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Low Sidelobe Pattern Synthesis Using Iterative Fourier Techniques Coded in MATLAB
Will P.M.N. Keizer
Clinckenburgh 32, 2343JH, Oegstgeest, Netherlands
Tel: +31 713011249; E-mail: [email protected]
Abstract - Theiterative Fourier technique for the synthesis of low sidelobe patterns for linear
arrays with uniform element spacing is described. The method uses the property that for a
linear array with uniform 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. A brief outline of the
iterative Fourier technique for the synthesis of low sidelobe patterns for linear arrays method
will be given. The effectiveness of this method to realize low sidelobe sum and difference
patterns will be demonstrated for linear arrays equipped with 50 and 80 elements. This
demonstration of effectivity involves also the recovery of the original low sidelobe patterns asclose as possible in case of element failures. Included is a program listing of this synthesis
method coded inMATLAB. With a few minor modifications/additions the included
MATLAB program can also be used for the design of thinned linear arrays having a periodic
element spacing [6]. Since the computational part of the included MATLAB program is coded
as vector/matrix operations, this program can easily be extended for the synthesis of low
sidelobe patterns of planar arrays with a periodic element spacing including pattern recovery
in case of defective elements [1]-[2].
Keywords: antenna arrays, low sidelobe synthesis, FFT, MATLAB
1. INTRODUCTIONRecently several papers have been published about the iterative Fourier technique (IFT) used
for synthesis of low side patterns [1[-[2]. The results presented in both papers demonstrated
that this method is particular suited for the large arrays with periodic spacing of the array
elements and also that the IFT method is quite able to handle various design constraints
related to both the radiation domain as well as the aperture domain. The design examples
given in [1] showed that sum patterns and difference patterns featuring ultra low sidelobes can
be combined with multiple ultra low level sector nulling. These examples covered amplitude-
only weighting and complex weighting of the aperture illumination. In [2] synthesis resultsbased on the IFT method were presented for a large planar array corrupted by defective
elements. It was shown that for each of the three monopulse patterns a very acceptable
recovery of the original very low sidelobe performance was possible when the array was
corrupted by a large number of defective elements.
This paper deals with the synthesis of low sidelobe sum and difference patterns for periodic
linear arrays using the IFT method. A description of the IFT pattern synthesis method will be
given including a listing of the MATLAB code that was used to get the results presented in
this paper. It will be demonstrated that using the IFT method a partial or even complete
recovery of the original sidelobe performance in case the array suffers from defective
elements, is feasible.
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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. At each iteration, the newly calculated AF is adapted to the
sidelobe requirements, which then is used to derive a new set of excitation coefficients. Onlythose excitation coefficients belonging to the array are used to calculate a new AF.
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).
The presented results are related to linear arrays consisting of 50 elements and 80 elements
located in periodic grid with half-wavelength inter-element spacing and comprise sum and
difference patterns.
2. FORMULATION OF THE IFT METHOD FOR LOW SIDELOBE PATTERNSYNTHESIS
The far-field F(u) of a linear array withMelements arranged along a periodic grid at distance
dapart, can be written as the product of the embedded element patternEFand the array factor
AF = (1)
= 1
=0(2)
whereAm is the complex excitation of the mth element, kis wavenumber (2/), is thewavelength, u = sinand angular coordinate measuredbetween far-field direction and the
array normal.
Equation (2) forms a finite Fourier series that relates the element excitation coefficientsAmof
the array to its AF through a discrete inverse Fourier transform. AF is periodic in u-dimension
over the interval d/. Since AF is related to the to the element excitations through a discrete
inverse Fourier transform, a discrete direct Fourier transform applied on AF over the period
/dwill yield the element excitationsAm. These Fourier transform relationships are used in an
iterative way to synthesize low sidelobe pattern for arrays with a periodic element
arrangement.
