Convolution - Signal Processing of Random Physiological Signals - page 98

6

f 2 ( t )

4

2

0 2 t

FIGURE 10.25 : Result from the second translation. The second value after integration of the
convolution function, f 3 , with the limits between 0 < t < 2, is shown by the shaded area in the

figure

The next step is to repeat the translation step by moving the pulse function one
increment to the right; hence, the limits of integration are changed to the interval of
0 < t < 2, as given by (10.18).

t

e −2 τ ∂τ

f 3 ( t ) = 6 (10.18)

0

The result of the second translation and including the multiplication and integra-
tion steps is shown in Fig. 10.25. Note that the width of the area is equal to the width of
the pulse function. As the pulse is translated to the next interval between 0 < t < 3, it

should be noted that the common area is reduced to the interval between 1 3 (the

width of the pulse). Hence, for this example, the area is smaller than the previous one.
As the pulse is continuously translated, the value of the integrated area approaches zero.
The process repeats the three steps:

< t <

1. Translation along the horizontal x -axis

2. Multiplication of the two functions, f 1 ( τ ) and f 2 ( t − τ )

3. Integration of area bound by the limits of integration