scheme for ultrasound system has been introduced and explained in detail in [@08:Guo] based on the transmit beampattern design technique for MIMO radar systems [@04:Fuhrmann; @Stoica2007a].

In the following we consider the practical power constraint where all array elements have the same power. Therefore, the covariance matrix $\R$ belongs to the following set $\mathcal{R}$: $$\mathcal{R} \triangleq \{\mathbf{R}
\hspace{.1cm}|\hspace{.1cm} \mathbf{R}\succeq {\mathbf{0}} , R_{mm} =
\frac{\gamma}{M}, m=1,2,...,M \},$$ where $\gamma$ is the total transmitted power and $R_{mm}$ is the $m$th diagonal element of $\R$ corresponding to the power emitted by $m$th transducer. The healthy tissue and the tumor regions are represented by two sets of discrete control points $\r$: $$\begin{split}
\Omega_S &= \{ \r_1,\r_2,\ldots,\r_{N_S}\} \\
\Omega_T &= \{ \r_1,\r_2,\ldots,\r_{N_T} \},
\end{split}$$ where $N_S$ and $N_T$ denote the number of points in the healthy tissue region and the tumor regions, respectively. Without loss of generality, let $\r_0$ be a representative point which is taken to be the center of the tumor region $\Omega_T$. The objectives for this optimization problem can be summarized as follows: Design the waveform covariance matrix ${\mathbf{R}}$ so as to

-   maximize the gap between the power at the tumor center $\r_0$ and the power at the control points $\r$ 