Analyzing Power Spectra of Series: Frequency Scaling

[nastia]

Homework Statement

I am analyzing power spectrum of the series obtained using different approaches. I have 100 points series. First method is obtaining power spectrum by taking squared FT of series and divided by the period. Second method is taking IFT from ACVF of the series. In both cases, in terms of frequency, spectrums are the same, but the amplitudes are different, amplitudes are significantly higher for the first case (about 1000 times).
Also, as an additional question to that, I padded with 100 zeroes the original series in order to get the series 200 points long and the power spectrum is again the same, but with different amplitude

Homework Equations

The Attempt at a Solution

I know, that in first case we take FT out of series with length 100 points, in the second and third cases the length of series 200 points. And it has something to do with frequency scaling. But how exactly it is related and why? I can't seem to find the answer.

 

[mighty2000]

Hello,

Thank you for sharing your analysis and questions about the power spectrum of your series. As you have observed, the amplitudes of the power spectrum can vary depending on the method used for calculation. This is due to the different techniques used to obtain the power spectrum and the inherent assumptions and limitations of each method.

In the first method, you are taking the squared FT of the series and dividing by the period. This is known as the periodogram method and assumes that the series is stationary (meaning the statistical properties of the series do not change over time) and that the data is evenly spaced. The amplitude of the power spectrum in this method is affected by the length of the series and the spacing between data points. In your case, since you have 100 points in your series, the amplitude will be higher compared to the second method where you have 200 points.

In the second method, you are taking the IFT from the ACVF of the series. This method is known as the autocorrelation function (ACF) method and assumes that the series is a linear combination of sinusoids. The amplitude of the power spectrum in this method is affected by the shape of the autocorrelation function, which can vary depending on the length of the series. In your case, since you have 200 points in your series, the amplitude will be lower compared to the first method where you have 100 points.

The third method, where you padded the original series with 100 zeroes, also affects the amplitude of the power spectrum. By adding zeroes to the series, you are essentially increasing the spacing between data points, which can lead to a decrease in the amplitude of the power spectrum. This is because the frequency resolution of the power spectrum is inversely proportional to the length of the series. In other words, the longer the series, the higher the frequency resolution and the lower the amplitude of the power spectrum.

I hope this helps to clarify the relationship between the length of the series and the amplitude of the power spectrum in different methods. Keep in mind that there are also other factors that can affect the amplitude, such as the shape of the series and the presence of noise. It is important to carefully consider the assumptions and limitations of each method when interpreting the results of your analysis.

Best of luck with your research!