International Journal of Engineering Trends and Technology (IJETT) – Volume 21 Number 5 – March 2015
Optimization of Reduction in Side Lobe Level
for Antenna Arrays using Iterative Fourier
Transform Technique
V.Karthik#1, T.Asha Rani*2, V.Praveen Kumar#3, V.Bhaskar Reddy*4, R.Divya Kanti#5
1234
5
Student, Department of ECE, Lendi Institute of Engineering and Technology, Vizianagaram, India
Assistant professor, Department of ECE, Lendi Institute of Engineering and Technology, Vizianagaram, India
Abstract-Recently, the Iterative Fourier transform
(IFT) method has been introduced for the synthesis of
low side lobe patterns for array antennas with periodic
element arrangement. The IFT method makes use of the
property that for an array with periodic element
spacing, an inverse Fourier transform relationship
exists between the array factor and the element
excitations. This property is used in an iterative way to
derive the array element excitations from the
prescribed array factor. This paper presents Iterative
Fourier Technique (IFT) for the synthesis of low Side
lobe patterns for linear array with uniform element
spacing. A comparative study is done with Array
Element failure is introduced at various location of
linear array in distributed and concentrated form.
Quantitative results are presented to show effectiveness
of the algorithms for failure correction in terms of
pattern recovery. The MATLAB simulation results are
presented for 100 to 150 elements arranged in regular
grid for antenna sum and difference pattern.
Index Terms—
Antenna arrays, low side lobe synthesis, IFT,
MATLAB
I.
INTRODUCTION
The antenna is the basic part or the back bone of
the wireless communication. It is mainly used as
device to transmit and receive signal. To get better
facility from this device it is necessary to modify or
synthesize the geometric configuration or the other
parameter of the device. This modification of the
parameter of the antenna for getting our desired
requirement is known as antenna synthesis. The
Radiation Pattern of an antenna array depends
strongly on the weighting method and the geometry
of the array[4].
Previously, there were some of the antenna pattern
synthesis such as:
1) Integral Equation Method
2) Geometry Theory of Diffraction
These are used in wired type antennas. Now-a-days
,as we have wide development in communication, so
we have chosen wireless method for synthesis i.e.,
Iterative Fourier Transform.
ISSN: 2231-5381
Recently several papers have been published about
the Iterative Fourier Transform (IFT) used for
synthesis of low side patterns. The results presented
in this paper demonstrated that this method is
particular suited for the large arrays with periodic
spacing of the array elements. The IFT method is
quite able to handle various design constraints related
to both the radiation as well as the aperture
domain[3].
The IFT approach uses the property that for an array
having a uniform spacing of the elements, an inverse
Fourier transform relationship exists between the
array factor (AF) and the element excitations.
Because of this relationship, a direct Fourier
transform performed on AF will yield the element
excitations. The underlying approach relies on the
repeatedly use of both types of Fourier transforms[3].
At each iteration, the newly calculated AF is
adapted to the side lobe requirements, which then is
used to derive a new set of excitation coefficients.
Only those excitation coefficients belonging to the
array are used to calculate a new Array Factor[3]. A
key characteristic of this iterative synthesis method is
that the algorithm itself is very simple, highly robust,
and very easy to implement in software requiring
only a few lines of code when programmed in
MATLAB as will be shown. The computational
speed is very high because the core calculations are
based on direct and inverse fast Fourier transforms
(FFTs)[2].
The presented results are related to linear arrays
consisting of 100 to 150 elements located in periodic
grid with half-wavelength inter-element spacing and
comprise sum and difference pattern[3].
II . ITERATIVE FOURIER TRANSFORM
The far-field F(u) of a linear array with M
elements arranged along a periodic grid at distance d
apart, can be written as the product of the embedded
element pattern EF and the array factor AF
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International Journal of Engineering Trends and Technology (IJETT) – Volume 21 Number 5 – March 2015
F(u)=EF(u)AF(u)
(1)
(2)
A(u)=
Where Am is the complex excitation of the mth
element, k is wave number (2π/λ), λ is the
wavelength, u = sinθ and θ angular coordinate
measured between far-field direction and the array
normal. Equation (2) forms a finite Fourier series that
relates the element excitation coefficients Am of the
array to its AF through a discrete inverse Fourier
transform. AF is periodic in u-dimension over the
interval d/λ. Since AF is related to the element
excitations through a discrete inverse Fourier
transform, a discrete direct Fourier transform applied
on AF over the period λ/d will yield the element
excitations Am. These Fourier transform relationships
are used in an iterative way to synthesize low side
lobe pattern for arrays with a periodic element
arrangement.
