Correlation Functions - Signal Processing of Random Physiological Signals - page 80

10

8

6

4

2

0

6543210123456
Time ( t )

FIGURE 9.5 : Incorrect autocorrelation results. Note the lack of symmetry in the graph of the
incorrect calculation results of Table 9.2

For the Direct Numerical Method via polynomial multiplication, use the following

steps:

1. directly tabulate values of the function in the second row with the increment of

time above the function as in Table 9.1;

2. reverse tabulate the same function in the third row;

3. do normal polynomial multiplication (from right to left);

4. write the products in columns following normal multiplication procedures;

5. when there are no more products, sum the columns; and

6. multiply each sum by the interval width.

It is important to “Reverse tabulate” the second function; otherwise, the incorrect
answer is obtained as shown in the calculations of Table 9.2. The graph of the incorrect
results is shown in Fig. 9.5. Note the lack of symmetry.

9.5 CROSS-CORRELATION FUNCTION
For pairs of random data from two different stationary random processes, the joint
statistics of importance include the cross-correlation and cross-spectral density functions.
The cross-correlation theorem for periodic functions f 1 ( t ) and f 2 ( t ) applies to two
different periodic functions, but with the stipulation that the functions must have the