LinearArraySidelobeSynthesis.pdf

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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.




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.






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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.






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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.






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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.


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.






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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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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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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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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

LinearArraySidelobeSynthesis.pdf

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Transcript of LinearArraySidelobeSynthesis.pdf