2.2 Dsp00108-averaging time series  (Page 9/14)

The Fourier transform

In order to perform the spectral analysis, I will perform a Fourier transform on the time series to transform that data into the frequency domain. Then I willplot the data in the frequency domain.

(This module will not provide technical details on the Fourier transform. That information will be forthcoming in a future module.)

Keeping it simple

To keep this explanation as simple as possible, I will stipulate that all of the sinusoids contained in the time series are cosine functions. There are nosine functions in the time series.

(If the time series did contain sine functions, the process would still work, but the explanation would be more complicated.)

A brief description of the fourier transform

Before I get into the results, I will provide a very brief description of how I performed the Fourier transform for these experiments.

The following steps were performed at each frequency in a set of 400 uniformly spaced frequencies across the frequency range from zero to the foldingfrequency.

The steps were:

• If the time series was shorter than 400 points, extend it to 400 points by appending zero-valued points at the end.
• Select the next frequency of interest.
• Generate a cosine function, 400 samples in length, at that frequency.
• Multiply the cosine function by the time series.
• Compute the average value of the time series produced by multiplying the cosine function by the time series.
• Save the average value. Call it the real value for later reference.
• Generate a sine function, 400 samples in length, at the same frequency.
• Multiply the sine function by the time series.
• Compute the average value of the time series produced by multiplying the sine function by the time series.
• Save the average value. Call it the imaginary value for later reference.
• Compute the square root of the sum of the squares of the real and imaginary values. This is the value of interest. Plot it.

Why does this work?

No matter how many sinusoidal components are contained in the time series, only one (if any) of those sinusoidal components will match the selected frequency.

Multiply by the cosine and average the product

When that matching component is multiplied by the cosine function having the selected frequency, the new time series created by the multiplication willconsist of a constant value plus a sinusoid whose frequency is twice the selected frequency.

The computed average value of this time series will converge on the value of the constant with the quality of the estimate depending on the number of pointsincluded in the average.

Multiply by the sine and average the product

Since the sinusoids in the time series are stipulated to be cosine functions, when the sinusoid with the matching frequency is multiplied by the sinefunction, the new time series will consist of a constant value of zero plus a sinusoid whose frequency is twice the frequency of the sine function.

The computed average of this time series will converge on zero with the quality of the estimate depending on the number of points in the average.