# Simultaneous Coded Plane-Wave Imaging Using an Advanced Ultrasound Forward Model
## 2 Materials and Methods
### Temporal Plane-Wave Encoding  
As \(\Delta x<p\), the forward model for one PW insonification is underdetermined Section 2.3. To increase the size of the observation space, coded PWs are emitted simultaneously. For the kernel \(\mathbf{\Psi}\), the number of observations is equal to the number of unknowns when \(N_{\text{pw}}=3\) PWs are emitted simultaneously inside the medium. The chosen emission angles \(\theta^{(k)}\) are the following: \(\theta^{(1)}=-10^{\circ}\), \(\theta^{(2)}=0^{\circ}\), and \(\theta^{(3)}=10^{\circ}\), which ensures that each angle provides new information about the TRF [18; 50].  
Extensive research, which we cannot detail here for the sake of brevity, shows that the waveforms \(e^{(k)}(t)\) and the corresponding filters \(f^{(k)}(t)\) that minimize the crosstalk (see Equation (18)) in accordance with the technical specifications of the ultrasonic emission system consist of a mix between chirps (Figure 4a-c) [24]. To compute \(a^{(k)}(t)\), we propose the following pipeline:  
* \(e^{1}(t)\) is designed as a chirp waveform emitted with an angle \(\theta^{(1)}\): \[e^{(1)}(t)=\eta(t)\text{cos}\big{(}2\pi f_{0}t+\pi\frac{B}{T}t^{2}\big{)},\] (28) where \(B=5.1\) MHz is the bandwidth of the ultrasound probe, \(T=18.55\) us is the waveform duration, \(f_{0}\) is the central frequency of the probe, and \(\eta(t)\) is a 20% Tukey window of duration \(T\). The instantaneous frequency of the signal grows with time and thus \(e^{1}(t)\) is called an 'up' chirp.
* In the same way as \(e^{(1)}(t)\), \(e^{(2)}(t)\) is designed as a 'down' chirp waveform emitted with an angle \(\theta^{(2)}\): \[e^{(2)}(t)=\eta(t)\text{cos}\big{(}2\pi f_{0}t-\pi\frac{B}{T}t^{2}\big{)},\] (29)
* \(e^{(3)}(t)\) is designed as a 'up-down' chirp, with slopes twice as steep as \(e^{(1)}(t)\) and \(e^{(2)}(t)\), emitted with an angle \(\theta^{(3)}\): \[e^{(3)}(t)=\eta(t)\text{cos}\big{(}2\pi f_{0}t+\pi\frac{2B}{T}t^{2}\big{)},\] (30) \[for\ 0<t<\frac{T}{2};\] \[e^{(3)}(t)=\eta(t)\text{cos}\big{(}2\pi f_{0}t-\pi\frac{2B}{T}t^{2}\big{)},\] \[for\ \frac{T}{2}<t<T;\] (d) The excitation signals \(a^{(k)}(t)\) are deduced from the relation \(e^{(k)}(t)=a^{(k)}(t)*h(t)\), using the following Wiener filter:\[\Psi^{(k)}(\
u)=\frac{\left(H^{(k)}(\
u)\right)^{*}}{\left|H(\
u)\right|^{2}+ \varsigma}, \tag{31}\] \[A^{(k)}(\
u)=\Psi(\
u)*E^{(k)}(\
u),\]  
where the parameter \(\varsigma\) can be empirically tuned to bypass zero division (outside the bandwidth of \(h(t)\)) [51], and \(A(\
u)\) and \(E(\
u)\) are the Fourier transforms of \(a(t)\) and \(e(t)\), respectively.  
The resulting excitation signals \(a^{(k)}(t)\) that allow the generation of echoes of shape \(e^{(k)}(t)\) (Figure 4a-c) are shown in Figure 4d-f. These excitation signals allow the negative effects to be overcome (i.e., main autocorrelation lobe broadening, compression side lobe level increased [52; 53]), and they were successfully implemented on chirps by [54; 55; 56].  
The optimal filters \(f^{(k)}(t)\) are computed using Equation (21). The filtering results of the \(\mathrm{e}^{(k)}(t)\) waveforms with the corresponding \(f^{(k)}(t)\) filters are shown in Figure 5a-c. Each crosstalk \(\zeta_{r}^{(k)}(t)\), computed using Equation (20), is illustrated in Figure 5d-f. The amplitude of the crosstalk noise is lower than the amplitude of the compressed pulse by more than 14 dB.  
Figure 4: (**a–c**) Examples of three orthogonal \(e^{(k)}(t)\) signals. This set of waveforms contains an ‘up’ chirp, a ‘down’ chirp, and a ‘up\(-\)down’ chirp. (**d–f**) Corresponding excitations \(a^{(k)}(t)\) are used to drive a transducer of IR \(h(t)\).