ECE251DN: Homework #3 Solutions

Transcription

1 ECE251DN: Homework #3 Solutions Problem (a) In this problem, we only have one null constraint. So N 1 j C = V(ψ ) = [e 2 ψ,..., 1,..., e j N 1 2 ψ ] T The weights of the least squares approximation to the desired beam pattern B d (ψ) is given by W o = [I C(C H C) 1 C H ] W d where then C H C = V H (ψ )V(ψ ) = N W o = [I 1 N V(ψ )V H (ψ )] W d For uniform weighing, the desired weighting W d is given by W d (m) = 1 N, m = N 1 2,...,,..., N 1. 2 Thus the weights for the least squares approximation to the desire beam pattern are given by ( 1 W o = N I 1 ) N 2 V(ψ )V H (ψ ) W d (b) Figure 1 shows the beam patterns for ψ = 3π/N, ψ = 2π/N, ψ = π/n and ψ =.5π/N. When ψ = 3π/N, the null constraint is outside the mainlobe, the approximated beam pattern has very similar performance as the desired one within the mainlobe region. When ψ = 2π/N, the null constraint overlaps with a null point of the desired beam pattern, so the least squared approximation is exactly the same as the B d (ψ). When ψ = π/n, the null constraint goes inside the mainlobe, which degrades the performance in the mainlobe. Similarly, when ψ =.5π/N, the null constraint goes further into the mainlobe and makes the mainlobe performance even worse. The matlab code for this problem is included at the end in Appendix B. 1

2 5 Beam pattern without null constraint B(ψ) ψ =3π/N ψ ψ =2π/N B(ψ) 5 B(ψ) ψ =1π/N ψ ψ =.5π/N ψ B(ψ) 5 B(ψ) ψ ψ Problem 3 Figure 1: Beam patterns for ψ = 3π/N, ψ = 2π/N, ψ = π/n, ψ =.5π/N. (a) The matlab code to generate snapshots with different values of is attached in Appendix C. The code computes both the spatial spectrum with uniform weighting and the spatial spectrum with a Hamming weighted beamformer. (b) For = 5, the spatial spectrum is plotted in Figures?? and?? in linear and db scale, respectively. For = 15, the spatial spectrum is plotted in Figures?? and?? in linear and db scale respectively. (c) The plots of the spatial spectrum of Hamming window based beamformer are also shown along with the plots of spatial spectrum of uniformly weighted beamformer data. Note that uniform weighting has smaller mainlobe and hence better resolution. It is able to resolve the closely spaced plane waves ( = 5 ). Also note that the beamformer have larger mainlobes as we steer away from broadside leading to broad peaks in the spatial spectrum away from broadside. The noise variance at the beamformer output has decreased by a factor equal to the array gain from variance 1 to variance 1/32 for the uniform weighting. The Hamming beamformer has not been properly normalized and hence the lower power indicated in the spatial spectrum. Need to be scaled by the energy in the Hamming window. 2

3 Delta = 5.6 Uniform Weighting Hamming Window.5 Spatial spectrum P(θ) Azimuthal angle ( θ ) Figure 2: Spatial Spectrum for = 5 Delta = 5 Uniform Weighting Hamming Window Spatial spectrum P(θ) Azimuthal angle ( θ ) Figure 3: Spatial Spectrum in db scale for = 5 A Matlab Code for Prob % matlab code for problem in the text clear all close all N = 21; M = (N-1)/2; n = (-M:M). ; psi = pi*(-1:.1:1); v = exp(j*n*psi); wd = ones(n,1)/n; Bd = wd * v; warning off; figure(1) subplot(3,2,[1 2]) plot(psi,2*log1(real(bd)), -b ); hold on; %plot([-pi pi], [1 1], :k ); plot([-pi pi], [ ], :k ); plot([ ], [ -1], :k ); hold off; %axis([-pi pi ]); 3

4 Delta = 15.6 Uniform Weighting Hamming Window.5 Spatial spectrum P(θ) Azimuthal angle ( θ ) Figure 4: Spatial Spectrum for = 15 Delta = 15 Uniform Weighting Hamming Window Spatial spectrum P(θ) Azimuthal angle ( θ ) Figure 5: Spatial Spectrum in db scale for = 15 title( Beam pattern ylabel( B(\psi) ); without null constraint ) xlabel( \psi ); % psi = 3*pi/N psi = 3*pi/N; v = exp(j*n*psi); wo = (eye(n) - 1/N * v*v ) * wd; B = wo * v; figure(1) subplot(3,2,3) plot(psi,2*log1(real(b)), -b ); hold on; %plot([-pi pi], [1 1], :k ); plot([-pi pi], [ ], :k ); plot([psi psi], [ -1], :k ); hold off; %axis([-pi pi ]); title(strcat( \psi_=,num2str(psi*n/pi), \pi/n )) xlabel( \psi ); ylabel( B(\psi) ); 4

