# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 2 Materials and Methods
### Temporal Encoding for Simultaneous Emission of Plane Waves  
A straightforward method that further increases the dimension of the observation space (\(\mathbf{y}^{(k)}\)) consists of the following:  
* Emit a PW in the direction \(\theta^{(k)}\) and receive its backscattered echoes \(\mathbf{y}^{(k)}\);
* Build the corresponding concatenated kernel \(\mathbf{\Lambda}^{(k)}\) using Equation (12);
* Repeat steps (a) and (b) \(N_{\text{pw}}\) times with a different angle \(\theta^{(k)}\);
* Build the total observation vector \(\tilde{\mathbf{y}}\) of size \(N_{\text{pw}}N_{\text{el}}N_{y}\times 1\) using: \[\tilde{\mathbf{y}}=\big{[}\mathbf{y}^{(1)}\|\mathbf{y}^{(2)}\| \cdots\|\mathbf{y}^{(N_{\text{pw}})}\big{]},\] (13) where each \(\mathbf{y}^{(k)}\) is computed using Equation (12);
* Build the enhanced kernel \(\tilde{\mathbf{\Lambda}}\) of size \(N_{\text{pw}}N_{\text{el}}N_{y}\times N_{z}N_{x}\) using: \[\tilde{\mathbf{\Lambda}}=\big{[}\mathbf{\Lambda}^{(1)}\|\mathbf{\Lambda}^{(2)}\|\cdots\| \mathbf{\Lambda}^{(N_{\text{pw}})}\big{]},\] (14) where each \(\mathbf{\Lambda}^{(k)}\) is computed using Equation (12). In Equation (14), it can be seen that there is a value of \(N_{\text{pw}}=\lceil(N_{z}N_{x})/N_{\text{el}}N_{y}\rceil\) for which the system given by Equation (15) stops being underdetermined, and the number of equations matches the number of unknowns. \[\tilde{\mathbf{y}}=\tilde{\mathbf{\Lambda}}\mathbf{\gamma}. \tag{15}\]  
However, the acquisition time needed to record echoes from \(N_{\text{pw}}\) observations is \(N_{\text{pw}}\) times greater than the time needed from one PW emission. The unwanted effect of the increase in acquisition time is that for fast events, the TRF changes so quickly that for \(N_{\text{pw}}\) successive emissions the TRF is not the same. Under these conditions, Equation (15) becomes less accurate. This drawback can be fixed by increasing the frame acquisition rate or by compensating the tissue displacement [43]. To overcome the previously stated problem, we propose to emit \(N_{\text{pw}}\) PWs simultaneously using quasi-orthogonal codes \(a^{(k)}(t)\) associated with each \(k^{th}\) PW. Using the principle of linear propagation of PWs, we can write the total received signal \(\tilde{y}_{i}(t)\) at the element \(i\) as follows:  
\[\tilde{y}_{i}(t)=\sum_{k=1}^{N_{\text{pw}}}y_{i}^{(k)}(t)=\sum_{k=1}^{N_{\text{ pw}}}e^{(k)}(t)*g_{i}^{(k)}(t), \tag{16}\]where \(g_{i}^{(k)}(t)\) is the impulse response of the medium seen by the \(i^{th}\) element, when in-sonified with the \(k^{th}\) PW. From Equation (16), it can be seen that the received signals \(\tilde{y}_{i}(t)\) contain a mix of echoes associated with each simultaneously emitted PW \(k\). To separate the contributions of each PW, we propose to use filters \(f^{(k)}(t)\) adapted to the typical \(e^{(k)}(t)\) echo waveform of the of \(k^{th}\) wave. We can write the expression of the filtered backscattered echoes for each emitted PW \(k\) as follows:  
\[\hat{g}_{i}^{(k)}(t)=f^{(k)}(t)*\tilde{y}_{i}(t), \tag{17}\]  
where \(\hat{g}_{i}^{(k)}(t)\) can be seen as an estimation of \(g_{i}^{(k)}(t)\). As we assume that all of the excitations \(a^{(k)}(t)\) occupy the same frequency bandwidth, in Equation (17) the signal \(\hat{g}_{i}^{(k)}(t)\) contains some unwanted residual signal related to the waveforms \(e^{l}(t)\) (\(l\
eq k\)), also known as the crosstalk noise. This noise can be evaluated by considering the case where all of the \(N_{\text{pw}}\) PWs that carry the codes \(a^{(k)}(t)\) are emitted successively. In this case, the received signals \(y_{i}^{(k)}(t)\) are not subject to crosstalk. Let us call \(\overline{\hat{g}}_{i}^{(k)}(t)\) the corresponding estimation of \(g_{i}^{(k)}(t)\); we then have:  
\[ \begin{split} y_{i}^{(k)}(t)&=e^{(k)}(t)*g_{i}^{(k) }(t),\\ \overline{\hat{g}}_{i}^{(k)}(t)&=f^{(k)}(t)*y_{i}^{( k)}(t).\end{split}