Convolutions
============

A convolution in the time domain becomes a multiplication in the
$s$-domain:

.. math::
   x(t)*h(t) \ltarrow X(s) H(s) 

For the Fourier transforms (discrete and continuous time) we also had
the property that multiplication in the time domain corresponds with
convolution in the frequency domain. A direct translation of this
property cannot be made because the $s$ domain is complexed valued and
convolution is not well defined in that case.

Just for completeness we give the Laplace transform of $x(t)y(t)$:

.. math::
   \frac{1}{2\pi j} \lim_{T\rightarrow\infty} \int_{c-jT}^{c+jT}
   X(\sigma)G(s-\sigma)d\sigma 

The integration is done along a vertical line $Re(\sigma)=c$ that lies
entirely in the ROC of the Laplace transform $X(s)$.