Our team had to use signal processing techniques to find the direction of desired incoming signals at an antenna rejecting interference from other signals.

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Direction Finding with Arrays in Matlab
By: Ryan Bull, Garrett Hoch, ​Surya Chandra, Yeshwanth Malekar
1.Introduction
The purpose of this project is to explore the different methods of array direction finding in
Matlab.
The direction finding methods that will be explored is the eigen beam method, periodogram,
Capon’s method, MUltiple SIgnal Classification (MUSIC), Maximum Entropy Method (MEM),
and Pisarenko Harmonic Decomposition (PHD). Determining the correlation matrix is a critical
first step for all of the methods listed above. For this reason, the process of calculating the
correlation matrix will be discussed in depth. By the end of this project the direction of an
incoming signal will be able to be determined with the above listed methods. The system will
be modeled after a uniform linear array with a varying number of elements.
2.Forming the Correlation Matrix
All of the methods described in this project first require calculating the correlation
matrix for the antenna array and incoming signal(s) system. The process for this is described
below. For a given antenna array, the array steering vector, A, is
where J is the number of elements in the array, and M is the number of incoming signals. A
gives the phase of incoming plane waves at each of the elements in the array. The signals
are in turn described by a vector,
where correspond to each input signal. The output of the array is therefore given bysM
where is the vector describing the noise inherent in the system. The covariance matrices(t)N
are then calculated for and by(t)X (t)N
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The covariance matrix can then manipulated depending on the method to be used to
determine the directions of each signal.
3. Method 1: Eigen Beam
The eigen beam method uses a simple eigenvalue decomposition to determine the directions
of the incoming signals. The covariance matrix can be decomposed by
where are the eigenvalues and is a matrix composed of the eigenvalues associatedλJ V λ
with each vector. There will be one eigenvalue for each input signal, while the other
eigenvalues correspond to the noise, and the total number of eigenvalues is the number of
elements in the array. The largest eigenvalues correspond to the signals. New array factors
are calculated for each eigenvalue, with the weights of the array factor given by the
eigenvector for the eigenvalue.
This was done in Matlab for a four­element uniform array with half­wavelength spacing. The
signal is a 1 V/m plane wave incident at 30° and ­50°. This gives eigenvalues and vectors of
When turned into weights for an array factor, the results are
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Figure 1: System with noise variance of .01
The signal array factor clearly has a maximum at 30° and ­50°, while the noise array factors
have nulls.
Figure 2: System with noise variance of .1
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The eigen beam method is clearly very resistant to noise. Changing the noise variance had
almost no change on the array factor.
4. Method 2: Periodogram
The periodogram uses the main beam of the array to determine the location of the signal.The
main beam is steered using beam steering across the desired angles. The received signal will
be the strongest when the main beam is aligned with it. This will give a peak in the power
received. The plot of the output power versus angle is a periodogram. The equations for the
periodogram are given below:
R is the same correlation matrix computed above. A gives the array steering vector. The
graph below shows an example that was produced in matlab. There are three different signal
shown here. The red line has has the signal incident at ­30 and 30 degrees. The nulls are
distinct in this case as the incident angles are far enough apart. The other two singal show
that as the angles of the incident signals becomes closer and closer the direction of the signal
is in accurate. The blue signal has the incoming signal at 10 and 30 degrees. The black line
has the signal incoming at 20 and 30 degrees, but it appears as if there is only a single signal
incoming at an angle of 25 degrees. This smearing of the signal is due to the beamwidth of
the main being large and not precise.
Figure 3: System with noise variance of .01
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Figure 4: System with noise variance of .1
The periodogram is very sensitive to noise. Increasing the noise greatly raised the noise floor.
With a little more noise the signals would not be seen at all. This is in addition to the
periodogram’s inability to differentiate between signals that are close together. These factors
combine to make the periodogram the least robust method of direction finding, but it is simple
to implement. The periodogram will be used to compare the effectiveness of all the other
methods described.
