Spectrum Analysis Basic Notation and Principles - A Simple Example
Shumway (1988) presents a simple example to clarify the underlying "mechanics" of spectrum analysis. Let us create a series with 16 cases following the equation shown above, and then see how we may "extract" the information that was put in it. First, create a variable and define it as:
x = 1*cos(2*π*.0625*(v0-1)) + .75*sin(2*π*.2*(v0-1))
You can produce this variable by typing in the formula in the long name formula field. This variable is made up of two underlying periodicities: The first at the frequency of n=.0625 (or period 1/n=16; one observation completes 1/16'th of a full cycle, and a full cycle is completed every 16 observations) and the second at the frequency of n=.2 (or period of 5). The cosine coefficient (1.0) is larger than the sine coefficient (.75). The spectrum analysis summary results spreadsheet computed by the Time Series module is shown below.
| Spectral analysis:VAR1 (shumex.sta)
No. of cases: 16 |
|||||
| t | Freq-
uency |
Period | Cosine
Coeffs |
Sine
Coeffs |
Period-
ogram |
| 0 | .0000 | 16.00 | .000 | 0.000 | .000 |
| 1 | .0625 | 8.00 | 1.006 | .028 | 8.095 |
| 2 | .1250 | 5.33 | .033 | .079 | .059 |
| 3 | .1875 | 4.00 | .374 | .559 | 3.617 |
| 4 | .2500 | 3.20 | -.144 | -.144 | .333 |
| 5 | .3125 | 2.67 | -.089 | -.060 | .092 |
| 6 | .3750 | 2.29 | -.075 | -.031 | .053 |
| 7 | .4375 | 2.00 | -.070 | -.014 | .040 |
| 8 | .5000 | -.068 | 0.000 | .037 | |
Let us now review the columns of this spreadsheet. Clearly, the largest cosine coefficient can be found for the .0625 frequency. A smaller sine coefficient can be found at frequency = .1875. Thus, clearly the two sine/cosine frequencies which were "inserted" into the example data file are reflected in the spreadsheet.