Computer Technology and Application 2 (2011) 370-373
Rapid Convergence of a New Wave Iterative Algorithm Used to Model a Patch Structure Hafedh Hrizi1, Lassaad Latrach1, Noureddine Sboui1, Ali Gharsallah1, Abdelhafidh Gharbi1 and Henry Baudrand2 1. Electronics Laboratory, Department of Physics, Faculty of Sciences in Tunis, El Manar 2092, Tunisia 2. Electronics Laboratory, ENSEEIHT, Toulouse 31071, France Received: March 25, 2011 / Accepted: April 15, 2011 / Published: May 25, 2011. Abstract: The wave iterative method is a numerical method used in the electromagnetic modeling of high frequency electronic circuits. The object of the authors’ study is to improve the convergence speed of this method by adding a new algorithm based on filtering techniques. This method requires a maximum number of iterations, noted Nmax, to achieve the convergence to the optimal value. This number will be reduced in order to reduce the computing time. The remaining iterations until Nmax will be calculated by the new algorithm which ensures a rapid convergence to the optimal result. Key words: Wave iterative method, rapid convergence, adaptive and autoregressive filtering (AARF), patch structure.
1. Introduction The iterative method is an integral method based on the wave concept and it is denoted WCIP (Wave Concept Iterative Process) [1-5]. It has shown efficiency for solving problems of electromagnetic diffraction and analysis of Radio Frequency circuits (RF). This method relies on the manipulation of waves instead of electromagnetic field. The studied circuit in Fig. 1 is a patch antenna placed in a metal cavity (perfect conductor), which has the interest to define the basis decomposition of electromagnetic fields and it serves as a mechanical support structure. The method is called iterative because it establishes a recurrent relation between the incident and reflected waves. Although this method is absolutely convergent, the Lassaad Latrach, doctor, research fields: circuits and electronic HF systems. Noureddine Sboui, professor, research fields: circuits and electronic HF systems. Ali Gharsallah, professor, research fields: physics and electronics. Abdelhafidh Gharbi, professor, research field: physics. Henry Baudrand, professor, research field: physics. Corresponding author: Hafedh Hrizi, Ph. D, research fields: circuits and electronic HF systems. E-mail:
[email protected].
number of iterations is relatively high and it needs much time especially for structures requiring a fine mesh. To improve the convergence speed of this method, we use an Adaptive and AutoRegressive Filtering algorithm AARF. The AutoRegressive Filter ARF is used to estimate and predict the following of an uncompleted input sequence while the Adaptive Filter [6-9] ensures a rapid convergence towards the optimum result with minimum error. We propose in this research to improve the algorithm of the classical 70
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Fig. 1 Structure of a patch antenna discretized in 64×64 pixels.
Rapid Convergence of a New Wave Iterative Algorithm Used to Model a Patch Structure
wave iterative method, by adding a new algorithm based on the AARF technique. Our goal is to reduce the computing time through the reduction of the number of iterations used in the iterative method so that we improve the convergence speed of this method. The paper is organized as follows: Section 2 discusses the theoretical part in which we present the new wave iterative algorithm. Section 3 presents results and discussions. Section 4 gives conclusions.
2. Theoretical Study The new proposed algorithm is composed of two functional blocks as in Fig. 2. The first block is an ARF filter whose role is to predict the remaining samples of an uncompleted input sequence having a length equal to Nmin. The iterations from 1 to Nmin are calculated by the classical iterative method while the prediction of the signal samples in the following iterations from Nmin+1 to Nmax is realized by the filter ARF using a Gaussian white noise b(n) :
x ( n)
¦ a(i) x(n i) b(n) m
(1)
of the Adaptive Filter, P is the adaptation step, and e(n) d (n) y (n) is the error value relative to a reference signal d (n) .The output in the iteration n is (4) P as in the The best choice of the adaptation step next condition (5) provides the stability and the convergence of the LMS algorithm to the optimal results with minimum error: y ( n)
h T ( n) X ( n )
0 P 2 / LV x2
(5) We notice that the choice of the adaptation step depends on the power V x2 of the input signal and on the order L of the adaptive filter. Thus, the algorithm of the new method will be noted as A-WCIP (Adaptive Wave Concept Iterative Process). We introduce an input sequence whose length is Nmin, the iterations of this sequence are calculated by the classical iterative method. The new algorithm predicts the result of the remaining iterations after Nmin using the base formed by the Nmin values until achieving the convergence to the optimal value with Nmax iterations (Fig. 3).