Implementation of the IFT algorithm for the synthesis
of low side lobe patterns for linear arrays using
amplitude only element weighting proceeds as
follows[5].
1. Start the synthesis with a uniform excitation for M
elements in case of the sum pattern.
2. Compute AF from {Am} using a K-point inverse
FFT with K>M.
3. Adapt AF to the prescribed side lobe constraints.
4. Compute {Am} for the adapted AF using a Kpoint direct FFT.
5. Truncate {Am} from K samples to M samples by
making zero all samples outside the array.
6. Make the phase of the M samples of {Am} equal
to the phase of initial excitation at Step 1.
7. Set the magnitude of the excitations violating the
amplitude dynamic range constraint to the lowest
permissible value.
8. Enforce the optional defective element constraint.
Take element failures into account by setting their
excitation values to zero.
9. Repeat Steps 2-9 until the prescribed side lobe
requirements for AF are satisfied or the allowed
number of iterations is reached.
The above algorithm refers to the amplitude only
synthesis[5].
III. SIMULATION RESULTS
A linear array consisting of 100 elements spaced
0.5 wavelength apart and characterized by an
isotropic embedded element pattern. For this array a
sum and difference pattern was synthesized using
Side lobe Synthesis both featuring a -40 dB
ISSN: 2231-5381
maximum peak side lobe level. The dynamic range of
the amplitude of element excitations, defined as the
ratio of its maximum value to its minimum value
|Amax|/|Amin|, was not allowed to exceed 19 dB.
Fig.1(a) shows the results of the synthesis for the sum
pattern and Fig.1(b) those of the difference pattern.
The synthesis was carried out with 4096-point direct
and inverse FFTs[3].
Fig.1(a) depicts the low side lobe sum pattern with
a maximum peak side lobe level (SLL) of -40 dB.
The peak levels of the side lobes are not uniform but
fall off for the far-out side lobes. The result of Fig(a)
was obtained after 29 iterations as can be seen from
Fig(b).The same figure shows how the maximum
peak SLL decreases during the iteration process.
Fig.1(c ) depicts the sum taper that is responsible for
the pattern of Fig.1(a). In this figure one can see that
the element excitations obey the dynamic range
requirement of 19 dB. Fig.1(d) illustrates how the
number of far-field directions of the side lobe region
exceeding the SLL requirement of -40 dB, decreases
with increasing iteration number. This figure shows
also the widening of the main beam due to the drop in
maximum peak SLL as the synthesis progresses. The
number of far-field directions contained in the main
lobe region, Fig.1(d), is a direct measure of its
width[3].
Fig.1 Sum Pattern of 100 element array
As can be seen from Fig. 2(a) the maximum peak
SLL of the difference pattern is also -40 dB. To
obtain this result, 29 iterations were required.
Information about the decrease of the maximum peak
SLL as function of the iteration number is shown in
Fig. 2(b). Fig. 2(a) provides also numerical
information about the taper efficiency and the relative
angle sensitivity Kr . The relative angle sensitivity
qualifies the difference slope and has a maximum
value of one for the odd linear taper, which features
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International Journal of Engineering Trends and Technology (IJETT) – Volume 21 Number 5 – March 2015
the steepest difference slope but suffers from the
highest side lobe level[3] .
When the dynamic amplitude range of the element
excitations is increased from 19 dB to 24 dB, then the
low side lobe synthesis reveals uniform -40 dB peak
side lobes for the sum pattern. The larger dynamic
range for the taper has no influence on taper
efficiency; it provides equal level peak side lobes to
the array factor. The 3 dB larger dynamic range for
the difference taper has only a marginal effect on the
difference pattern as can be noted from Fig. 2(a) and
on the number of iterations[3].
