Download generalized regression neural network based phase

Generalized Regression Neural Network based Phase
Characterization of a Reflectarray Employing
Minkowski Element of Variable Size
Selahattin Nesil1, Filiz Günes1, Ufuk Özkaya1 and Bahattin Türetken2
1
Electronic and Communication Engineering, Yıldız Technical University, Istanbul, TURKEY
e-mail: [email protected], [email protected], [email protected]
2
National Research Institute of Electronics and cryptology (UEKAE), The Scientific and Technological Research
Council of Turkey (TUBITAK), Istanbul, TURKEY, e-mail: [email protected]
Abstract
In this paper, a simple, accurate and fast Generalized Regression Neural Network (GRNN) model is carried out
to determine the phase characterization of a reflectarray unit cell design employing Minkowski shape element for
different geometry parameters within the frequency range 10-12 GHz. The recent aim of reflectarray antenna design is
to have a smaller gradient (slower slope) of the reflected wave phase by varying the element’s size and substrate
thickness. The efficient relationship is achieved between the actual and reconstructed data with this model. This model
can provide a fast, accurate interface between the antenna design and global optimization algorithms.
1. Introduction
A reflectarray antenna is a low profile reflector consisting of a planar array of microstrip patches, with a certain
tuning to produce prescribed beam shape and direction when illuminated by a primary source [1]. Due to its properties
of being of being flat, light weight, low cost, and high gain, reflectarray antenna is rapidly becoming an attractive
alternative to the traditional parabolic reflector antenna in the applications where high gain antenna is needed.
The most important and critical step of the reflectarray design is its element characterization. If the element
design is not optimized, the Reflectarray will not scatter the signal from the feed effectively to form an efficient far –
field beam. In order to satisfy requirements as the capability to radiate a shaped beam or multibeams, or also to enhance
the frequency behavior and bandwidth, it is necessary to use advanced element configurations, showing several degrees
of freedom. The management of different parameters and the need of satisfying requirements that could be also in
opposite each other could however make the design of a Reflectarray (RA) quite complex and therefore it is not
feasible for the current computer technology to have a complete, rigorous solution to include all the mutual coupling
effect of all different elements since the Reflectarray generally consists of too many elements. Thus the commonly used
approach is to obtain the element phase information using the mathematical waveguide simulator by adopting the
infinite - array approach [1].
In this work, phase behavior of a single patch of a novel shape called Minkowski which is the first iteration of
the fractal’s shape and shown to provide low insertion loss and acceptable phase range [2], is modeled as a function of
its geometric parameters using the Generalized Regression Neural Network (GRNN). The resultant GRNN model is
simpler, more accurate and faster compared to the work in [3]. GRNN is a one-pass learning algorithm with a highly
parallel structure. Even with sparse data in a multidimensional measurement space, the GRNN algorithm provides
smooth transitions from one observed value to another [4]. This model can be utilized for global optimization of both
the phase behavior of the unit cell and the overall performance of the antenna.
In the next section, the black-box phase modeling of a single Minkowski radiating element is briefly
explained The third section gives fundamentals of the GRNN while GRNN model of the phase characterization for
the Minkowski element is taken place in the fourth section and the paper ends with the conclusions.
2. Black- Box Phase Modeling of the Minkowski Radiating Element
In the black-box modeling of the Minkowski radiating element (Fig.1), the geometry parameters (m, η , h) and
operation frequency are considered as the input parameters and phase of the reflection is outputted from the GRNN
which is used as a function approximator (Fig.2). Here m is the length of the patch, h is the thickness of the substrate
and η is the iteration factor defined as
η =
s
,
m /3
0 ≤ η ≤1
(1)
where m , s are depicted in Fig.1.
3. GRNN (Generalized Regression Neural Network)
In the literature, the fundamentals of the GRNN can be obtained from Specht, 1991[4]; Nadaraya–Watson kernel
regression [5, 6], Tsoukalas and Uhrig, 1997 [7], also Schioler and Hartmann [8]. A diagrammatic of the GRNN is
depicted in figure 2. A general regression neural network (GRNN) does not require an iterative training procedure. It
can approximate any arbitrary function between input and output vectors, drawing the function estimate directly from
the training data. Furthermore, it is consistent; that is, as the training set size becomes large, the estimation error
approaches zero, with only mild restrictions on the function. The GRNN is used for estimation of continuous variables,
as in standard regression techniques.
The GRNN is composed of four layers: Input layer, pattern layer, summation layer, and output layer. The total
number of parameters equal to the number of input units in the first layer. The first layer is fully joined to the second,
pattern layer. In pattern layer, each unit symbolizes a training pattern, and its output measures the distance of the input
from the stored patterns. Each pattern layer unit is linked to the two neurons in the summation layer: S- summation
neuron and D- summation neuron. Here, the sum of the weighted outputs of the pattern layer is calculated by the Ssummation and the un-weighted output of the pattern neurons is computed by the D-summation. The linkage weight
between the S-summation neuron and the ith neuron in the pattern layer is yi ; the target output value reciprocal to the
ith input pattern. The linkage weight for D-summation is unity. The output layer just divides the output of each Ssummation neuron by the output of each D-summation neuron, supplying the predicted value to an unknown input
G
vector x as
n
G
yˆ i ( x ) =
∑
G G
y i e x p [ − D ( x , xi )]
i =1
n
∑
G G
ex p [− D ( x , xi ) ]
(2)
i =1
in which the number of training patterns is indicated by n and the Gaussian D function in (3) is expressed as
G G
D( x , xi ) =
p
∑(
j =1
x j − xij
ζ
)2
(3)
G
G
in which p shows the number of element of an input vector. The xj and xij represent the jth element of x and xi ,
respectively. The ζ is generally regarded as the spread factor, whose optimal value is often determined experimentally
for the problem under investigation. If the spread factor ζ becomes larger, the function approximation will be smoother.