5 % psi = 2*pi/N psi = 2*pi/N; v = exp(j*n*psi); wo = (eye(n) - 1/N * v*v ) * wd; B = wo * v; figure(1) subplot(3,2,4) plot(psi,2*log1(real(b)), -b ); hold on; %plot([-pi pi], [1 1], :k ); plot([-pi pi], [ ], :k ); plot([psi psi], [ -1], :k ); hold off; %axis([-pi pi ]); title(strcat( \psi_=,num2str(psi*n/pi), \pi/n )) xlabel( \psi ); ylabel( B(\psi) ); % psi = pi/n psi = pi/n; v = exp(j*n*psi); wo = (eye(n) - 1/N * v*v ) * wd; B = wo * v; figure(1) subplot(3,2,5) plot(psi,2*log1(real(b)), -b ); hold on; %plot([-pi pi], [1 1], :k ); plot([-pi pi], [ ], :k ); plot([psi psi], [ -1], :k ); hold off; %axis([-pi pi ]); title(strcat( \psi_=,num2str(psi*n/pi), \pi/n )) xlabel( \psi ); ylabel( B(\psi) ); % psi =.5*pi/N psi =.5*pi/N; v = exp(j*n*psi); wo = (eye(n) - 1/N * v*v ) * wd; B = wo * v; figure(1) subplot(3,2,6) plot(psi,2*log1(real(b)), -b ); hold on; %plot([-pi pi], [1 1], :k ); plot([-pi pi], [ ], :k ); plot([psi psi], [ -1], :k ); hold off; %axis([-pi pi ]); title(strcat( \psi_=,num2str(psi*n/pi), \pi/n )) xlabel( \psi ); ylabel( B(\psi) ); warning on; B Matlab Code for Prob. 3 % File: tp3.m % Created: Tue 1/19/4 15:58:32-7 % Last Updated: Thu 1/26/4 1:15:3-7 % Description: ECE 251D - Fa4 - HW3 - Prob. 3 % clear all; close all; DELTA = 5; % offset of third plane wave N = 32; % number of elements in the ULA numpts = 512; M = 1; % number of samples L = 6; % number of plane waves 5

6 antenna = [:N-1] ; % N x 1 THETAL = [ DELTA ]; % Angle of incidence THETAL = THETAL * pi/18; % in radians s = sqrt([ ]); % signal powers data = zeros(m, N); P = zeros(numpts, 1); PH = zeros(numpts, 1); X = hamming(2*n-1); % hamming window Y = X(N:end); thetas Vtheta = linspace(,pi,numpts); % 1 x numpts, x-axis for plotting = exp(j*pi*antenna*cos(thetas)); % N x numpts for iter=1:l V(:,iter) = exp(j*pi*cos(thetal(iter))*antenna); end for m=1:m, y = zeros(n,1); for l=1:l, y = y + s(l)*randn*v(:,l); % generate signal end y = y + randn(n,1); % add noise y is N x 1 %f = (Vtheta *y)/n; % numpts x 1; f = fftshift(fft(y, numpts))/n; % this is the fft way P = P + abs(f).^2; y = y.*conj(y); %f = (Vtheta *y)/n; % numpts x 1 f = fftshift(fft(y, numpts))/n; % this is the fft way PH = PH + abs(f).^2; end P = P/M; PH = PH/M; %x = [1:numPts]*18/numPts; x = acos(linspace(-1,1,numpts))*18/pi; hold off %plot(x, P, r-. ) semilogy(x, P, r-. ) hold on %plot(x, PH, g-- ) semilogy(x, PH, g-- ) axis tight v = axis; axis([v(1) v(2) 1.25*v(4)]) legend( Uniform Weighting, Hamming Window, Location, Best ) xlabel( Azimuthal angle ( \theta ) ) ylabel( Spatial spectrum P(\theta) ) grid on; return 6

7

8 3.1.1a (1) W d = Vbs( u) u= (Eq ) Null constraint c P = I C C H H ( C) C C = V bs ( u) u= 3 32 (Eq ) (Eq ) W n = P c W d (Eq ) Null within beamspace region causes little deviation from B d

9 3.1.1 (2) b). Nulls outside Beamspace region are harder to form, causing large deviations from desired pattern have to suppress beam pattern over large region to get the null.

10 3.1.1 (3) c) Steering causes large drop-off in side lobes. Null is formed with little degradation to Bd.

11

12

13