I am working on ultrasound scan and the processing of the signal received by the probe. I made the model I wanted and as I do not have an ultrasound scan machine I want to simulate the signal processing.
I will do that with the Python package Scipy and the function [scipy.signal.lsim2][1] But here is the problem. To use this function I need the Z-transform of a sequence... And I only know few things about Z-transform. I only know a bit about the Laplace Transform and I think I understood that the Z-transform is the discrete version of Laplace transform.
Let me explain what is my problem. Let's say $u_s$ is the processed signal and $u_e$ the signal sent by the probe to the computer. I got this differential equation :
$$ \frac{du_s(t)}{dt} = -\frac{u_e(t)}{ \tau } - \frac{u_s(t)}{\tau'}$$
What I need is the transfer function $ \frac{u_s(t)}{u_e(t)}$ . In the Laplace domain I know I would have :
$$\frac{U_s}{U_e} = \frac{-1}{p \tau + \tau \tau'} $$
I know that to apply the Z-transform I need to consider $u_e$ and $u_s$ as sequences by sampling them. So now I need to get the Z-transform. I consider the sequences $(u_{sn})$ and $(u_{en})$. By using the Taylor series (I'm french, I dont know if it's the correct name) the differential equation become : Let's take $n \in \mathbb{N}$
$$ \frac{u_{s(n+1)} - u_{sn}}{T_e} = -\frac{u_{en}}{\tau} - \frac{u_{sn}}{\tau'}$$
But then I don't know what to do to get the sequence equivalent of the transfer function. Could you help me please ? Also, I didn't know how to post this message, I thought that MathStack would be the better place.
Thanks in advance. [1]: https://docs.scipy.org/doc/scipy/reference/generated/scipy.signal.lsim2.html