[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 , normalized to the so called standard beamwidth

for the following spatial separations

• ₁ = 2_B, ₂ = 0, ₃ = 2_B (two beamwidth separation)
• ₁ = _B, ₂ = 0, ₃ = _B (one beamwidth separation)
• ₁ = 0.5_B, ₂ = 0, ₃ = 0.5_B (one half beamwidth separation)
• ₁ = 0.1_B, ₂ = 0, ₃ = 0.1_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)*10(SNR/20));

• Plot the power spectra in 3_B 3_B using about 200 equally spaced points.
• Help on OCTAVE available at: http://www.gnu.org/software/octave/