Correlation Functions - Signal Processing of Random Physiological Signals - page 76

3. Positive incremental translation along the horizontal x -axis (one delay)

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

5. Integration of area

Steps 3 through 5 are repeated until 2 N delays are completed
The process might be synopsized as determining the area under the curves (prod-
uct) as the folded function is slid along the horizontal axis. Note the direction of trans-
lation; the conventional notation is as follows:

a. Movement to the right for positive ( + ) time

b. Movement to the left for negative ( − ) time

At this point, let us go through an example to visualize the autocorrelation process.
The function f 1 is shown in Fig. 9.1 as two repeated graphs. To simplify the example,
the change of variable will not be preformed or shown.
The second step is to translate or move one of the functions in the negative direction
without reversing the function. Figure 9.2 shows what the graph of the two functions
would look like. Note that x

0 is in the middle of the axis.
The next step is that of moving the translated function in a positive direction by 2
delays (from 0 to

=

2) along the x -axis as shown in Fig. 9.3. Note that the overlapping
area is the area of both functions between 0 and

+

+ 2onthe x -axis.

Autocorrelation first step

2

1.5

1

0.5

Original f

0

0 1

2

3

4

5

6

Time ( t )

FIGURE 9.1 : Starting the autocorrelation process. The function f 1 is shown as two separate
3-D objects