# Transfer Function

In control theory the transfer function describes the relationship between the input and the output of a linear time-invariant (autonomous) system. If a system represented by the equation \begin{align} a_0 \, y^{(n)} + a_1 \, y^{(n-1)} + \dots + a_{(n-1)} \, y + a_n = b_0 \, x^{(m)} + b_1 \, x^{(m-1)} + \dots + b_{m-1} \, x + b_m \end{align} has the input also called driving function $\sum_i b_i \, x^{(m-i)}$ (where powers of $x$ denote derivatives when written in brackets), then the output is given by the solution $y$ to the ODE. In compact notation a general systems can be describes by \begin{align} \sum_{i=0}^n a_i \, y^{(n-i)} = \sum_{i=0}^m b_i \, x^{(m-i)} \end{align}

The transfer function $G(s)$ of a linear time-invariant system of differential equations is defined as the ratio between the laplace transform of the ouput (response function) and the laplace transform of the input (driving function), when all initial conditions are zero. \begin{align} G(s)=\left[ \frac{\mathcal{L}[output]}{\mathcal{L}[input]} \right]_{\text{zero i.c.}}=\frac{Y(s)}{X(s)}=\frac{\sum_i b_i \, y^{(m-i)}}{\sum_i a_i \, x^{(n-i)}} \end{align} By the use of Laplace transform it is possible to represent the system dynamics by algebraic equations.

## Transfer function for an Impulse input

For an impulse input the transfer function takes a very simple form. Acording to the property $\delta *x =x$ of the convolution we simply get \begin{align} Y(s)=\int_0^{\infty} \left\{ \delta*g \right\} \, e^{-st} dt=\int_0^{\infty} \left\{g \right\} \, e^{-st} dt=G(s) \\ \end{align} \begin{align} Y(s)=G(s) \\ \end{align} The inverse Laplace transform of $g(t)=\mathcal{F}^{-1}G(s)$ is called the impulse-response function or weighting function of the system. Notice, that we can determine the weighting function by measuring the response of a system to a unit impulse. In this case the weighting function is the inverse Laplace transform of the system response \begin{align} g(t)=\mathcal{F}^{-1}Y(s) \end{align}