The synthesis procedure starts with the calculation of AF using an initial set for theMelementexcitation coefficients. The calculation of AF is carried out with a K-point inverse FFT with
K>Mand using zero padding. This is followed by an adaptation of the sidelobe region of AF
to the sidelobe requirements. Only the sidelobe samples of AF that exceed the sidelobe
thresholds are corrected, the other samples of AF are left unchanged. After this correction, a
direct K-point FFT is performed on the adapted AF to get an updated set of excitation
coefficients. From those Kexcitation coefficients only theMsamples belonging to the array
are retained. When an amplitude-only synthesis has to be performed, the phases of the
retained M excitation coefficients are made equal to the phases of the initial element
excitations at the start of the synthesis after which a new updated AF will be calculated. This
process is repeated until the new updated AF will satisfy the sidelobe requirements.
Constraints for the element excitations can be quite easily incorporated in the IFT synthesis
method. Dynamic range constraints for the array illumination are included by setting a lower
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limit to the amplitude values of the updated element excitation coefficients at each iteration.
The occurrence of defective elements across the aperture can be included by setting the
associated excitation coefficients to zero. The inclusion of defective elements in the synthesis
provides the possibility to restore the original sidelobe performance as close as possible. Both
element coefficient constraints are executed near the end of each iteration just before the next
iteration starts with the calculation of a new array factor.
Implementation of the IFT algorithm for the synthesis of low sidelobe patterns for linear
arrays using amplitude-only element weighting proceeds as follows.
1. Start the synthesis using a uniform excitation for M elements in case of the sum patternand an odd linear taper when the synthesis involves the difference pattern.
2. Compute AF from {Am} using a K-point inverse FFT with K>M.3. Adapt AF to the prescribed sidelobe constraints.4. Compute {Am} for the adapted AF using a K-point 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 bysetting their excitation values to zero.
9. Repeat Steps 2-9 until the prescribed sidelobe requirements for AF are satisfied or theallowed number of iterations is reached.
The above algorithm refers to the amplitude-only low sidelobe synthesis. In order to apply
phase-only synthesis the present Step 6 has to be replaced by: "Make the amplitude of the M
samples of {Am} equal to one". When Step 6 is deleted, the synthesis is of the complex weight
type.
Any low sidelobe AF consisting ofKsamples can be realized with Kelement excitations. The
objective of the IFT low sidelobe synthesis method is to arrange that theMelements of the
array take completely over the contribution to AF of the excitations of the (K-M) elements
located outside the aperture. This means that when the synthesis is successful when the
excitations of(K-M) elements outside the aperture have become zero.
The IFT method is implemented in theMATLAB program SidelobeSynthesis. This program
determines for a uniform linear array the element excitation coefficients producing an array
factor that matches the user defined peak sidelobe requirements. The capabilities ofSidelobeSynthesis are demonstrated by the results obtained for an 80-element array and a 50-
element array. These results include both sum and difference patterns. Also pattern recovery
in case of element failures is addressed for both types of far-field patterns. The pattern
recovery examples refer to the 50-element array having 8 failed elements.
3. EXAMPLESThe first example concerns a linear array consisting of 80 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 SidelobeSynthesis both featuring a -45 dB maximumpeak sidelobe level. The dynamic range of the amplitude of element excitations, defined as
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the ratio of its maximum value to its minimum value |Amax|/|Amin|, was not allowed to exceed
21 dB. Fig.1 shows the results of the synthesis for the sum pattern and Fig. 2 those of the
difference pattern. The synthesis was carried out with 4096-point direct and inverse FFTs.
Fig. 1a depicts the low sidelobe sum pattern with a maximum peak sidelobe level (SLL) of -
45.02 dB. The peak levels of the sidelobes are not uniform but fall off for the far-out
sidelobes. The result of Fig. 1a was obtained after 73 iterations as can be seen from Fig.1b.The same figure shows how the maximum peak SLL decreases during the iteration
process. Fig. 1c depicts the sum taper that is responsible for the pattern of Fig. 1a. In this
figure one can see that the element excitations obey the dynamic range requirement of 21 dB.