5. Method 3: Capon’s method
Unlike the periodogram method Capon’s uses nulls in the array pattern to detect the direction
of a signal. Capon’s method uses statistics, specifically, maximum likelihood estimate. This
method allows for the distinction of the estimated signal while lowering the magnitude of all
other sources. Each antenna in the array is weighted as to maximize the
signal­to­interference ratio. Once the array is weighted properly the equation below can be
used to calculate the direction of the signal.
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The graph below shows both the periodogram and Capon’s method. The signal has nulls at
0,10 and ­60 with the weights of 2, 4, and 1. The periodogram cannot distinguish between the
signals at 0 and 10 while Capon’s method can. The noise floor with Capon’s method is also
much lower than the periodogram method making the peaks more distinct.
Figure 5: System with noise variance of .01
Figure 6: System with noise variance of .1
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Capon’s method is fairly resistant to noise. However, it can be seen that the added noise
makes the difference between closely spaced peaks much smaller. With too much noise this
method would be unable to differentiate between the two close peaks.
6. Method 4: MUltiple SIgnal Classification (MUSIC)
MUSIC makes two major assumptions; noise is uncorrelated and the signals are uncorrelated
or only mildly correlated. This is one of the most popular and studied methods for determining
the direction of arrival of signals. The equation used is below:
where is again the matrix of the eigenvectors. The graph shows that there is even furtherV λ
refinement in the peaks of closely spaced incident angles. The noise floor is further lowered
as well. The weights of the signals do not affect the detection as much either. Even with the
better performance this method is not very robust.
Figure 7: System with noise variance of .01
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Figure 8: System with noise variance of .1
MUSIC is very resistant to noise. Increasing the noise variance had very little effect on the
plot. The two close peaks are still very easy to identify.
7. Method 5: Maximum Entropy Method (MEM)
The MEM method places poles at the location of the the incoming signal in a rational function
as shown below. With no zeros in the transfer function the peaks are very distinct. For
different values of n the results will differ. The magnitudes of the incoming signal is reversed
from the actual weights present of the incoming signal.
where n is the nth column of the covariance matrix.
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Figure 9: System with noise variance of .01
Figure 10: System with noise variance of .1
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MEM is fairly resistant to noise. The peaks are not as sharp with the increased noise, but they
are still all readily identifiable.
8. Method 6:Pisarenko Harmonic Decomposition (PHD)
The pisarenko Harmonic decomposition uses the smallest eigenvector to minimize the mean
squared error of the array output. The weights of the system are normalized to 1. Using the
equation below the following graph is obtained. It can be seen that the peaks of the incoming
signals are very sharp, but there are some erroneous peaks at other angles. The equation for
the PHD method is given below
where is the eigenvector associated with the smallest noise eigenvalue. This method is alsoe1
highly variable to the sampling rate. Sampling at different rates has dramatic effects on the
results, and these sampling issues are likely the cause of the extra peaks.
Figure 11: System with noise variance of .01
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Figure 12: System with noise variance of .1
Increasing the noise variance actually had the effect of removing one of the extraneous peaks
around 30 degrees. However, the other extraneous peak around ­40 degrees is much more
pronounced. The real peaks are still easily differentiated.
9. Conclusion
The goal of this project was to show multiple methods of direction finding or direction of arrival
methods for antenna arrays. Ideally some of these methods would have been implemented
using software defined radios. Due to time constraints this was not done. Instead, Matlab was
used to simulate an eight element array with three incoming signals. The basis for direction
finding is determining the correlation matrix. Once the correlation matrix is found the
implementation of any of the 6 methods is relatively simple.
There are trade offs for each method that will need to be considered when deciding
what method to use. The periodogram method is the most simple method but it cannot make
the distinction between closely spaced signals. The more complex methods: Eigen beams,
Capon’s, MUSIC, MEM, and PHD are more successful in interpreting close signals and lower
the noise floor of all other signals. They are also less sensitive to increased noise. The eigen
beam and MUSIC methods seem to be the most robust for noise response, followed by
Capon’s and MEM, and finally the PHD method. The PHD method is notable in that in
addition to its increased noise sensitivity, it also has issues with sampling creating extra
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peaks. For robust direction finding with multiple signals, either the eigen beam or MUSIC
methods should probably be used. If there is only one incoming signal, a simpler method like
the periodogram is sufficient.
