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The 2014 edition of the Peyresq summer school just finished. Before the coda, let us summarize the very last talk of Jamal Najim on his series on large random matrices. Jamal’s talk was focused on applications of results he described us in the previous day. I will focus on the application to direction of arrival estimation (DOA), although Jamal also discussed MIMO communications.
In DOA the goal is to estimate the directions to r sources from n measurements that mix the sources linearly with phase factors and additive noise: $\vec{y} = \sum_{\ell=1}^{r} \vec{a}(\varphi_\ell)s_\ell + \sigma \vec{w}$
The classical MUSIC algorithm uses the fact that one can obtain a characterization of the signal subspace from the measurements~: $\vec{a}^* (\varphi) \bigl( I_N - \Pi_N \bigr) \vec{a} (\varphi) = 0 \Leftrightarrow \varphi \in \{\varphi_1, ..., \varphi_r\}$
where $\Pi_N$ is the ortho projection on the span of the $\vec{a}(\varphi_\ell$. When the sources are drawn at random, this argument is replaced by an estimation using the ortho projection on the eigenvectors of the measurements covariant matrix since~: $\frac{1}{n} \mathbb{E}\{Y_N Y_N^*\} = A_N(\vec{\varphi}) \mathbb{E}\{ \frac{S_N S_N^*}{n}\} A^*_N(\vec{\varphi}) + \sigma^2 I_N$.
This naturally leads to replacing $\Pi_N$ with the projection on the first r eigenvectors of the sample covariance matrix. Except that this is not a super good idea: modeling the sources by complex gaussian variables, one obtains a multiplicative spiked model with a rank r perturbation for the observations and we have learned with Jamal that the sample covariance matrix is not a good estimator of the covariance in this case. Fortunately the theory of large random matrices tells us how we can construct a consistent estimator which turns out to involve if a simple de-biasing correction. Jamal then showed with simulations how and when this method improves on the original MUSIC.

This was the conclusion of an absolutely marvelous week at Peyresq. I could not seriously express with words how much I enjoyed the week: I learned a lot of new things, discussed with many, got constructive feedback on my own work and this in turn generated many ideas I now want to work on. Every workshop should be like this instead of gigantic waste of time, energy and money (a whole week at Peyresq will set you back much less than the registration at those big conferences). There is always a bit of saudade when such an excellent week ends. But here is an enormous thank you to :
. Patrick and Cédric who organized the event and made sure it was nothing short of perfect
. The Peyresq local team who makes it such a wonderful place to stay at
. All participants, speakers and students: you guys stay cool and keep up the good work!