Fig. 1d illustrates how the number of far-field directions of the sidelobe region exceeding the
SLL requirement of -45 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. 1d, is a
direct measure of its width.
As can be seen from Fig. 2a the maximum peak SLL of the difference pattern is also -45.02
dB. To obtain this result, 94 iterations were required. Information about the decrease of the
maximum peak SLL as function of the iteration number is shown in Fig. 2b.Fig. 2a provides also numerical information about the taper efficiency and the relative angle
sensitivity Kr[3]. The relative angle sensitivity qualifies the difference slope and has a
maximum value of one for the odd linear taper, which features the steepest difference slope
but suffers from the highest sidelobe level.
When the dynamic amplitude range of the element excitations is increased from 21 dB to 24
dB, then the low sidelobe synthesis reveals uniform -45 dB peak sidelobes for the sum
pattern, see Fig. 3a, a result that was obtained after 50 iterations, Fig. 3b. The larger dynamic
range for the taper has no influence on taper efficiency; it provides equal level peak sidelobes
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. 4a and on the number of iterations.
The second array to be considered is a 50-element linear array having a 0.5uniform 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 sidelobe synthesis is -21 dB for
both patterns. Fig. 5 shows the synthesized results for the sum pattern and Fig. 6 those for the
difference pattern. The sum pattern with maxim peak SLL of -21.19 dB matches the SLL
requirement of -21 dB after only three iterations. For the difference pattern 10 iterations were
needed to get a maximum peak SLL of -21.26 dB.
The same 50-element linear array is described in [4]. But instead of using amplitude-onlysynthesis, the authors applied complex weighting to get a maximum peak SLL of -21 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 [4] only the sum pattern was considered and the synthesis of this
pattern required about 1000 iterations to get a maximum peak SLL of -21 dB. The synthesis
method in [4], 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 [4] is different from that of the IFT method.
Reference [4] describes the recovery of the original sidelobe performance of the sum patternwhen the array is corrupted by defective elements. In [4] the elements with the indices 8, 20,
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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 -21 dB to -11dB. Pattern recovery in [4] revealed a maximum peak
SLL performance of -14 dB.
Fig. 7a shows the effect of the same 8 defective elements on the synthesized sum pattern of
Fig. 5a. As can be seen the original maximum peak SLL has increased to -11.20 dB and thetaper efficiency has decreased to 0.804. Fig. 7b shows the corresponding taper corrupted by
the 8 defective elements.
Fig. 8 shows the effect of the 8 defective elements on the difference pattern results of Fig. 6.
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 -17.33
dB.
Pattern recovery in case of element failures has also been performed with the IFT method
using the same 8 defective elements as in [4]. Fig. 9 shows the result of the recovery for the
sum pattern of the 50-element 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.9a demonstrates that a maximum peak SLL of -16.63 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
32 iterations using amplitude-only weighting. The achieved maximum peak SLL of -16.63 dB
is more than 2 dB lower than the corresponding result of -14 dB realized for the recovered
pattern in [4].
Complete recovery of the difference pattern could be achieved by raising the dynamic range
from 6 dB to 8 dB. Fig. 10a shows that a maximum peak SLL of -21.29 dB is obtained after
only 8 iterations.
The data for the taper efficiency and relative angle sensitivity listed in Figs. 1-10 are
calculated using the equations given in [5] for these parameters.
The computer time to calculate the results was quite modest. The synthesis of the sum pattern
of Fig. 1a required only 0.11 sec CPU time and 0.14 sec was needed for the difference pattern
of Fig. 2a. 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 R2008b.
4.
CODING IN MATLAB
Fig. 11 lists the coding in MATLAB of the program SidelobeSynthesisthat is used to obtain
the results of Figs. 1-10. The computational part of the program is coded as matrix or vector
operations instead offor-loops to minimize computational times. The IFT method is
contained in the inner function lowSLLift, Lines 61-129 of the listing in Fig. 11.Input data used by the program is contained in the Lines 5-11. Lines 5-6 specify the array,
Line 7 the maximum peak SLL that may not to be exceeded and the Line 8 the dynamic range
of amplitude taper defined as the ratio of maximum amplitude and the minimum amplitude of
the array excitation coefficients.
