Convolution - Signal Processing of Random Physiological Signals - page 104

Convolution Results B

100

90

80

70

60

50

40

30

20

10

0

1 2 3 4 5 6 7 8 9 10

Time ( t )

FIGURE 10.29 : Graph of Results B

It should be noted that the procedures and results shown in Tables 10.2 and
10.3 and Figs. 10.28 and 10.29 are comparable, but the procedure that locates discon-
tinuities along one of the steps in the approximation gives better results. In the exam-
ple, the results labeled A (Table 10.2 and Fig. 10.28) are better for the discontinuity
at t = 0.

10.3 CONVOLUTION ALGEBRA
Convolution is referred to as a “Linear Process” in the arithmetic sense; therefore, let
us examine those mathematical properties that make convolution a linear process. Re-
call the mathematical properties of the “Real Numbering System” whose properties
are

1. associative,

2. commutative, and

3. distributive.

Consider the convolution of two functions, one of the functions being the result
of a previous convolution process. First, “Convolution is Associative,” which means that