Convolution - Signal Processing of Random Physiological Signals - page 100

However, when the functions are of infinite duration, the computations become lengthy
as the overlapped area becomes very large. There are alternative methods that can be
applied for simple problems or when a computer is available.

10.2.1 Direct Calculation
10.2.1.1 Strips of Paper Method
One of the numerical methods is to use two strips of paper and to follow the following
steps:

1. tabulate values of the two functions at equally spaced intervals on two strips of

paper;

2. reverse tabulate one function; f ( − λ ), which is equivalent to folding a function;

3. place strips beside each other and multiply across the strips and write the

products;

4. slide one strip, and repeat step three, until there are no more products;

5. sum the rows; and

6. multiply each sum by the interval width.

10.2.1.2 Polynomial Multiplication Method
Another method is the polynomial multiplication method (Table 10.1). For this method,
the values of the two functions are placed in two rows one function above the other
(without any reverse tabulation) as in Table 10.1. It should be noted that the functions
are aligned to the left and not to the right as in normal multiplication. The alignment
and multiplication from left to right serves as equivalence to folding of a function about
the y -axis. The number of terms in the results should equal the sum of the number of
terms in the two functions. The final results of the convolution operation are obtained by
summing the columns and multiplying the resulting summations by the interval width.
At this point, it is worth discussing an area of possible confusion: that of estab-
lishing the proper relationship of the time variable to the sample values employed in the