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:

*.*

- 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

*µ*_{1 }= −2*µ*,_{B}*µ*_{2 }= 0,*µ*_{3 }= 2*µ*(two beamwidth separation)_{B }*µ*_{1 }= −*µ*,_{B}*µ*_{2 }= 0,*µ*_{3 }=*µ*(one beamwidth separation)_{B }*µ*_{1 }= −0*.*5*µ*,_{B}*µ*_{2 }= 0,*µ*_{3 }= 0*.*5*µ*(one half beamwidth separation)_{B }*µ*_{1 }= −0*.*1*µ*,_{B}*µ*_{2 }= 0,*µ*_{3 }= 0*.*1*µ*(one tenth beamwidth separation)_{B }

for

- The MVDR spectrum,
*S*_{MVDR} - The Fourier spectrum,
- the MUSIC spectrum,
*S*_{MUSIC}

- 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 −3
*µ*≤_{B }*µ*≤ 3*µ*using about 200 equally spaced points._{B } - Help on OCTAVE available at: http://www.gnu.org/software/octave/

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