References
Haupt, Randy L. ​Antenna Arrays: A Computational Approach​. Hoboken, NJ: Wiley­IEEE,
2010.
Haupt, Randy L. ​Direction Finding Arrays or Angle of Arrival Estimation​. Colorado School of
Mines Antennas Course Lecture. 2015.
Appendix A: Matlab Code
Eigen Beam
%% DATA
f = 2*10^9; % frequency
c = 3 *10^8; % speed of light
j= sqrt(­1);
lambda = c/f; % wave length
k = 2*pi/lambda; % wave number
d = lambda/2; % distance between adjacent array elements
% Locations of the linear array elements
x1= 0;
x2=x1 + d;
x3=x2 + d;
x4=x3 + d;
%%%%%THREE SETS OF DATA USED%%%%%
% STEER ANGLE
thetaV{1}=20;
thetaV{2}=[­50;30];
thetaV{3}=[­50;30];
% SIGNALS
S1V{1} = 1;
S1V{2} = [1 0;0 1];
S1V{3} = [1 0 ; 0 2];
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%LOOPING THROUGH EACH SET OF DATA
for data=1:3
%steer angle of the signal
theta = (thetaV{data}) * pi/180;
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%Signal Vector V/m
S1 = (S1V{data});
% Steering Vector
A = [
exp(­j*k*x1*sin(theta'));
exp(­j*k*x2*sin(theta'));
exp(­j*k*x3*sin(theta'));
exp(­j*k*x4*sin(theta')) ];
% Noise
noise_var = .01;
N = sqrt(noise_var) * randn(size(A,1),size(S1,1));
X1 = A * S1; % Output of the antenna array without noise
X = X1 + N; % Output of the antenna array with noise
Rn = noise_var * eye(size(A,1)); % Noise Covariance Matrix
Rs = (1/size(X1,1))*(X1*X1'); % Signal Covariance Matrix
Rxx = Rs + Rn; % Signal+Noise Covariance Matrix
%Display the data index being used
disp('For Data ');
disp(data);
[V,D] = eig(Rxx); % Eigen Decomposition
%Eigen vectors to polar form
[PHASE,AMPLITUDE]= cart2pol(real(V),imag(V));
EIGENVALUES = eig(D);
%Printing the Eigen values for current data for reference
EIGENVALUES
AMPLITUDE
PHASE
%% EIGEN BEAMS
%%%%%%%%%% PLOTS%%%%%%%%%%%%%%%
th = linspace(­pi/2,pi/2,100);% Sampling theta
color = ['r','b','g','c'];
for loop=1:size(A,1)
figure(data)
% Each eigen vector used as weights for antenna array elements
% Starting with the largest eigen value
w = V(:,size(A,1)+1­loop);
%Array Factor corresponding to that eigen weights
AF = w(1)* exp(j*k*x1*sin(th)) + ...
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w(2)* exp(j*k*x2*sin(th)) + ...
w(3)* exp(j*k*x3*sin(th)) + ...
