a pod is at ion
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Project on Non-linear Apodisation
Techniques
for
Sidelobe Suppression
Submitted by
Satam Choudhury
Electronics and Communication Engg.
National Institute of Technology, Silchar
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Acknowledgement
I would like to express my heartfelt gratitude for my supervisor, Dr. Amit Kumar
Mishra, Dept. of Electronics and Communications Engg., Indian Institute of Technology,
Guwahati, for guiding me through my summer project on Non-linear Apodisation
Techniques for Sidelobe Suppression. My special thanks to Mr. Rajiv Panigrahi, Dept. of
Electronics and Communications Engg., Indian Institute of Technology, Guwahati ,for
constantly clarifying my doubts and also guiding my project work.
I am beholden to the Ms. Jharana Rabha of System Simulation Laboratory for
providing me with the necessary facilities for accomplishing the project.
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We have considered a sinusoidal signal comprising of 3 components, 2, 2.5 and 5 kHz
and the signal is sampled at a rate 1.5 times the Nyquist rate. The spectrum analysis is done
using the following techniques:
Dual Apodisation
fig.1
fig.2
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The two versions of the image are the IPRs shown in Fig 1. and Fig 2. (Hanning). The
resultant dual-apodized IPR, shown below has the narrow mainlobe of the sinc IPR and the
small sidelobes of the Hanning IPR. The combination of a uniformly weighted image with a
normalized version of the same image computed with any weighting function chosen to
produce low sidelobes is the basic principle behind Dual apodisation. The combination
technique uses a min function applied to the detected image on a pixel-by-pixel basis.
Applied to complicated images, DA results in an effective weighting function that can differ
from sample to sample in the image. Thus it results in a weighting function and an IPR that
are both nonlinear and spatially varying.
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Complex Dual Apodisation
To make use of the complex values we operate on the real (I or in-phase) and
imaginary (Q or quadrature) components of the undetected images separately. If the value
of a component has a sign that is different for the sinc IPR than for the Hanning IPR, then
there must be some weighting function intermediate between uniform weighting and
Hanning weighting for which the value of the component would be zero.
CDA provides significantly lower sidelobes than DA for the one-dimensional case (1-D).
SVA
We shall now discuss spatially variant apodization (SVA), which allows each pixel in an
image to receive its own frequency domain aperture amplitude weighting function from a
continuum of possible weighting functions. SVA effectively eliminates finite-aperture induced
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sidelobes from uniformly weighted SAR data while retaining nearly all of the good mainlobe
resolution and clutter texture of the unweighted SAR image.
(simulated at 2-times Nyquist frequency)
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(simulated at 1.5 times Nyquist frequency)
We observe that for non-integer multiples of Nyquist sampling frequency, the
resolution is not accurate and the sidelobe suppression is not complete.
There is another important limitation of Spatially Variant Apodisation. At higher
smapling rates, the mainlobe resolution is not good and the sidelobes are not suppressed as
well.
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However the proposed algorithm intends to address this problem of sidelobe
suppression at non-integer multiples of Nyquist sampling frequency.
1. At first, the zero crossing points are determined. For every three consecutive zerocrossing points, there is a maxima( appositive maxima) and a minima(a negative
minima). These maxima are determined for a uniform weighting function. The same
process is carried out for two or more windowed-weighted functions and the points of
maxima are determined. The maxima points are compared and those points which
occur in the all the weighted functions are retained while other maxima( the
sidelobes) are ignored.
2. A particular case may occur when two closely placed frequency components overlapeach other. This may be illustrated as follows:
The following signal is composed of 6 frequency components: 2, 2.9, 3.5,3.8,4, 5 kHz
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In this case we observe that there is overlap of two closely spaced frequencycomponents. It is obvious that in between zero-crossing points there is more than 1 maxima.
So we need to determine the total positive minima in that interval and subdivide the interval
into [zero-crossing point 1, 1st
+ve minima], (1st
+ve minima, 2nd
+ve minima](nth +ve
minima, zero crossing point 2]. The same process is repeated for other weighted window
functions and the maxima are compared. This process can help us in considerably resolving
closely spaced frequency components.
