[Solved] Array-sensor Homework07

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 = −2µ_B, µ₂ = 0, µ₃ = 2µ_B (two beamwidth separation)
• µ₁ = −µ_B, µ₂ = 0, µ₃ = µ_B (one beamwidth separation)
• µ₁ = −0.5µ_B, µ₂ = 0, µ₃ = 0.5µ_B (one half beamwidth separation)
• µ₁ = −0.1µ_B, µ₂ = 0, µ₃ = 0.1µ_B (one tenth beamwidth separation)

for

• The MVDR spectrum, S_MVDR
• The Fourier spectrum,
• the MUSIC spectrum, S_MUSIC

2. Repeat the above problem with an SNR = 20

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 −3µ_B ≤ µ ≤ 3µ_B using about 200 equally spaced points.
• Help on OCTAVE available at: http://www.gnu.org/software/octave/