Understanding TSA::periodogram()

Question

I have some data sampled at regular intervals that looks sinusoidal and I would like to determine the frequency of the wave, to that end I obtained R and loaded the TSA package that contains a function named 'periodogram'.

In an attempt to understand how it works I created some data as follows:

x<-.0001*1:260

This could be interpreted to be 260 samples with an interval of .0001 seconds

Frequency=80

The frequency could be interpreted to be 80Hz so there should be about 125 points per wave period

y<-sin(2*pi*Frequency*x)

I then do:

foo=TSA::periodogram(y)

In the resulting periodogram I would expect to see a sharp spike at the frequency that corresponds to my data - I do see a sharp spike but the maximum 'spec' value has a frequency of 0.007407407, how does this relate to my frequency of 80Hz?

I note that there is variable foo$bandwidth with a value of 0.001069167 which I also have difficulty interpreting.

If there are better ways of determining the frequency of my data I would be interested - my experience with R is limited to one day.

Answer 1

The periodogram is computed from the time series without knowledge of your actual sampling interval. This result in frequencies which are limited to the normalized [0,0.5] range. To obtain a frequency in Hertz that takes into account the sampling interval, you simply need to multiply by the sampling rate. In your case, the spike you get at a normalized frequency of 0.007407407 and a sampling rate of 10,000Hz, this correspond to a frequency of ~74Hz.

Now, that's not quite 80Hz (the original tone frequency), but you have to keep in mind that a periodogram is a frequency spectrum estimate, and its frequency resolution is limited by the number of input samples. In your case you are using 260 samples, so the frequency resolution is on the order of 10,000Hz/260 or ~38Hz. Since 74Hz is well within 80 +/- 38Hz, it is a reasonable result. To get a better frequency estimate you would have to increase the number of samples.

Note that the periodogram of a sinusoidal tone will typically spike near the tone frequency and decay on either side (a phenomenon caused by the limited number of samples used for the estimation, often called spectral leakage) until the value can be considered comparatively 'negligeable'. The foo$bandwidth variable then indicates that the input signal starts to contain less energy for frequencies above 0.001069167*10000Hz ~ 107Hz, which is consistent with the tone's decay.