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In this Presentation
Fundamentals of Antenna Arrays Linear Arrays Planar Arrays Phased Arrays Adaptive Arrays
Beamforming (Spatial Filtering) Signal Models
Conventional Beamforming Optimal Beamforming MATLAB Illustration
Space-Time Adaptive Processing
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Fundamentals of Antenna Arrays Many applications require radiation characteristics (like
gain, directivity) that may not be achievable by a singleelement
Antenna array is a geometric arrangement of antennaelements
Resulting radiation pattern is a vector sum of individualpatterns
Antenna arrays provide more directivity by thephenomena of wave interference
Directive Gain in a given direction is a measure of abilityof an antenna/array to radiate power in that given
direction
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Fundamentals of Antenna Arrays
As the length of the antenna aperture increases, beamwidth decreasesAs the number of antenna elements increase, directive gain increases
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Linear Arrays
Linear array is a lineararrangement of antenna elementswith equal spacing d betweensuccessive elements
Uniform LA is LA with equalcurrent excitation and uniformprogressive phased shift betweenelements
The electric field at a far observation point is (assumingisotropic elements) is independent of
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Radiation Pattern of an 8-Element UniformLinear Array with d = 0.5
Polar Plot E(sin) vs direction cosine (sin)
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Array Controls
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Increasing Array Sizeby Adding Elements
S
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Increasing Array Sizeby Separating Elements
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Amplitude Tapering (Windowing) and Phase Quantization
In Antenna Arrays the current exitations (aperture distribution) are tapered i.emultiplied by a window sequence to reduce the side lobe level. Reduction of SideLobe Level is obtained at the cost of increase in beamwidth
In practice phase shifters are implemented as part of TR modules, using finitenumber of bits
Due to quantization error (difference between desired phase and actualquantized phase) the sidelobe levels are affected
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Planar Arrays Planar Arrays have antenna elements
placed on a plane according to somegrid configuration (rectangular, circular)
Planar arrays can control thebeamshape in both planes (, ) andform pencil beams whereas linear array
only controls the pattern in one plane
Total electric field at a far fieldobservation point is given by
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Radiation pattern of an 8 8-element uniform-amplitude and spaced square planar array
Spherical Radiation Pattern
Radiation Pattern Translation fromSpherical coordinates into U,V space
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Phased Array Antennas
Array antennas synthesize narrow directive beams thatmay be steered mechanically or electronically
Electronic steering is achieved by controlling the phaseof the electric current feeding the array elements, thus
the name phased array
Phase relation is maintained using a network of powerdividers and phase shifters
Direction is selected by adjusting the phase differenceprovided by each phase shifter (usually done using amicroprocessor)
Li Ph d A
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Linear Phased ArrayScanned Every 30deg, N=15, d=/4
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Adaptive Arrays
An adaptive array not only steers the beamsbut also the nulls
Nulls are steered towards the direction ofjammers and nullify their detrimental effects
Adaptive arrays first sample the environmentto estimate the interferences
Next a weight vector is calculated to modify
the sidelobes for effective null steering andsupression of interferences
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Signal Models In spatial array processing there are three types of signals
desired target signal
jammer signal noise signal
Thus the total signal received by the array is
x = xs + xj + xn Jammer and Noise are classified as interference. The
undesired interference signal is
xu = xj + xn Both jammer and noise are characterized as zero mean
normally distributed. Hence the covariance matrix of thisundesired signal would be
Ru = E[xuxuH] = Rn + Rj
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Signal Models Desired Target Signal
Signal is narrowband x(t) = e j2ft
Antenna Array has receiverbehind each element. These
receivers digitize the receivedsignal