The program assumes an even number of radiating elements.
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From Line 113 it can be noted that the adaptation of AF samples to the peak SLL requirement
uses a value that is 40 dB lower than the required peak SLL. Experimentally , it was found
that an adaptation with a much lower value than the requirement provides a substantial
reduction of the number of iterations of the IFT synthesis and furthermore it creates also
substantial deeper nulls for the sidelobes.
At each iteration, the program computes the width of the main beam to determine which far-
field directions of AF are contained in the sidelobe region. This computation is required since
the main beam width widens as the sidelobes are reduced. Due to this widening less sidelobes
samples of AF have to be checked and adapted to the sidelobe requirements (Step 3). The
main beam width computation involves the angular location of the first null at both sides of
the main lobe region, Lines 79-93. The indices of all AF samples located in the sidelobe
region and violating the peak level sidelobe requirement, are contained in the vector
indSynth, Line 100.The synthesis is successful when the number of sidelobes that exceed the sidelobe
requirement becomes zero. Checking for the zero value of this number occurs at Line 109.
The dynamic range requirement applied on the amplitude of the element excitations is
arranged by the vector operation of Line 123. The defective element constraint responsible for
the recovery of the original pattern as close as possible, involves also one vector operation,
Line 125. To be become effective, this line has to be uncommented (= deleting the comment
symbol %).
The statements of Lines 135-149 are responsible for the computation of the taper efficiency
and the relative angle sensitivity. It must be noted that these calculations are only valid when
the element spacing is 0.5 wavelengths.
5. CONCLUSIONWith the examples presented in this paper is demonstrated that the iterative Fourier technique
is ideally suited for the synthesis of low sidelobe 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 and
earlier by those of [1]-[2]. The IFT method features a very high computational speed, is robust
and can be quite simply implemented in MATLAB. With a few minor modifications the IFT
method can also be used for the design of thinned arrays having periodic element spacing [6].
The IFT method is not a new method; it was already described more than 20 years ago in [7].
This paper came to the attention of this author near the end of the year 2007 when he was
investigating the possibility to use non-uniform FFTs for low sidelobe pattern synthesis of
aperiodic arrays. Reference [8] that deals with non-uniform FFTs, provides a description of
the IFT method for the synthesis antenna patterns developed by C. Carroll and B. Vijaya
Kumar [7]. A few years after the publication of [7] Bucci et al. [9] presented the intersection
approach for array synthesis, an iterative method that uses forward and backward
transformation applied to the aperture domain and aperture domain. These forward and
backward transformations, which can be FFTs, are operated in the same way as the FFTs with
the IFT method. The method used in [4] is based on the approach described in [9] and makes
use of forward and backward FFTs.
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However, even Carroll and Vijaya Kumar were not the first to have proposed the IFT method
for antenna applications. The origin of the IFT method goes back to the near-field alignment
of phased array antennas which technique was introduced around 1980. The way the IFT
method is used for low-sidelobe synthesis is comparable to the near-field alignment of phased
array antennas by which the computed far-field derived from planar a near-field measurement
is transformed back to the aperture plane. This method is used in an iterative way to negatethe uncertainty of the back projected field that is caused by the truncation (Gibb's
phenomenon) of the far-field plane wave spectrum to the part located in visible space. This
truncation is necessary to filter out the contribution of evanescent modes, located in invisible
space, to the aperture field. Using the reconstructed radiating aperture phase and amplitude
data from the backward transformation, the phase and amplitude settings of the T/R modules
are adjusted followed by a new near-field measurement resulting in a new far-field
computation. This new computed far-field is anew subjected to a backward transformation to
get more precise phase and amplitude settings for the T/R modules. By repeating this process
a few times the remaining phase and amplitude errors of the array elements at the end of the
phased array alignment are then of the order of the residual uncertainty caused by the
backward transformation of the incomplete plane wave spectrum. This type of phased arrayalignment makes us of inverse and direct FFTs in an identical way as is done with the IFT
method.
