Convolution

For each \(t\), the convolution weighs past values of \(f\) with \(g\) read backwards: as \(\tau\) grows, \(g(t-\tau)\) looks at instants ever closer to the origin. That is why it describes how a system accumulates the effect of an input \(f\) filtered by a response \(g\). It is commutative, associative and distributive over addition, though \(f*g\) is not the product \(fg\).

The convolution theorem states \(\mathcal L\{f*g\}=F(s)\,G(s)\). Hence \(\mathcal L^{-1}\{F(s)G(s)\}=f*g\): instead of partial fractions you can integrate. Convolving with \(1\) is integrating from \(0\): \((f*1)(t)=\int_0^t f\), which corresponds to dividing by \(s\).

Guiding example: \(\dfrac{1}{s^2(s-1)}=\dfrac1{s^2}\cdot\dfrac1{s-1}\) is the transform of \(t*e^{t}=\int_0^t\tau e^{t-\tau}d\tau=e^{t}-t-1\). In equations, if \(y''+y=g(t)\) with zero initial conditions, then \(Y=\dfrac{G(s)}{s^2+1}\) and \(y=\sin t*g\) for any forcing \(g\).

Typical mistakes: thinking \(\mathcal L\{fg\}=F\,G\) (that holds for \(f*g\)) or that \(f*1=f\) (it actually integrates). In the integral, \(\tau\) is the dummy variable and \(t\) is fixed; the result is a function of \(t\).