in the healthy tissue region $\Omega_S$;

-   while guaranteeing a certain power level for control points $\r$ in the tumor region $\Omega_T$.

Mathematically, this problem is formulated as (see [@08:Guo]) $$\label{eq:sidelobe min}
\begin{aligned}
\underset{\R,t}{\textrm{max}}  \hspace{.5cm} &t& \\
\textrm{s.t.}  \hspace{.5cm} &\a^H(\r_0) \R \a(\r_0) -  \a^H(\r) \R \a(\r) \geq t, \forall \r \in \Omega_S &\\
& \a^H(\r) \R \a(\r)  \geq (1 - \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T& \\
& \a^H(\r) \R \a(\r) \leq  (1 + \delta) \a^H(\r_0) \R \a(\r_0), \forall \r \in \Omega_T&\\
&\R \in \mathcal{R}&
\end{aligned}$$ where $t$ denotes the gap between the power at $\r_0$ and the power at the control points $\r$ in the healthy region $\Omega_S$. The parameter $\delta$ is introduced here to control the required certain power level at the control points in the tumor region. For instance, if we set $\delta = 0.1$, then we aim for having power at the tumor region $\Omega_T$ to be within 10% of $p(\r_0)$, i.e., the power at the tumor center. This is an SDP problem which can be solved efficiently in polynomial time using any SDP solver, e.g., `CVX` [@cvx2013; @gb08].

Robust Waveform Diversity based Ultrasound System
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The convex optimization problem and consequently its optimal 