah, H. Baudrand, A. Gharbi, ‘‘Glabal Modeling of Microwave Active Circuits by an efficient iterative procedure”, IEE Proc-Microw. Antenna Propag. Vol 148, No. 3, June 2001. [10] N. Sboui, A. Gharsallah, H. Baudrand, A. Gharbi, ‘‘Design and Modeling of RF MEMS Switch by Reducing the Number of Interfaces’’, Microw. and Opt. Technol. Lett vol. 49,No. 5 pp 1166-1170, May 2007. [11] N. Sboui, L. Latrach, A. Gharsallah, H. Baudrand, A. Gharbi, “A 2D Design and Modeling of Micro strip Structures on Inhomogeneous Substrate”, Int. Journal of RF and Microwave Computer –Aided Engineering vol. 19, No. 3 pp 346-353, May 2009. [12] N. Sboui, A. Gharsallah, H. Baudrand, A. Gharbi, “Global Modeling of periodic Coplanar Waveguide Structure For Filter Applications Using an Efficient Iterative Procedure”, Microwave and Opt. Technol. Lett Vol. 43 No 2, pp 157-160, 2004. [13] N. Sboui, A. Gharsallah, A. Gharbi, and H. Baudrand, “Analysis of double loop meander line by using iterative process”, Microw. Optical Technical Letters, vol.26, pp. 396-399 June 2000. [14] L. Latrach, N. Sboui, A. Gharsallah, H. Baudrand, A. Gharbi, “A Design and Modelling of Microwave Active Screen Using a Combination of the Rectangular and Periodic Waveguides Modes”, Journal of Electromagnetic Waves and Applications, Volume 23, Numbers 11-12, 2009 [15] L.Latrach, N. Sboui, A. Gharsallah, H. Baudrand, A. Gharbi, “Analysis and Design of Planar Multilayered FSS with Arbitrary Incidence”, Applied Computational Electromagnetic Society journal vol. 23, No. 2 pp 149-154, June 2008. [16] H. Hrizi, N. Sboui, A. Gharsallah, A. Gharbi and H. Baudrand, ‘‘Improving the Convergence of the Wave Iterative Method by Filtering Techniques’’, ACES JOURNAL, VOL. 26, NO. 10, OCTOBER 2011. [17] S. HAYKIN, “Adaptive filter theory, Prentice Hall”, 1991. [18] B. ANDERSON and J. MOORE, “Optimal Filtering, Prentice Hall”, 1979. [19] J. G. Proakis and D. G. Manolakis, “Digital Signal Processing: Principles, Algorithms and applications”, Third Edition, Macmillan, Chap. 12, 1996. [20] W. Byrne, P. Flynn, R. Zapp, M. Siegel, “Adaptive Filter Processing in Microwave Remote Heart Monitors”, IEEE Trans. On Biomed. Enginee. Vol. BME-33, NO. 7, July 1986. [21] J. Luukko, K. Rauma, “Open-Loop Adaptive Filter for Power Electronics Applications”, IEEE Trans. On Indust. Electronics Vol. 55, N. 2, Feb. 2008. [22] A. Ogunfunmi, A. M. Peterson, “On the Implementation of the Frequency-Domain LMS Adaptive Filter”, IEEE Trans. On Circuits, Systems-II Analog, and Digital Signal Processing, Vol. 39, No. 5, May 1992. [23] M. Godavarti, O. Alfred Hero, “Partial Update LMS Algorithm”, IEEE Trans. On Signal Processing Vol. 53, N. 7, July 2005. [24] J. Daniel Allred, H. Yoo, V. Krishnan, W. Huang, V. David Anderson, “LMS Adaptive Filters Using Distributed Arithmetic for High Throughput”, IEEE Trans. On Circuits and Systems-I REGULAR PAPERS, VOL. 52, NO. 7, JULY 2005.

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