then the robust design scheme is applied where its power distribution in the worst-case model is evaluated and compared to the nonrobust formulation.

*Notation:* Boldface (lower case) is used for column vectors, $\mathbf{x}$, and (upper case) for matrices, $\mathbf{X}$. $ \| \a \|_{\mathbf{W}} \triangleq \sqrt{\a^H {\mathbf{W}} \a}$ where ${\mathbf{W}} \succ {\mathbf{0}}$. ${\mathbf{x}}^T$ and ${\mathbf{x}}^H$ denote transpose and Hermitian transpose. ${\mathbf{R}} \succeq {\mathbf{0}}$ signifies positive semidefinite matrix and ${\mathbf{R}}^{1/2}$ a matrix square-root, e.g., Hermitian. The set of complex numbers is denoted by $\mathcal{C}$.

*Abbreviations:* Semi-infinite programming (SIP); multiple-input multiple-output (MIMO); Semidefinite program (SDP); linear matrix inequality (LMI).

system model {#sec:system model}
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We consider an array of $M$ acoustic transducers to heat target points. These transducers are located at known positions $\th_m$, for $m = 1,2,\dots,M$, around the tissue at risk, cf., [@08:Guo; @14:Shariati2014c]. We parameterize an arbitrary point in 3D space using Cartesian coordinates $\mathbf{r} = [x \: y \: z]^T$.

Let $x_m(n)$ denote the baseband representation of narrowband discrete-time signal transmitted at the $m$th transducer, at sample $n = 1, \dots, N$. Then the baseband signal received at a generic location $\r$ equals the superposition of signals from all $M$ transducers, i.e., $$\label{eq:baseband_recieved}
\begin{aligned}
y(\r,n) &= \sum_{m=1}^M  a_m(\r) x_m(n)  