4.1.5. Convolutions

A convolution in the time domain becomes a multiplication in the \(s\)-domain:

\[x(t)*h(t) \ltarrow X(s) H(s)\]

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 \(x(t)y(t)\):

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

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