Program for Uniform Weighting, Hanning weighting and Dual apodisation
clc;
clear all;
close all;
N= 256;
l = 64;%length of seq
t=sqrt(-1);
n = 0:1:l-1;
eps=power(10,-10);
%frequency components
F1 = 5000;%in Hz
F2 = 2000;
F3= 2500;
Fs = 3*F1; %sampling freq
f1 = F1/Fs;
f2= F2/Fs;
f3= F3/Fs;
%given signal
x = sin(2*pi*f1*n)+sin(2*pi*f2*n)+sin(2*pi*f3*n);
f= (-Fs/2):(Fs/N):(Fs/2-(Fs/N));%range
x1= [x zeros(1,N-l)];%zero padding
y1= fftshift(fft(x1));
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m1= abs(y1); figure(1);
plot(f,m1);grid on;title('Uniform weighting'); ylabel('Amplitude');
%windoww= (hann(l))';
x2= x.*w;
x3= [x2 zeros(1,N-l)];
%fft of x1 and x2
y3= fftshift(fft(x3));
m3= abs(y3);
figure(2);plot(f,m3);grid on;title('Hanning Weighting'); xlabel('frequency in
Hz');ylabel('Amplitude');
%DA
for k= 1:N
mag1(k)=min(m1(k),m3(k));end
figure(3);plot(f,mag1);grid on;title('DA'); xlabel('frequency in
Hz');ylabel('Amplitude');
Program for Complex Dual Apodisation
clc;
clear all;
close all;
N= 256;
l = 64;%length of seq
t=sqrt(-1);
n = 0:1:l-1;
eps=power(10,-10);
%frequency components
F1 = 5000;%in Hz
F2 = 2000;
F3= 2500;
Fs = 3*F1; %sampling freq
f1 = F1/Fs;
f2= F2/Fs;f3= F3/Fs;
%given signal
x = sin(2*pi*f1*n)+sin(2*pi*f2*n)+sin(2*pi*f3*n);
f= (-Fs/2):(Fs/N):(Fs/2-(Fs/N));%range
x1= [x zeros(1,N-l)];
y1= fftshift(fft(x1));
m1= abs(y1);
w= (hann(l))';
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x2= x.*w;
x2= [x2 zeros(1,N-l)];
y2= fftshift(fft(x2));
m2= abs(y2);
I1=real(y1);
I2=real(y2);
Q1=imag(y1);
Q2=imag(y2);
I=zeros(1,N);
Q=zeros(1,N);
%I-Q indepenent comparison
for k = 1:N
if(sign(I1(k))~=sign(I2(k)))
I(k)=0;
else
I(k)=min(abs(I1(k)),abs(I2(k)));end
if(sign(Q1(k))~=sign(Q2(k)))
Q(k)=0;
else
Q(k)=min(abs(Q1(k)),abs(Q2(k)));
end
end
mag=I+t*Q;
figure(3);plot(f,abs(mag),'r');grid on;title('CDA'); xlabel('frequency in Hz');
Program for Spatially Variant Apodisation
clc;clear all
close allN= 256;l = 64; %length of seqt=sqrt(-1);n = 0:1:l-1;
%frequency componentsF1 = 2000;F2 = 5000;
F3= 3000;
Fs = 4*F2; %sampling freqf1 = F1/Fs;f2 = F2/Fs;f3= F3/Fs;
xn = sin(2*pi*f1*n)+sin(2*pi*f2*n);
f= (-Fs/2):(Fs/N):(Fs/2-(Fs/N));%range
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x1= [xn zeros(1,N-l)];%zero paddingy= fftshift(fft(x1));
p=sign(angle(y));
y1=p.*y;I1=real(y);Q1=imag(y1);n1=length(y1);I=zeros(1,N);Q=zeros(1,N);w1= zeros(1,N);w2=zeros(1,N);
for k=3:N-2w1(k)=-I1(k)/(I1(k-2)+I1(k+2)+eps);
if(w1(k)
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