$M \times M$ covariance matrix of the signal $\x(n)$. As equation suggests, the transmit beampattern is dependent on the waveform covariance matrix $\R$ and the array steering vector $\a(\r)$. In the following we analyze how one can form and control the beampattern by optimizing the covariance matrix $\R$, so as to heat up the tumor region of the tissue while keeping the power deposition in the healthy tissue minimal. In this work, we consider schemes which allow for the lowest possible power leakage to the healthy area.

Once an optimal covariance matrix $\R$ has been determined, the waveform signal $\x(n)$ can be synthesized accordingly. One simple approach is $\x(n) = \R^{1/2} \mathbf{w}(n)$, where $\mathbf{w}(n)$ is a sequence of independent random vectors with mean zero and covariance matrix $\mathbf{I}$. For detailed discussion see [@Stoica2007b; @08:Fuhrmann][@12:He ch. 14].

A significant challenge to this approach, however, is that the *true* steering vector $\a(\r)$ in does not exactly match the model in for a host of reasons: array calibration imperfections, variations in transducing elements, tissue inhomogeneities, inaccurately specified propagation velocity, etc. We will therefore consider the aforementioned design problem subject to model uncertainties in the array steering vector at any given point $\r$. We 