The IFT method proposed in this paper is derived from the near-field phased array alignment
technique.
6. REFERENCES[1] W.P.M.N. Keizer, "Fast low sidelobe synthesis for large planar array antennas utilizing
successive fast Fourier transforms of the array factor,"IEEE Trans. Antennas
Propagat., vol. 55, no. 3, pp. 715-722, March 2007.
[2] W.P.M.N. Keizer, "Element failure correction for a large monopulse phased array
antenna with active amplitude weighting,"IEEE Trans. Antennas Propagat., vol. 55,
no. 8, pp. 2211-2218, August 2007.
[3] E.T. Bayliss, "Design of monopulse antenna difference patterns with low sidelobes,"
Bell Syst. Tech. J., pp. 623-650, May-June 1968.
[4] Y. Yang and H. Stark, "Design of self-healing arrays using vector-space projections,"
IEEE Trans. Antennas Propagat., vol. 49, no. 4, pp. 526-534, April 2001.
[5] D.K. Barton and H.E. Ward, "Handbook of radar measurement," Prentice-Hall Inc,
1969.[6] W.P.M.N. Keizer, "Linear array thinning using iterative Fourier techniques,"IEEE
Trans. Antennas Propagat., vol. 56, no. 8, pp. 2211-2218, August 2008.
[7] C.W. Carroll and B.V.K. Vijaya Kumar, "Iterative Fourier transform phased array radar
pattern synthesis," Proc. of SPIE, vol. 827, pp. 73-84, 1987.
[8] S. Bagchi and S.K. Mitra, "The nonuniform discrete Fourier transform and its
applications in signal processing," Kluwer Academic Publishers, Boston, 1999, pp. 153-
154.
[9] O.M. Bucci, G. Franceschetti, G. Mazzarella and G. Panariello, "Intersection approach
to array pattern synthesis," IEE Proc., vol. 137, Pt. H, no. 6, Dec. 1990, pp. 349-357.
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Fig. 1 Synthesized -45 dB SLL sum pattern realized for an 80 element linear array withdynamic range ofAmax/Amin = 21 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -45 dB SLL requirement and number of far-field directions contained
in the mainlobe region during the iteration process.
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Fig. 2 Synthesized -45 dB SLL difference pattern realized for an 80 element linear array withdynamic range ofAmax/Amin = 21 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -45 dB SLL requirement and number of far-field directions contained
in the two mainlobes during the iteration process.
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Fig. 3 Synthesized -45 dB SLL sum pattern realized for an 80 element linear array withdynamic range ofAmax/Amin = 24 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -45 dB SLL requirement and number of far-field directions contained
in the mainlobe region during the iteration process.
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Fig. 4 Synthesized -45 dB SLL difference pattern realized for an 80 element linear array withdynamic range ofAmax/Amin = 24 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -45 dB SLL requirement and number of far-field directions contained
in the two mainlobes during the iteration process.
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Fig. 5 Synthesized -21 dB SLL sum pattern realized for a 50 element linear array withdynamic range ofAmax/Amin = 6 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -21 dB SLL requirement and number of far-field directions contained
in the mainlobe region during the iteration process.
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Fig. 6 Synthesized -21 dB SLL difference pattern realized for a 50 element linear array with
dynamic range ofAmax/Amin = 6 dB for the taper. The result is obtained by amplitude-only
synthesis.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d)
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Fig. 7 Sum pattern of Fig. 1 when corrupted by 8 defective elements with indices 8, 20, 23,
24, 25, 26, 27, 29. (a) Normalized array factor. (b) Sum taper with 8 defective (non-radiating)
elements.
Fig. 8 Difference pattern of Fig. 2 when corrupted by 8 defective elements indices 8, 20, 23,
24, 25, 26, 27, 29. (a) Normalized array factor. (b) Difference taper with 8 defective (non-
radiating) elements.
