**2. Inverse problems**

To see easily the two categories of inverse problems, first a very simple example is given. Consider the electrical circuit of **Figure 1**.

$$R\mathbb{C} = \theta \longrightarrow H(\omega) = \frac{\mathbb{I}}{\mathbb{T} + \overline{\theta \mathbb{W}}\omega} \to h(t) = \exp\left[-t/\theta\right] \to f(t) = h(t) \* f(t)$$

**Figure 1.**

*A simple electrical circuit example to show two different expressions of inverse problems modeling: Ordinary differential equation (ODE) or Integral equation (IE).*

Using the notations used on the figure, we can easily obtain the following ODE:

$$g(t) + \theta \frac{\partial \mathbf{g}(t)}{\partial t} = f(t) \tag{1}$$

Then, using the Fourier transform (FT), we obtain easily the following integral equation:

$$f(t) = \int f(\tau)h(t-\tau) \,\mathrm{d}\tau\tag{2}$$

These two simple equations describe the same linear inverse problem, where we can distinguish the following mathematical problems:

	- Simple: Given the characteristics of the system (either the parameter *<sup>θ</sup>* or equivalently the impulse response *h*(*t*)) and the output *g*(*t*) estimate the input *f* (*t*);
	- Blind: Given the output *<sup>g</sup>*(*t*) estimate both the system, parameter *<sup>θ</sup>* or the impulse response *h*(*t*), and input *f* (*t*).

For general vocabulary and examples, see [2, 4, 5].
