# Fresnel

## Fresnel simulator

USTB comes with a simple ultrasound simulator. It is based on the Fresnel approximation for rectangular transducers and LTI system theory. Its performance and accuracy cannot compare with that of other simulation tools, such as Field II, but its integration with USTB makes using it a breeze.

### Theory

A rectangular element, with width $$a$$ and height $$b$$, and orientation $$(\theta_0,\phi_0)$$ is placed at $$\mathbf{s}=(s_x,s_y,s_z)$$. The impulse response from that element to the evaluation point $$\mathbf{r}=(r_x, r_y, r_z)$$ can be modelled as

$$h(t,\mathbf{s},\mathbf{r}) = \frac{\delta(t-r/c_0)}{4\pi r} \, D(\theta,\phi)$$

where $$r$$ is the distance between $$\mathbf{s}$$ and $$\mathbf{r}$$, $$c_0$$ is the speed of sound,

$$D(\theta,\phi) = sinc \left( \frac{ka}{2} \tan \theta \right) sinc \left( \frac{kb}{2} \tan \phi \sec \theta \right)$$

is the element directivity, $$k$$ is the wavenumber, and

$$\theta=\arctan\left(\frac{r_x-s_x}{r_z-s_z}\right)-\theta_s,$$
$$\phi=\arcsin\left(\frac{r_y-s_y}{r}\right)-\phi_s.$$

We define the transmitted signal $$e(t)$$ as gaussian-modulated RF signal

$$e(t) = \cos \left(2\pi f t \right) e^{-2.77 \left(1.1364 \, t \, \Delta f\right)^2}$$

where $$f$$ is the center frequency in Hertz, and $$\Delta f$$ is the signal bandwidth in Hertz.

These examples show how to use the built-in Fresnel simulator.