An example is presented in this page. Both the functions, h(t) and f(t), are pulses.
Given the two waveforms shown in Fig. 21, obtain the convolution integral of h(t) and f(t). Let
y(t) = f(t) * h(t).
SOLUTION :
Convolution Integral Approach: Analytical Technique
Given h(t) and f(t), the convolution integral of h(t) and f(t) can be obtained as presented below. From Fig. 21,
Replace t by λ. Then we get that
Function f(t - λ) is a waveform that is obtained by reflecting f(t) and shifting to the right as the value of t increases from zero onwards. That is,
For t < 0, f(t - λ) lies on the left side of vertical axis and there is no overlap of h (λ) and f(t - λ) and the convolution integral has zero value.
It can be seen from the equation for f(t - λ) that it has a value of unity when (t - 2) < λ< t. For 0 < t < 1, h( λ) that it has a value of unity. Hence the convolution integral can be obtained as shown in Fig. 22.
When t =1, there is overlap of h ( λ) and f(t - λ) as shown in Fig. 23 and the convolution integral equals unity at t =1, as seen from equation (6.7).
For 1 < t < 2, part of f(t - λ) overlaps with the whole of h ( λ) , as shown in Fig. 24.
We can evaluate the convolution integral for 1 < t <3, as shown below.
For 2 < t < 3, part of f(t - λ) overlaps with a part of h ( λ) , as shown in Fig. 25.
Equation (6.9) shows how the convolution integral can be evaluated. When t =3, there is no overlap of h ( λ) and f(t - λ) as shown in Fig. 26 and the convolution integral equals zero at t =3, as seen from equation (6.9).
Laplace Transforms Approach
The Laplace transform of a time-shifted pulse is presented below.
Hence for this problem,
The inverse transform of Y(s) yields the convolution integral.
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