w(4)* exp(j*k*x4*sin(th)) ;
AFabs(:,loop) = abs(AF);
end
AFF = 20*log10((AFabs)/max(AFabs(:))); %dB value of the AF
% Plotting signals and noise components seperately
for pp = 1 : size(A,1)
hold on
if pp<=size(S1,1)
plot(th*(180/pi),AFF(:,pp),color(pp),'LineWidth',2);
Legend{pp}=strcat('Signal',num2str(pp));
else
plot(th*(180/pi),AFF(:,pp),strcat('.­',color(pp)),'LineWidth',1);
Legend{pp}=strcat('Noise',num2str(pp­size(S1,1)));
end
end
%Formatting the plots
hold off
leg = legend(Legend,'Location','SE','FontSize',12);
set(leg,'FontSize',12);
xlabel('theta (degrees)','FontSize',18);
ylabel('AF (dB)','FontSize',18);
title('Eigen Beams','FontSize',18)
hold off
axis([­90 90 ­50 0]);
pause
end
Data for the remaining methods
%% DATA
f = 2*10^9; % frequency
c = 3 *10^8; % speed of light
j= sqrt(­1);
lambda = c/f; % wave length
k = 2*pi/lambda; % wave number
d = lambda/2; % distance between adjacent array elements
% Locations of the linear array elements
x1= 0;
x2=x1 + d;
x3=x2 + d;
x4=x3 + d;
x5=x4 + d;
x6=x5 + d;
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x7=x6 + d;
x8=x7 + d;
%steer angle of the signal
theta = [­60;
0;
10] * pi/180;
%Signal V/m
S1 = [1 0 0;
0 2 0;
0 0 4];
% Steering Vector
A = [ exp(­j*k*x1*sin(theta'));
exp(­j*k*x2*sin(theta'));
exp(­j*k*x3*sin(theta'));
exp(­j*k*x4*sin(theta'));
exp(­j*k*x5*sin(theta'));
exp(­j*k*x6*sin(theta'));
exp(­j*k*x7*sin(theta'));
exp(­j*k*x8*sin(theta'))];
% Symbol version of steering vector
syms t
A_theta = [
exp(­j*k*x1*sin(t));
exp(­j*k*x2*sin(t));
exp(­j*k*x3*sin(t));
exp(­j*k*x4*sin(t));
exp(­j*k*x5*sin(t));
exp(­j*k*x6*sin(t));
exp(­j*k*x7*sin(t));
exp(­j*k*x8*sin(t))
];
%Noise
noise_var = .01;
N = sqrt(noise_var) * randn(size(A,1),size(S1,1));
X1 = A * S1; %Output of antenna without noise
X = X1 + N; %Output of the antenna array with noise
Rn = noise_var * eye(size(A,1)); % Noise Covariance Matrix
Rs = (1/size(X1,1))*(X1*X1'); % Signal Covariance Matrix
Rxx = Rs + Rn; % Signal+Noise Covariance Matrix
[V,D] = eig(Rxx); %Eigen Decomposition
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Periodogram
%% PERIODOGRAM
%%%%%%%%%% PLOTS%%%%%%%%%%%%%%%
color{1} = 'r';
color{2} = '­­b';
color{3} = '­­black';
color{4} = 'g';
th = linspace(­pi/2,pi/2,100); % Sampling theta
for i = 1:size(th,2)
% Computing steering for each theta
At = double(subs(A_theta,{t},{th(i)}));
% Computing Periodogram Value
P(i) = ((At'*(Rxx)*At));
end
% Plot in dB
Pdb(1,:) = 10*log10(abs(P)/max(abs(P)));
plot(th*(180/pi),(Pdb),char(color{data}),'LineWidth',1.2);
hold on
end
Capon
%%%%%%%%%% PLOTS %%%%%%%%%%%%%%%
%% PERIODOGRAM + CAPON
color{1} = '­­r';
color{2} = '­­b';
color{3} = 'black';
color{4} = 'g';
th = linspace(­pi/2,pi/2,207);% Sampling theta
% Calculating Periodogram and Capon values at each theta sample
for i = 1:size(th,2)
At = double(subs(A_theta,{t},{th(i)}));
P(i) = ((At'*(Rxx)*At));
CAP(i) = (1/(At'*inv(Rxx)*At));
end
% Plotting in dB scale
Pdb(1,:) = 10*log10(abs(P)/max(abs(P)));
Pdb2(1,:) = 10*log10(abs(CAP)/max(abs(CAP)));
plot(th*(180/pi),(Pdb),char(color{1}),'LineWidth',2);
hold on
plot(th*(180/pi),(Pdb2),char(color{3}),'LineWidth',2);
hold off
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% Formatting the plots
leg=legend('PERIODOGRAM','CAPON');
