from my knowledge, Power Spectral Density (PSD) should stay relatively constant with the total time sampled (or aka. N-points sampled), however I have having trouble obtaining this result.
As I know from Discrete Fourier Transform (DFT), the amplitude normalization is 1/N. (e.g Amplitude Spectrum = DFT/N). However, from various sources, the PSD is defined as (DFT * DFT-conjugate / N).
How can this be possible? It is true that the Amplitude Spectrum has a 1/N normalization constant, then shouldn't the PSD have a 1/N^2 normalization constant (since DFT is proportional to N and so is its conjugate).
More specifically, I am trying to calcuated the PSD of a continuous electric field wave using the Eq. 9 of this paper. However I can't make sense of it's constants infront of the DFT since the factors of N's cancel out leaving behind only the summation of the window function squared. I tested this result and found that the PSD does not stay relatively constant with sampling size.
In summary, I have having troubles since my PSD varies with the amount of total time of the signal sampled. Any help would be great, thanks!
I've found the PSD of a time-series does increase linearly with the number of points sampled, N, however, an appropriately FITTED function (or some sort of averaging) allows the PSD to remain constant with N. One would then take the PSD at a point on this fitted function.
This is a direct result of conserving the area of a curve, AKA Plancherel's theorem.