Example 11.2.1. - The Spatial Matched Filter or Steering Vector Beamformer.
A signal received by a ULA with M = 20 elements and spacing contains both a signal of interest at with an array SNR of 20 dB and thermal sensor noise with unit power . The signal of interest is an impulse present only in the 100th sample.
clc; clear; close all;
M = 20; % Number of Array Elements. N = 200; % Number of Signal Samples. n = 1:N; % Sample Index Vector. lambda = 1; % Incoming Signal Wavelength in (m). d = lambda/2; % Interelement Distance in (m). SNR = 20; % Array Gain in dBs. phi_s = 20; % Signal Direction angle in deg. u_s = (d/lambda)*sin(phi_s*pi/180); % Normalized Spatial Frequency of the signal of interest. % Desired Signal Definition: s = zeros(M,N); % Array snapshot (n=100) that contains the signal of interest (an impulse). s(:,100) = (10^(SNR/20)*exp(-1i*2*pi*u_s*(0:M-1).'))/sqrt(M);
Run this experiment many times (Monte-Carlo runs) to come up with an average output vector y1: To reproduce perfectly Fig. 11.10 you should set NumExperiments = 1; As you can see, Monte-Carlo Averaging improves the output SNR a lot!
NumExperiments = 100; y1 = zeros(1,N); x1 = zeros(1,N); for k=1:NumExperiments % Uncorrelated noise samples at each array element with a Gaussian distribution: w = (randn(M,N)+1i*randn(M,N))/sqrt(2); % The two signals are added to produce the overall array signal: x = s + w; % Examining the signal at a single sensor in Figure 11.10 (a), we see that the signal is not visible % at n = 100 since the element level SNR is only 7 dB (full-array SNR minus M in decibels). % The output power of this sample for a given realization can be more or less than the expected % SNR due to the addition of the noise. However, when we apply a spatial matched filter using % the Spatial Matched Filter or Steering Vector Beamformer. c_mf = exp(-1i*2*pi*u_s*(0:M-1).')/sqrt(M); y = c_mf'*x; y1 = y1 + y; end y_average = 1/NumExperiments*y1; max(10*log10(abs(y_average).^2))
ans = 19.9101
figure('NumberTitle', 'off','Name','Figure 11.10'); subplot(2,1,1); % This plots the instantaneous power for every element (M waveforms). plot(n,10*log10(abs(x).^2)); ylim([-20 25]); grid on; title('Instantaneous Signal Power at each Element (Ensemble of 20 Elements)'); xlabel('Sample Number'); ylabel('Output Power (dB)'); subplot(2,1,2); plot(n,10*log10(abs(y_average).^2),'*-'); grid on; ylim([-30 25]); title('Instantaneous Signal Power at the Output of the Steering Vector Beamformer'); xlabel('Sample Number'); ylabel('Output Power (dB)');