Mandatory assignment 2 (Part 2) in INF5410 - Signal Processing in Space and Time

High Resolution Beamforming

Problem 1-5 and 7 are mandatory. Problem 6, 8 and 9 are voluntay.

The problem is to estimate the spatial spectrum for an M=10 element uniform linear array with half-wavelength spacing. The input consists of two incoherent signals at 0 and -10 degrees in additive spatially white noise. The signal to noise ratio for both sources is 0 dB, and N=100 samples are available of the input. The signal and noise model is described on page 73 of the paper.

The MATLAB code for the generation of the input is found in ~inf5410/www_docs/2005V/proj2b.m.

1. Estimate the spatial correlation matrix. Estimate and plot the spatial correlation as a function of lags 0 to M-1.
   

2. Estimate the spatial spectrum using the conventional method (Figure 4b). Discuss why the sources are not separated.
   

3. Estimate the spatial spectrum for the same signal using the minimum variance beamformer (Capon's beamformer) (Figure 5). Discuss the differences from the conventional beamformer.
   

4. Plot the distribution of the eigenvalues of the correlation matrix and explain it on the basis of the signal and noise model.
   

5. Estimate the spectrum using the MUSIC algorithm (figure 5) assuming that the number of signals is known. Discuss the differences from the previous estimates.
   

6. Estimate the spatial spectrum by the eigenvector method (Johnson & Dudgeon eq. 7.8a). Discuss the differences from the MUSIC beamformer.
   

7. Incorrect estimate of the number of sources.
   Estimate the spatial spectrum with the MUSIC method (and eigenvector method) when the number of signals is incorrectly estimated. Let the estimate of the number of signals be 0 ,1, and 3. Discuss the differences between the estimates for the various cases.  (Which spatial spectrum estimator is the eigenvector method equivalent to when 0 signals are assumed to be present?)
   

8. Sparse array and adaptive beamforming.
   Analyze the best and the worst 6 element symmetrical thinned arrays that were found in project 1 in the same signal scenario. Compute the coarrays. Estimate the spatial correlation matrix and spatial spectrums using conventional and adaptive beamformers. Discuss the results.
   

9. Coherent sources.
   Modify the signal generator so that it generates coherent signals instead. Find the properties of the previous beamformers for coherent signals (You may have to change the angles of incidence also). Implement the various forms of averaging of the correlation estimate and see if this gives the methods a better ability to handle coherence.