Convolution of Causal Signals

After having defined causality we can define the convolution in continous time:

$\displaystyle y(t)$	$\textstyle =$	$\displaystyle h(t) * x(t) = \int_{-\infty}^{\infty} h(t - \tau) x(\tau) d\tau$	(74)
$\displaystyle y(n)$	$\textstyle =$	$\displaystyle h(n) * x(n) = \sum_{n = -\infty}^\infty h(n) x(m - n)$	(75)

Note the reversal of integration of the function $h$

. This is characteristic of the convolution. At time $t=0$

only the values at $h(0)$

and

are evaluated (see Fig. 13). Note that both functions are zero for $t<0$

(causality!). At $t>0$

the surface which is integrated grows as shown in Fig. 13 for $t=1$

What happens if $x(t)=\delta(t)$ ? Then Eq. 75:

$\displaystyle y(t) = \int_{-\infty}^{\infty} h(t - \tau) \delta(\tau) d\tau = h(t)$ (76)

provides us with the function $h(t)$

itself. This will be used later to determine the impulse response of the filter.