If spread factor ζ is too large, then a lot of neurons will be required to fit a fast changing function. Too small a spread
means many neurons will be needed to fit a smooth function, and the network may not generalize well.
4. Worked Example
In this work, a Minkowski patch with the resonant frequency 11 GHz is used, which is printed on the singlelayer substrate RF-35 with the thickness swept within 1.370mm < h < 3mm. The geometry of resonant element patch
is swept with a patch variation, n = ±2 from their resonant size m=5.41mm which is n = 0. For example, n = +2 means
the patch is 20% larger than its resonant size and vice versa. The geometry of Minkowski is created using the iteration
factor η given by (1) which is varied with the values 0.15, 0.3, 0.45, 0.6, 0.75 and 0.9. The data for training and
validation is generated from the theoretical analysis implemented by the H- wall waveguide simulator using available
full-wave simulation tool Computer Simulation Technology Microwave Studio (CST MWS) as given in [2]. Thus, 750
data is obtained at the frequencies of 10-10.5-11-11.5-12 GHz and 450 data at the 10-11-12 GHz are used for training
and the network is validated with the rest 300 data at the frequencies 10.5 and 11.5 GHz. Scattering plot between the
actual and reconstructed data is n takken place in the figure 3 where one can observe that both the actual and
reconstructed data match, thus one caan decide the modeling process results in an accuraate model the In figure 4,
reflection phase variations can be examiined with respect to the percentage patch length %n annd the substrate thickness h
for the iteration factor value η=0.45 which
w
is built up using the reconstructed data at the reesonant frequency 11GHz.
Figure 5 gives phase variations of the reflected
r
waves with respect to the percentage patch length
l
%n and the iteration
factor η using the actual and reconstruccted data. These phase variations of the reflected wave are compared using the
actual and reconstructed data for an optiimal substrate thickness h=1.524 mm in figure 6.
Figure 2 Black-Box Phase Modeling of Minnkowski with GRNN
Figure 1 Minkowski Shape
Reconstructed Data for f = 11 GHz, ŋ= 0,45
Phase of S11 (degree)
200
100
0
-100
3
2,5
-200
2
-300
-2
1,524
-1
0
% n patch variation
1,370
1
h substrate
thickness
2
Figure.3 Scattering plot for the GRNN
Figure 4 Reflection Phase Variations
V
w.r.t patch
variation and Subsstrate thickness
Reconstructed Data for f = 10.5 GHz, h = 1,524 mm
50
50
0
-50
-100
-150
-200
-250
-300
0,75
0,45
-2
-1
İteration Factor η
0
% n patch variation
1
0,15
2
(a)
Phase of S11 (degree)
Phase of S11 (degree)
Actual Data for f = 10.5 GHz, h = 1,524 mm
0
-50
-100
-150
-200
-250
-300
0,9
0,75
0,6
0,45
-2
0,3
-1
0
% n patch variation
0,15
1
İteration Factor η
2
(b)
Figure 5 Reflection Phase Variations w..r.t patch variation and iteration factor with (a) the Acttual Data; (b) the
Reconstructed Data
Actual and Reconstructed Data for f = 10.5 GHz , h = 1,524 mm
50
Actual Phase at η=0,15
Actual Phase at η=0,3
Phase of S11 (degree)
0
Actual Phase at η=0,45
-50
Actual Phase at η=0,6
-100
Actual Phase at η=0,75
Actual Phase at η=0,9
-150
Reconst. Phase at η=0,15
-200
Reconst. Phase at η=0,30
Reconst. Phase at η=0,45
-250
Reconst. Phase at η=0,6
-300
Reconst. Phase at η=0,75
Reconst. Phase at η=0,9
-350
-2
-1
0
1
2
% n patch variation
Figure 6 Comparison of the Actual Data and the Reconstructed Data for Reflection Phase Variations w.r.t patch
variation using the different iteration factors
5. Conclusion
It can be concluded that the GRNN works as an accurate and effective interpolator of the complex
relationships among geometrical parameters of the Minkowski radiator and the phase of the reflected field, making this
technique suitable to be applied for the global optimization of both the phase behavior of the unit cell and the overall
far-field pattern of reflectarray antennas with together the algorithms.
6. References
1. J. Huang, and A. Encinar, Reflectarray Antennas, John Wiley & Sons Inc., Hoboken, NJ, 2007.
2. F. Zubir, M. K. A. Rahim, O. Ayop, A. Wahid and H. A. Majid, “Design and Analysis of Microstrip Reflectarray
Antenna with Minkowski Shape Radiating Element,” Progress In Electromagnetics Research B, Vol.24, 317-331, 2010.
3. D. Caputo, A. Pirisi, M. Mussetta, A. Freni, P. Pirinoli, R.E. Zich, “Neural Network Characterization of
Microstrip Patches for Reflectarray Optimization,” Proceedings of the European Conference on Antennas and
Propagation, EuCAP, 23-27 March 2009, pp 2520-2522.
4. D. F. Specht, “A general regression neural network,” IEEE Trans. Neural Netw. 2 (6) (1991) 568–576.
5. E.A. Nadaraya, “On estimating regression,”Theory Probab. Appl. 10 (1964) 186–190.
6. G.S. Watson, Smooth regression analysis, Sankhya, Ser. A 26 (4)
7. Tsoukalas, L. H., and R. E., Uhrig, 1997. Fuzzy and Neural Approache in Engineering: New York, John Wiley and
Sons, Inc., 87p.
8. H. Schioler, U. Hartmann, “Mapping neural network derived from the Parzen window estimator”, Neural Netw. 5 (6)
(1992) 903–909.1964) 359–372