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Fig. 9 Recovery of the sum pattern of Fig. 7 corrupted by 8 defective elements. Thedynamic range of the taperAmax/Amin was raised from 6 dB to 18 dB.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iterative
synthesis. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -16.50 dB SLL requirement and number of far-field directions
contained in the mainlobe region during the iteration process.
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Fig. 10 Recovery of the difference pattern of Fig. 8 corrupted by defective 8 elements. Thedynamic range of the taperAmax/Amin was increased from 6 to 8dB.
(a) Normalized array factor. (b) Behaviour of the maximum peak SLL during the iteration
process. (c) Amplitude taper responsible for the array factor of (a). (d) Number of far-field
directions violating the -21 dB SLL requirement and number of far-field directions contained
in the two mainlobes during the iteration process.
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Fig. 11 Listing of the MATLAB function SidelobeSynthesis
function SidelobeSynthesis2
% ------------------ Start Input Parameters -------------------------------4
noEl = 80; % # of array elements, only even number!dx = 0.5; % Normalized element spacing6specSLL = -45; % Required SLL (dB)
ratioIllum = 21; % Ratio maximum/minimum value Illumination (dB)8noIter = 130; % Maximum # of iterations10noFF = 4096; % # of FF directions
12% ------------------ End Input Parameters ---------------------------------
14u = (-noFF/2:noFF/2-1)/dx/noFF; % u-coordinates FF
16patternType = {'Sum Pattern';'Difference Pattern'};
18IllumEven = ones(1,noEl); % Uniform illuminationIllumOdd = 2*(0:noEl-1)/(noEl-1) - 1; % Odd linear illumination20minIllum = 10^(-ratioIllum/20); % Minimum value illumination
22monResults(1:noIter,1:4) = 0; % Container for storing
% intermediate synthesis results24maxAF = 1;
26for imk = 1:2
28if imk==1
30IllumInit = IllumEven; % Initial illumination sum pattern at start
32indPat = 0;
34else
36
IllumInit = IllumOdd; % Initial illumination difference pattern38indPat = 1;
40end
42IllumS = IllumInit;
44Phase = exp(1i*(angle(IllumInit)));
46tic
48[Illum, AFabs] = lowSLLift;
50toc
52plotGraphics(Illum./max(abs(Illum)))
54monResults(1:iCount,1:end) = 0; % Restore initial state
56end
58% ------ Low sidelobe synthesis using iterative FTT technique ---------
60function [Illum, AFabs]= lowSLLift
62for ikk = 1:noIter
64Illum = IllumS;
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Fig. 11 Listing of the MATLAB function SidelobeSynthesis
66AF = ifftshift(ifft(Illum,noFF)); % AF computed by FFT
68AFabs = abs(AF); % Absolute value AF
70maxAF = max(AFabs);
72
AFabs = AFabs/maxAF; % Normalize |AF|74AF = AF/maxAF; % Normalize AF (complex)
76% ------- Find all FF nulls --------------------------------------
78minVal = sign(diff([inf AFabs inf]));
80indMin = find(diff(minVal+(minVal==0))==2); % Indices FF nulls
82% -------- Find all FF peaks -------------------------------------
84indPeaks = find(diff(minVal) SLL requirement104
monResults(ikk,1:4) = [ikk noSLLdir max_SLL ...106(indMin(indNullL+1+indPat)-indMin(indNullL))];
108if noSLLdir==0, return, end
110% ----- Adapt AF to SLL constraints -----------------------------