i 1
where x(n) is an estimation of x (n) in the iteration n and a(1), a(2),…, a(m) are the coefficients in the following transfer function of the ARF filter where m is the order of this filter: H (z)
1 /(1
¦ a ( k ).z k )
k m
(2)
Fig. 2
Functional block.
k 1
The second block is an Adaptive Filter based on the LMS (Least Mean Square) algorithm [8-9]. This ensures the convergence toward the correct value with minimum error. The LMS algorithm allows the updating of the Adaptive Filter transfer function coefficients h (n ) in every new iteration: h ( n 1) h ( n ) PX ( n ) e ( n ) (3) The coefficients h( n )
T
[ h0 ( n ), h1 ( n ),..., hL 1 ( n )]
are
defined in the iteration n and the coefficients h(n 1) are in the iteration n 1 . The is X ( n )
input
[ x ( n ), x ( n 1),..., x( n L 1)]
vector T
, L is the order
371
Fig. 3 Algorithm of the new A-WCIP approach.
Rapid Convergence of a New Wave Iterative Algorithm Used to Model a Patch Structure 1
3. Results and Discussion The input signal x (n ) in the theoretical part will be designated by the coefficients of the diffraction matrix
S21
372
ARF
A-WCIP
WCIP
0.8 0.6
S11 and S21. The coefficients S21 are represented in function of the number of iterations with Nmax is equal to 1,000 iterations and Nmin is 100 iterations. We visualize that
0.4 0.2 1
the predicted output signal of the ARF Filter is likely to diverge as it is shown in Fig. 4. We notice that the
100 199 298 397 496 595 694 793 892 99 Iterations
Fig. 4 S21 variations by both methods with Nmax = 1000, Nmin = 100 iterations.
output signal of the Adaptive Filter, which will be the final output of our new A-WCIP system, is close to the classical WCIP curve. The Adaptive Filter takes as input the output of the ARF Filter and its final output (AARF) must converge to the optimal values of S21 with minimum error (Fig. 4). Thus, we improve the convergence of the wave iterative method because the new used algorithm A-WCIP provides good results and it doesn’t take much time to converge to the optimal result. From Table 1, we observe a large gain in convergence time when calculating S11 and S21 values by the two methods (the classical WCIP method and the new A-WCIP method). This gain of time is provided by the new method with Nmax = 1,000 iterations and with two values of Nmin: 25 and 100 iterations. (The patch Structure is discretized in 512 × 512
pixels).
We
use
a
machine
having
a
microprocessor Intel(R) Pentium(R) DualCore CPU 2 × 2.16 GHz and 3 GB of RAM. We can notice that the new A-WCIP method
Table 1 Comparison of time between the two methods with Nmax=1000 iterations, frequency = 10 GHZ. Nmin A-WCIP time(mn) WCIP time(mn) Gain of time (%) 25 0.61016667 24.0116667 97 100 2.4112 24.0116667 89
4. Conclusions In our study, the convergence speed of the iterative method has been improved when modeling a patch antenna structure. The classical algorithm has to calculate only a minimum number Nmin of iterations that can be reduced from 1,000 to 25 iterations. The remaining iterations are treated by the new proposed algorithm to achieve convergence to the optimal value with Nmax = 1,000 iterations. With our new A-WCIP method, we have a very fast convergence with Nmax = 1,000 iterations in comparison with the classical WCIP method. Finally, we have a very significant reduction in computing time which ensures a rapid convergence with a minimum average error hence the efficiency and robustness of our new approach.
provides the same results as the classical WCIP
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