The combined output of thereceivers is a N-dimensionalsignal
x is complex baseband signalreceived at left most element
V is spatial array vector
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Signal Models Noise Signal
In the receiver array each element produces thermal noise Modelled as zero mean Gaussian random process
The noise covariance matrix is Rn = n2I, n
2 =kTnB
Jamming Signal Jammers are modeled as spatial point sources that constantly
transmit high power omni-directional interference signal
The signal covariance matrix is equal to
Where j
2 jammer noise power
Vj is array manifold vector associated with jammer
direction of arrival If we are dealing with N jammers, then the covariance
matrices would add, because we assume jammers are
mutually uncorrelated
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Conventional Beamformer The beamformer output y of our
array is
The complex weights wi arechosen to control the sidelobe level to steer the main beam towards
an angle 0
However this data independentbeamformer may not providenulls in the direction of interferersand hence suboptimal SINR
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Optimal Beamformer (MVDR) A minimum variance distortionless response beamformer (MVDR) also
called an optimal beamformer accomplishes two objectives Minimize the array output interference power Get the target desired signal without any distortion
Controllable parameters are array weights wi (weight vector w) Array output in vector notation is
Output is a combination of desired signal and interference components
The interference output is the sum of noise and jammer outputs or
If we want to minimize the interference output power, then we mustminimize
subject to the constraint
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Optimal Beamformer (MVDR) The constrained minimum variance distortionless response can be
achieved with the weight vector
where is the desired target steering vector
The desired weight vector can be calculated to be
with
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Matlab Illustration(Both Jammers in Sidelobes, with No Weighting)
N=16 antennas elements are used
Desired target located at 00
Jammers located at 180 (with SNR of 50dB) and -330 (with SNR of 30dB)
Two dotted vertical lines indicate the angles of arrival of the two jammers
-80 -60 -40 -20 0 20 40 60 80-60
-50
-40
-30
-20
-10
0
Angle of Arrival (degrees)
NormalizedArrayResponse(dB)
Unadapted Array Pattern
-80 -60 -40 -20 0 20 40 60 80-60
-50
-40
-30
-20
-10
0
Angle of Arrival (degrees)
NormalizedArrayResponse(dB)
Distortionless Beamformer Array Pattern
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Adaptive Beamforming Optimal Beamforming only sounds good only in theory
Obtaining Rin (interference covariance matrix) requires
infinite number of samples. Hence we can onlyestimate it. A sampled interference covarianceestimation is
Various adaptive beamforming methods are based oncollecting data from which correlation matrix isestimated Block Adaptive Implementation
Uses block of data to estimate the adaptive beamforming weight vectorand is known as Sample Matrix Inversion (SMI)
Sample by Sample Adaptive Implementation RLS Algorithm Steepest Descent Method
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Training Data
Before the beamforming system can be used,it must be trained with target- free samples
Training Data are of two types Target Free training data xin = xn + xj Target in training data xin = xn + xj + xs
In applications like radar, target free trainingdata is always available by takingmeasurements at ranges shorter or longerthan the target
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Space-Time Adaptive Processing (STAP)
STAP is concerned with the two-dimensional processing ofsignals in both the spatial and temporal domains to optimallydiscriminate targets from both clutter and jamming
Detection of slowly moving targets by air- and spaceborneMTI radar (moving target indication) is heavily degraded bythe motion induced Doppler spread of clutter returns.
Space-time adaptive processing (STAP) can achieve optimumclutter rejection via implicit platform motion compensation.
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General STAP Architecture
Space TimeEnvironment issampled in spatialdomain by using
array of antennaelements
Also sampled in
temporal domain bytransmitting a seriesof pulses to obtaindoppler information
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Space-Time vs Spatial Processing Degrees of Freedom
In spatial array processing, thedegrees of freedom equals numberof antenna elements N
In STAP, every antenna transmits atrain of M pulse and applies a
complex weight on each echo afterreceiving them. Hence the degrees offreedom equals NM.