112AF(indSLL(indSynth)) = 10^((specSLL-40)/20);
114IllumS = fft(ifftshift(AF)); % Illumination derived from AF
116% ---- Adapt Illumination to aperture domain constraints ---------
118IllumS = abs(IllumS(1:noEl)); % Truncate # elements illumation
120IllumS = IllumS/max(IllumS);
122IllumS(IllumS
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Fig. 11 Listing of the MATLAB function SidelobeSynthesis
end132
function plotGraphics(Illum)134
AF = 20*log10(AFabs);136
maxAFunif = max(abs(ifftshift(ifft(IllumEven,noFF))));
138 taperEff = ((maxAF/maxAFunif)^2)*sum(abs(IllumEven).^2)/...sum(abs(Illum).^2);140
Kr = 0;142
if indPat ==1 % Difference pattern144
coordX = ((0:noEl-1) - (noEl-1)/2)/(noEl-1); % Position array elements146
% Relative angle sensitivity148
Kr = abs(sum(coordX.*Illum))/sqrt(sum(abs(Illum).* ...abs(Illum))*sum(coordX.*coordX));150
end152
fonts = 11;154
iCount = max(monResults(:,1));156
hFig = figure('Tag','Array');158
aX1 = axes('position',[.07 .58 .35 .35],'Units','Normalized');plot(u,AF,'Parent',aX1);160
set(aX1,'Ylim',[specSLL-20 0])162xlabel('u','Fontsize',fonts+1);ylabel('Normalized Far Field (dB)','Fontsize',fonts+1)164title([patternType{1+indPat},' ', num2str(noEl), ...
' Elements Linear Array'],'Fontsize',fonts+1)166
txtStr{1} = {['element spacing = ',sprintf('%3.1f',dx), ' \lambda'];...168'isotropic element pattern';...
['taper efficiency = ',sprintf('%5.3f',taperEff)];...170['max SLL = ',sprintf('%6.2f',monResults(iCount,3)),' dB'];...['SLL Req. = ',sprintf('%6.2f',specSLL),' dB']};172
txtStr{2} = {['element spacing = ',sprintf('%3.1f',dx), ' \lambda'];...174'isotropic element pattern';...
['taper efficiency = ',sprintf('%5.3f',taperEff)];...176['K_{r} = ',sprintf('%5.3f',Kr)];...['max SLL = ',sprintf('%6.2f',monResults(iCount,3)),' dB']};178
text(0.66,0.84,txtStr{1+indPat},'Parent',aX1,'Units','Normalized',...180'Fontsize',fonts-1);
182aX2 = axes('position',[.07 .1 .35 .35],'Units','Normalized');plot(20*log10(abs(Illum)),'Parent',aX2);184set(aX2,'XLim',[1 noEl])ylabel('Magnitude (dB)','Fontsize',fonts+1)186xlabel('Element Index','Fontsize',fonts+1)title(['Illumination ',patternType{1+indPat}, ' ',num2str(noEl),...188
' Elements Linear Array'],'Fontsize',fonts+1)190
aX3 = axes('position',[.53 .58 .35 .35],'Units','Normalized');plot(monResults(1:iCount,1),monResults(1:iCount,3),'Parent',aX3);192ylabel('Max. SLL (dB)','Fontsize',fonts+1)xlabel('# of Iterations','Fontsize',fonts+1)194set(aX3,'XLim',[1 iCount])
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Fig. 11 Listing of the MATLAB function SidelobeSynthesis
title('Progress max SLL during Synthesis','Fontsize',fonts+1)196text(0.66,0.90,['# of iterations = ',sprintf('%5.0f',iCount)], ...
'Parent',aX3,'Units','Normalized','Fontsize',fonts-1);198
axes('position',[.53 .1 .35 .35],'Parent',hFig);200Hndl = plotyy(monResults(1:iCount,1),monResults(1:iCount,2), ...
monResults(1:iCount,1),monResults(1:iCount,4));202
ylabel(Hndl(1),'# of SL Directions violating SL Requirements', ...'Fontsize',fonts+1)204
xlabel(Hndl(1),'# of Iterations','Fontsize',fonts+1)set(Hndl(1),'XLim',[1 iCount])206ylabel(Hndl(2),'# of FF Directions contained in Mainlobe', ...
'Fontsize',fonts+1)208set(Hndl(2),'XLim',[1 iCount])end210
end