set(leg,'FontSize',14);
xlabel('theta (degrees)','FontSize',18);
ylabel('P(theta) dB','FontSize',18);
q = char(39);
title(strcat('CAPON',q,'S METHOD'),'FontSize',18);
%axis([­90 90 ­20 0]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
MUSIC
%% PERIODOGRAM + MUSIC
%%%%%%%%%% PLOTS%%%%%%%%%%%%%%%
color{1} = '­­r';
color{2} = '­­b';
color{3} = 'black';
color{4} = 'g';
th = linspace(­pi/2,pi/2,304);% Sampling theta %304
% V*V' calculated for Noise eigen vectors only
VVt = V(:,1:end­3)*V(:,1:end­3)';
% Calculating Periodogram and Music values at each theta sample
for i = 1:size(th,2)
At = double(subs(A_theta,{t},{th(i)}));
P(i) = ((At'*(Rxx)*At));
MUS(i) = (At'*At/((At'*VVt*At)));
end
% Plotting in dB scale
Pdb(1,:) = 10*log10(abs(P)/max(abs(P)));
Pdb2(1,:) = 10*log10(abs(MUS)/max(abs(MUS)));
plot(th*(180/pi),(Pdb),char(color{1}),'LineWidth',2);
hold on
plot(th*(180/pi),(Pdb2),char(color{3}),'LineWidth',2);
hold off
% Formatting the plots
leg=legend('PERIODOGRAM','MUSIC');
set(leg,'FontSize',18);
xlabel('theta (degrees)','FontSize',18);
ylabel('P(theta) dB','FontSize',18);
q = char(39);
title('MUSIC','FontSize',18);
axis([­90 90 ­30 0]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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MEM
%% PERIODOGRAM + MEM
color{1} = '­­r';
color{2} = '­­b';
color{3} = 'black';
color{4} = 'g';
th = linspace(­pi/2,pi/2,198); % Sampling Theta
RT = Rxx;
RTinv = inv(RT);
% Calculating Periodogram and MEM values at each theta sample
for i = 1:size(th,2)
At = double(subs(A_theta,{t},{th(i)}));
P(i) = ((At'*(Rxx)*At));
% Calculating only for noise signals
MEM(i) = (1/((At'*RTinv(:,end­4:end)*RTinv(:,end­4:end)'*At)));
end
% Plotting in dB scale
Pdb(1,:) = 10*log10(abs(P)/max(abs(P)));
Pdb2(1,:) = 10*log10(abs(MEM)/max(abs(MEM)));
plot(th*(180/pi),(Pdb),char(color{1}),'LineWidth',2);
hold on
plot(th*(180/pi),(Pdb2),char(color{3}),'LineWidth',2);
hold off
% Formatting the plots
leg=legend('PERIODOGRAM','MEM');
set(leg,'FontSize',18);
xlabel('theta (degrees)','FontSize',18);
ylabel('P(theta) dB','FontSize',18);
title('MEM','FontSize',18);
axis([­90 90 ­30 0]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
PHD
%% PERIODOGRAM + PHD
color{1} = '­­r';
color{2} = '­­b';
color{3} = 'black';
color{4} = 'g';
%%%% PLOT CHANGES VERY MUCH WITH CHANGE IN SAMPLE RATE
th = ­pi/2:.011:pi/2; % Sampling Theta
%%%%
RT = Rxx;
RTinv = inv(RT);
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% Calculating Periodogram and Music values at each theta sample
for i = 1:size(th,2)
At = double(subs(A_theta,{t},{th(i)}));
P(i) = ((At'*(Rxx)*At));
% Only Noise eigen vector with least eigen value is considered
PHD(i) = (1/((At'*(V(:,1)))))^2;
end
% Plotting in dB scale
Pdb(1,:) = 10*log10(abs(P)/max(abs(P)));
Pdb2(1,:) = 10*log10(abs(PHD)/max(abs(PHD)));
plot(th*(180/pi),(Pdb),char(color{1}),'LineWidth',2);
hold on
plot(th*(180/pi),(Pdb2),char(color{3}),'LineWidth',2);
hold off
% Formatting the plots
leg=legend('PERIODOGRAM','PHD');
set(leg,'FontSize',18);
xlabel('theta (degrees)','FontSize',18);
ylabel('P(theta) dB','FontSize',18);
title('PHD','FontSize',18);
axis([­90 90 ­30 0]);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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