Tag Archives: ULA

Fundamentals of Direction of Arrival Estimation

Direction of Arrival (DOA) estimation is a fundamental problem in communications and signal processing with application in cellular communications, radar, sonar etc. It has become increasingly important in recent times as 5G communications uses DOA to spatially separate the users resulting in higher capacity and throughput. Direction of Arrival estimation can be thought of as the converse of beamforming. As you might recall from the discussion in previous posts, in beamforming you use the steering vector to receive a signal from a particular direction, rejecting the signals from other directions. In DOA estimation you scan the entire angular domain to find the required signal or signals and estimate their angles of arrival and possibly the ranges as well.

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Multicarrier Beamforming at mmWave

Some Background

We have previously discussed beamforming for single carrier signals. Now we turn our attention to multicarrier signals particularly at mmWave where the bandwidths are two orders of magnitude (100x) higher than at sub 6GHz band. We want to investigate that whether there is any distortion in the array response due to high signal bandwidths at mmWave.

But let us start with the case that we have discussed so far i.e. 1GHz single carrier case and a Uniform Linear Array (ULA). We then add two other carriers at 1.2GHz and 0.80GHz, quite an extreme case, stretching the bandwidth to 400MHz. Antenna spacing is still λ/2=0.15m corresponding to the center frequency of 1GHz.

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Fundamentals of a Uniform Linear Array (ULA)

Introduction

A Uniform Linear Array (ULA) is a collection of sensor elements equally spaced along a straight line. The most common type of sensor is a dipole antenna that can transmit and receive Electromagnetic Waves over the air. Other types of sensors include acoustic sensors that may be used in air or under water. The requirements of a ULA are different for different applications but the most common requirement is to improve the Signal to Noise Ratio (SNR) and to improve its response (Gain) in a particular direction. The second property means that the array accepts a signal from a particular direction and rejects the signal from another direction just as required in Radar.

Excess Path Length

The graphical representation and the mathematical framework for the problem are shown below. It is assumed that Electromagnetic Waves (Rays) arrive at the array in the form of a plane wave. This means that there is a large distance between the transmitter and receiver (the receiver is in the far field of the transmitter). The array elements are separated by a distance ‘d’ which must be less than or equal to half the wavelength (similar to the concept of minimum sampling frequency in DSP). Now we can see that the second ray travels an excess distance dcos(θ). Similarly, the third and fourth rays travel an excess distance of 2dcos(θ) and 3dcos(θ) respectively. In array processing it is this excess distance between the arriving rays that is important, absolute distance from the source does not matter (unless you are interested in large scale effects such as path loss). This excess distance between the different rays determines if the signals are going to add constructively or destructively.

Plane Wave Impinging Upon a ULA
Plane Wave Impinging Upon a ULA (four elements)

Mathematical Model of a ULA

Mathematical Framework for ULA (four elements)
Mathematical Framework for ULA (four elements)

Simulation Methodology

Given below is the MATLAB code for the scenario shown in the figure above. We have considered two methods, one employing a ‘for-loop’ and another using matrix manipulation. The second method is usually preferred as it is much faster and also allows us to directly apply techniques from linear estimation theory. We have plotted the array pattern for four cases with N=2,4,6 and 8. It is seen that as the number of array elements increases the Gain (or Directivity) of the array increases. In the case shown below we have considered that the four received signals are added with equal weights (w=1), but these weights can be adjusted to get various beam patterns (weights are typically complex quantities adjusting both phase and amplitude  of the signal). This is typically called Beamforming and we will discuss this in a future post.

Matlab Code

FOR LOOP IMPLEMENTATION

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% SIMPLE UNIFORM LINEAR ARRRAY
% WITH VARIABLE NUMBER OF ELEMENTS
% COPYRIGHT RAYMAPS (C) 2018
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
f=1e9;
c=3e8;
l=c/f;
d=l/2;
no_elements=4;
theta=0:pi/180:2*pi;
r=zeros(1,length(theta));
for n=1:no_elements
dx(n,:)=(n-1)*d*cos(theta);
r=r+exp(-i*2*pi*(dx(n,:)/l));
end
polar(theta,abs(r),'b')
title ('Gain of a Uniform Linear Array')

MATRIX IMPLEMENTATION

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
% SIMPLE UNIFORM LINEAR ARRRAY
% WITH VARIABLE NUMBER OF ELEMENTS
% MATRIX IMPLEMENTATION
% COPYRIGHT RAYMAPS (C) 2018
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
clear all
close all
f=1e9;
c=3e8;
l=c/f;
d=l/2;
no_elements=4;
theta=0:pi/180:2*pi;
n=1:no_elements;
n=transpose(n);
A=(n-1)*(i*2*pi*d*cos(theta)/l);
X=exp(-A);
w=ones(1,no_elements);
r=w*X;
polar(theta,abs(r),'r')
title ('Gain of a Uniform Linear Array')

MATLAB Plot

Uniform Linear Array with Varying Array Length

Note: For a Uniform Linear Array with N elements and half wavelength inter-element spacing the Half Power Beam Width (HPBW) can be estimated as 1.78/N Radians [source]. For the four element case shown above the formula gave a HPBW of 25.49 degrees whereas our simulation yielded 26.20 degrees. For ten element case the formula gave a HPBW of 10.19 degrees whereas the simulation result was 10.20 degrees. Similarly the result for 20 elements is also quite accurate. So we can say that the formula does help us to get a ballpark estimate and gives progressively more accurate results as the number of elements is increased. For a general case where the inter-element spacing is not equal to half wavelength the formula is 0.89*(wavelength/total aperture length).

Lastly, for those who still do not know what Half Power Beam Width also known as 3dB Bandwidth means, it is the width of the main lobe in degrees 3dB down from the peak value of the radiation pattern.