This problem intends to compare the resolution capabilities of the MUSIC algorithm, MVDR beamformer and classical Fourier based periodogram when applied to an azimuth angle-ofarrival estimation task. We consider a linear array consisting of M = 12 uniformly spaced antenna elements. Three equally powered, uncorrelated, plane wavefronts are impinging at the array. We have N = 100 snapshots available and the Signal-to-Noise ratio is SNR = 10dB (white gaussian noise, uncorrelated with the signals). The transmitted signals are Q-PSK modulated and have unit power, i.e. they take on the four values:
.
1. Use MATLAB or OCTAVE to plot the power spectra as a function of the spatial frequency µ, normalized to the so called standard beamwidth
for the following spatial separations
• µ1 = −2µB, µ2 = 0, µ3 = 2µB (two beamwidth separation)
• µ1 = −µB, µ2 = 0, µ3 = µB (one beamwidth separation)
• µ1 = −0.5µB, µ2 = 0, µ3 = 0.5µB (one half beamwidth separation)
• µ1 = −0.1µB, µ2 = 0, µ3 = 0.1µB (one tenth beamwidth separation)
for
• The MVDR spectrum, SMVDR
• The Fourier spectrum,
• the MUSIC spectrum, SMUSIC
2. Repeat the above problem with an SNR = 20dB.
Hints:
• If the spatial frequencies of the impinging wavefronts are packed into a d × 1 column vector mu, then the array output for N snapshots can be calculated in MATLAB like
X = exp(i*([0:M-1]’)*mu’)*(sign(randn(d,N))+i*sign(randn(d,N)))/(sqrt(2)) + sqrt(d)*(randn(M,N)+i*randn(M,N))/(sqrt(2)*10ˆ(SNR/20));
• Plot the power spectra in −3µB ≤ µ ≤ 3µB using about 200 equally spaced points.
• Help on OCTAVE available at: http://www.gnu.org/software/octave/