Fig.3 Sum Pattern of 150 element array
For the difference pattern 28 iterations were needed
to get a maximum peak SLL of -40dB. The synthesis
method in , based on vector-space projections, has
strong similarities with the IFT method since both
methods rely on the use of forward and backward
Fourier transformations between the radiation domain
and the aperture domain. However the way element
constraints are implemented in the method of is
different from that of the IFT method[3].
Fig.2 Difference Pattern of 100 element array
The second array to be considered is a 150-element
linear array having a 0.5λ uniform inter-element
spacing and an isotropic embedded element pattern
for the elements. The magnitude of the taper for both
the sum and difference pattern was allowed to vary
over range less than 6 dB. The peak SLL requirement
for the amplitude-only low side lobe synthesis is -40
dB for both patterns. Fig. 3(a) shows the synthesized
results for the sum pattern. The sum pattern with
maximum peak SLL of -40 dB matches the SLL
requirement of -40 dB after only three iterations. But
instead of using amplitude-only synthesis, the authors
applied complex weighting to get a maximum peak
SLL of -19 dB. A second objective was that the width
of the main beam of the synthesized pattern must be
equal to that of the pattern produced by the uniform
sum taper. There were no requirements for the
dynamic range of taper. In only the sum pattern was
considered and the synthesis of this pattern required
about 1000 iterations to get a maximum peak SLL of
-19 dB[3].
ISSN: 2231-5381
Fig.4 Difference Pattern of 150 element array
The recovery of the original side lobe performance
of the sum pattern when the array is corrupted by
defective elements. In the elements with the indices
8, 20, 5, 23, 24, 25, 26, 27, 29 were made inoperable
by setting the amplitudes of their excitation
coefficients to zero. Due to the occurrence of these
defective elements the maximum peak SLL was
degraded from -19 dB to -11dB. Pattern recovery in
revealed a maximum peak SLL performance of -14
dB. The rise in maximum peak SLL due to the
occurrence of defective elements is for the difference
pattern smaller as for the sum pattern resulting in
maximum peak SLL of -19 dB[3].
Pattern recovery in case of element failures has
also been performed with the IFT method using the
same 8 defective elements shows the result of the
recovery for the sum pattern of the 150-element
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International Journal of Engineering Trends and Technology (IJETT) – Volume 21 Number 5 – March 2015
array. A better maximum peak SLL could only be
obtained by allowing a larger dynamic range for the
amplitude of the element excitation coefficients. Fig.
3(a) demonstrates that a maximum peak SLL of -19
dB is feasible for the sum pattern when the dynamic
range of 6 dB was raised to 18 dB. This recovered
sum pattern was obtained after 28 iterations using
amplitude-only weighting. The achieved maximum
peak SLL of -19 dB is more than 2 dB lower than the
corresponding result of -14 dB realized for the
recovered pattern[3].
Complete recovery of the difference pattern could
be achieved by raising the dynamic range from 6 dB
to 8 dB. Fig. 4(a) shows that a maximum peak SLL
of -19 dB is obtained after only 8 iterations[3].
Complete recovery of the difference pattern could
be achieved by raising the dynamic range from 6 dB
to 8 dB. Fig. 4(a) shows that a maximum peak SLL
of -19 dB is obtained after only 8 iterations modest.
The synthesis of the sum pattern of Fig. 3a required
only 1.359 sec CPU time and 0.103 sec was needed
for the difference pattern of Fig. 4a The computation
time of the implemented IFT method is directly
proportional to the number of iterations. The
computations were performed on a PC equipped with
an Intel Core 2 Quad Q6600 processor running at
2.40 GHz and using 2 GB of RAM memory. The
coding of the method was done in MATLAB
R2013b[3].
IV. CONCLUSION
From this paper we are concluding that the Iterative
Fourier technique is ideally suited for the synthesis of
low side lobe pattern for arrays with periodic element
spacing. The IFT method can deal with a wide range
of constraints both in the radiation domain as well as
in the aperture domain as has been proved by the
results of this paper. The IFT method features a very
high computational speed, is robust and can be quite
simply implemented in MATLAB.
MATLAB,‖ IEEE Antennas Propag. Mag.,vol. 51,
no. 2, pp. 137–150, Apr. 2009.
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