Simulation Results

Here we present computer simulations which show that the EM source location estimation algorithm for Gaussian distributed $f({\bf S})$ (from Section 4.1) performs better than CML, providing performance closer to that of ML estimation with known signal amplitudes. In these simulations two sources, impinging from 14^o and 16^o, were equipowered, mutually uncorrelated, and Gaussian distributed with mean 0. An equispaced linear array of 10 omnidirectional and matched sensors with half wavelength spacing was used. SNR was measured at a sensor. Twenty independent snapshots were employed for each estimate.

To assure results were not skewed by global convergence effects, the known signal ML algorithm was initialized using the true value of $\theta$. To avoid global convergence effects and assure convergence of the iteration process, the EM algorithm was initialized with the known signal ML solution. The CML algorithm, which minimizes $\sum_{k=1}^{N} \vert\vert {\bf x}_k - {\bf A} ( \theta ) {\bf s}_k \vert\vert^{2}$with respect to both $\theta$ and ${\bf S}$ using ${\bf s}_k = {\bf A} ( \theta )^+ {\bf x}_k$, was also initialized with the known signal ML solution.

For the first simulation, 1000 trials were used per point in SNR. For the EM algorithm, the temporally independent Gaussian signal amplitude model was used and the correct signal covariances ${\bf C}_k$ and means ${\bf d}_k$were assumed. Enough iterations were used to assure EM algorithm convergence. (For the higher SNR's, the number of iterations was very large - over 100 - although we used a generalized EM algorithm which significantly reduced computation per iteration. In practice, a smaller number of iterations could probably be used.)

  

Figure: 1: MSE in the estimate of $\theta _1$ vs. SNR for CML, Gaussian EM, and known signal algorithms.

Figure 1 shows the mean-square-error in the estimate of the 14^o source. The performance of the Gaussian EM algorithm is better than the deterministic algorithm, especially at lower SNR. From (20, 21) we see that with zero-mean signal amplitudes and at higher SNR, the ${\bf g}_k$ approach the projections of the data ${\bf x}_k$ onto the iterating signal subspace. Asymptotically (in SNR) the EM algorithm iteratively computes the solution to the joint location/signal estimation problem.

  

Figure: 2: MSE in the estimate of $\theta _1$ vs. SNR for CML, Gaussian EM, and known signal algorithms.

The second simulation illustrates the effect of using an incorrect prior distribution for the signal amplitudes. Again, the temporally independent Gaussian signal amplitude model, with zero mean, was correctly assumed. The SNR for each of the two sources was fixed at 8dB (i.e. 10^0.8 signal variance). The signal variance for the Gaussian prior distribution was varied from 10^-2 to 10³ (the model correctly assumed the sources were mutually uncorrelated and zero mean). 2000 trials were used. Again, enough iterations were used to assure EM algorithm convergence. Figure 2 shows the mean-square-error, in the estimate of the 14^o source, for the EM algorithm as a function of model variance. Also shown are the mean-squared-error for known signal ML and CML estimators (which are not a function of model variance). For model variance too small, performance is degraded. This should be expected since the marginalization over ${\bf S}$ is then based on a prior that does not represent the actual signal values. When the model variance is larger than that for the actual signal amplitudes, performance also degrades. However, this degradation is not as substantial, since the model still does represent the actual signal values. As the model variance increases, performance asymptotically approaches that realized with a noninformative prior (e.g. [4]). This illustrates the advantage of marginalization even when an accurate prior can not be identified.