, \hspace{.25cm} n = 1, \ldots, N& \\
&= \a^H(\r) \x(n)  , \hspace{.25cm} n = 1, \ldots, N,&
\end{aligned}$$ where the $m$th signal is attenuated by a factor $a_m({\mathbf{r}})$ which depends on the properties of the transducers, the carrier wave and the tissue. This factor is modeled as $$\label{eq:a_m(r)}
a_m(\r) = \frac{e^{-j2\pi f_c \tau_m(\r)}}{\|\th_m - \r\|^{\frac{1}{2}}},$$ where $f_c$ is the carrier frequency, and $\tau_m(\r) = \frac{\|\th_m - \r\|}{c}$ is the required time for any signal to arrive at location $\r$ where $c$ is the sound speed inside the tissue. Note that the root-squared term in the denominator in represents the distance dependent propagation attenuation of the acoustic waveforms. In , the narrowband signals are represented in vector form $\mathbf{x}(n) = [x_1(n) \: \ldots \: x_m(n) \: \ldots \: x_M(n)]^T \in \mathcal{C}^{M \times 1}$ and $\a({\mathbf{r}}) \triangleq [a_1(\r) \: \ldots \: a_m(\r) \: \ldots a_M(\r)]^T \in \mathcal{C}^{M \times 1}$ is the array steering vector as a function of $\r$.

At a generic location $\r$ in the tissue, the power of the transmitted signal, i.e., *the transmit beampattern*, is given by $$\label{eq:transmit_beampattren}
p(\r) = \mathbb{E} \{ | y(\r,n) |^2 \} = \a^H(\r)\R \a(\r),$$ where $$\R \triangleq \mathbb{E} \{ \x(n) \x^H(n)\}$$ is the 