# Convolution

### From bio-physics-wiki

In the article on the transfer function we have seen, that the relation between the input $x$ and the output $y$ of a system can be characterized by the transfer function $G$. We denote the Laplace transform of some quantity with capital letters. For zero initial conditions we can write \begin{align} G(s)=\frac{Y(s)}{X(s)} \end{align} in this case the relation between the input and the ouput is given by the equation. \begin{align} Y(s)=G(s)X(s) \end{align}

The convolution of $x$ and $g$ is denoted $x*g$ and is defined as the integral \begin{align} y(t)=\int_0^t x(\tau) \cdot g(t-\tau) d\tau \end{align}

The most important properties of the convolution are:

**Commutativity**

\[x * g = g * x \, \tag{1}\]

**Associativity**

\[x * (g * h) = (x * g) * h \tag{2}\]

**Distributivity**

\[x * (g + h) = (x * g) + (f * h) \,\tag{3}\]

**Multiplicative identity**

\[x * \delta = x \,\tag{4}\]

Furhter more associativity with scalar multiplication holds \[a (x * g) = (a x) * g \, \tag{5}\] for any real (or complex) number \({a}\,\).

## Convolution theorem

Let us now study the Laplace transfrom of the convolution \begin{align} Y(s)&=\int_0^{\infty} e^{-st} dt \left\{ \int_0^{\infty} x(\tau) \cdot g(t-\tau) d\tau \right\}\\ &=\int_0^{\infty} g(t-\tau) \, e^{-st} dt \left\{ \int_0^{\infty} x(\tau) d\tau \right\} \end{align} now we make the variable transformation $u=t-\tau$ to get \begin{align} &=\int_0^{\infty} g(u) \, e^{-s(u+\tau)} du \left\{ \int_0^{\infty} x(\tau) d\tau \right\}\\ &=\int_0^{\infty} g(u) \, e^{-su} du \, \int_0^{\infty} x(\tau) \, e^{-s\tau} d\tau \\ \end{align}

This fundamental result is called the *convolution theorem* and also holds for the Fourier transform of the convolution.

The convolution of two functions in the time domain is equivalent with their multiplication in the frequency domain.
\begin{align}
G(s)X(s)=\int_0^{\infty} \left\{ x*g \right\} \, e^{-st} dt \\
\end{align}
Let $\mathcal{F}$ denote the (Fourier or Laplace) transfrom, then we can write the **convolution theorem** as
\[\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}\]

It also works the other way around:

\[\mathcal{F}\{f \cdot g\}= \mathcal{F}\{f\}*\mathcal{F}\{g\}\]

By applying the inverse Fourier transform \(\mathcal{F}^{-1}\), we can write:

\[f*g= \mathcal{F}^{-1}\big\{\mathcal{F}\{f\}\cdot\mathcal{F}\{g\}\big\}\]

Further Reading:

- Gianfranco Cariolaro - Unified Signal Theory
- Riley, Hobson, Bence - Mathematical Methods for Physics and Engineering
- Katsuhiko Ogata - Modern Control Engineering