Progressive Phase Shift In spatial array processing only
progressive phase shift betweenantenna elements is used In STAP, progressive phase shift
between antenna elements andbetween successive pulses receivedfrom each antenna element is
exploited
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Signal Behavior in Space-Time Environment
Space Time Signal Environment
Receiver Noise has no structure inspace/frequency and thereforeappears as a uniform noise floor
Broadband Noise Jammers are
localized in AOA but spread acrossthe entire doppler spectrum
Appears as ridge of energylocalized in AOA but spreadacross all doppler shifts
Scatterers at an angle of w.r.tantenna boresight will have adoppler shift of
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Space Time Adaptive Processing
Angle doppler characteristics of the echo froma moving point target depend on both theradar platform motion and the target motion
If the target is stationary and directly on theboresight, the doppler shift will be zero andwill fold into the clutter
However if the target is moving it will separatefrom the clutter on the doppler axis andshown in the figure
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Optimal STAP Processing The spatial steering vector for a ULA of M sensors is
The temporal frequency steering vector assuming Ltransmitted/received pulses is
The two dimensional LM X 1 space time steering vector isgiven by
The optimal STAP weight vector is given by
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Thank You
Matlab Source Listing (beamformer m)
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Matlab Source Listing (beamformer.m)Input Section
close all
clear all
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% User Input Section
lambda = 0.03; % wavelength
d = lambda/2; % element spacing
dl = d/lambda;
N = 16; % # of array elements
aoa_max = asin(1/2/dl); % maximum "real space" AOA (radians)
window_on = false; % true or false
Nangle = 1000; % # of angles for evaluating beam pattern
Nm1 = Nangle-1;
t_aoa = pi/180*(0); % target AOA (radians)
j_aoa1 = pi/180*(18); % jammer #1 AOA (radians)j_aoa2 = pi/180*(-33); % jammer #2 AOA (radians)
SNR = 0; % signal to noise ratio (dB)
JSR1 = +50; % jammer #1 to noise ratio (dB)
JSR2 = +30; % jammer #2 to noise ratio (dB)
% End user input section%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Matlab Source Listing (beamformer m)
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Matlab Source Listing (beamformer.m)Computing the Optimal Weight Vector
p_n = 1; % noise powerp_t = p_n*(10^(SNR/10)); % target powerp_j1 = p_t*(10^(JSR1/10)) ; % jammer powerp_j2 = p_t*(10^(JSR2/10)) ; % jammer power% compute signal vectorstarget = sqrt(p_t)*exp(j*2*pi*(0:N-1)'*d*sin(t_aoa)/lambda);if (window_on)
t = target.*taylorwin(length(target),4,-30);elset = target;endj1 = sqrt(p_j1)*exp(j*2*pi*(0:N-1)'*d*sin(j_aoa1)/lambda);j2 = sqrt(p_j2)*exp(j*2*pi*(0:N-1)'*d*sin(j_aoa2)/lambda);
% compute covariance matrix with and without jammersR = p_n*eye(N);Rj = p_n*eye(N) + p_j1*j1*(j1') + p_j2*j2*(j2');disp(['Covariance matrix rank with jammers = ',num2str(rank(R))]);% compute beamformer weight vector with and without jammersw = R\conj(t);wj = Rj\conj(t);
Matlab Source Listing (beamformer m)
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Matlab Source Listing (beamformer.m)Plotting the Output Patterns
% compute and display beampatterntheta = -aoa_max + 2*aoa_max/Nm1*(0:Nm1);W = zeros(Nangle,1);Wj = zeros(Nangle,1);for p = 1:NangleW(p) = w.'*exp(-j*2*pi*(0:N-1)'*dl*sin(theta(p)));Wj(p) = wj.'*exp(-j*2*pi*(0:N-1)'*dl*sin(theta(p)));endWp = db(abs(W),'voltage');scale = 10*log10(N^2);% scale = 0;Wp = Wp - scale;Wjp = db(abs(Wj),'voltage') - scale;figure(1)plot(180/pi*theta,Wp)axis([-180*aoa_max/pi +180*aoa_max/pi -60 0])% gridxlabel('Angle of Arrival (degrees)');ylabel('Normalized Array Response (dB)')title('Unadapted Array Pattern')vline(180/pi*[j_aoa1, j_aoa2])
kappa = t'*transpose(inv(Rj))*conj(t);wj1 = wj/kappa;Wj1 = zeros(Nangle,1);for p = 1:NangleWj1(p) = wj1.'*exp(-j*2*pi*(0:N-1)'*dl*sin(theta(p)));endWjp1 = db(abs(Wj1),'voltage');figure(3)plot(180/pi*theta,Wjp1)axis([-180*aoa_max/pi +180*aoa_max/pi -60 0])% gridxlabel('Angle of Arrival (degrees)');ylabel('Normalized Array Response (dB)')title('Distortionless Beamformer Array Pattern')vline(180/pi*[j_aoa1, j_aoa2])