Power Spectral Density Confusion

[swraman]

Hi,

I am working on a project and came across a conceptual roadblock.

I am working with a PSD, let's say the units are V^2/Hz.

I choose a dF based on how many sine tones I want in the time signal I am going to create.

I sample the PSD curve at my frequency bin locations; so I have both my frequency vector (0:dF:fMax) and my PSD sampled at each frequency bin.

I multiply my sampled PSD by dF, then take square root of each bin to get a amplitude spectrum. I add a phase to each amplitude bin, and modify my resulting spectrum (amplitude+phase) so that it has Hermitian Symmetry.

IFFT, results in a purely real time signal.

All makes sense up till here.

Now, if I take this tiem frame and concatonate it with itself, I get a frame twice as big. If I FFT that 2x frame, I get a resulting FFT in which the magnitude of every other bin is 0 - which makes sense since the time signal wasn't generated with any of the 0-magnitude frequencies.

But, if I convert this FFT to a PSD, I will still see that every other point is 0 value.

This is my confusion - I thought that every PSD should be the same, regardless of what your sampling parameters are. It makes sense to me why every other bin is zero - but what bit of logic am I overlooking that would predict that my PSD will not match when sampled with 2N points? I thought PSDs were supposed to deal with the problem of sampling using different dFs.

 

[Baluncore]

When you concatenate two identical time sequences you generate even frequencies while eliminating the odd. All odd frequency bins will be zero.
When you stretched the spectrum by concatenation, gaps with zero energy opened at all the odd frequency bins.

The power spectrum is the square of the amplitude spectrum. If there is no amplitude in a particular frequency bin, then there can be no power. 0 * 0 = 0. You are looking at the energy in individual bins, not the sum of several adjacent bins, so the zeros from the amplitude spectrum are preserved in the power spectrum.

 

[swraman]

THanks - that part I do get. My question really is about PSDs. As I understand them, they are a way to give anyone a spectral profile and ensure that when they apply that profile, the FFT parameters they select will not affect the total power of the signal.

I guess another angle for the question I had is this. Am I correct in thinking that there is a big "flaw" in PSDs. They are intended to normalize power of a signal generated from a spectrum; so the process does not depend on your FFT parameters. But, the larger of a dF you use, the fewer frequency components will be in a time signal generated from the PSD.

FOr example, let's say I am using a vibration PSD (g^2/Hz) to generate an acceleration time signal to excite a spring/mass/damper that has a high-Q resonance at 67Hz. If my dF chosen is 15, I will generate a time signal that has content at 60Hz and 75Hz, but will not significantly excite the resonance at 67Hz unless I use a lower dF. Is this understanding correct?

 

[Baluncore]

I think the application of the window function is important in your situation. https://en.wikipedia.org/wiki/Window_function

A step where the ends of the time signal wrap around should add high frequency noise and spreads the bins. If you do not multiply your time data by a window function before the FFT you will have deep notch filters between each df bin. There are implications when you use an IFFT to generate a stimulus which has a particular power spectrum. A reverse window could be applied to spread the peaks and fill the notches.

 

[swraman]

I think the application of the window function is important in your situation. https://en.wikipedia.org/wiki/Window_function

A step where the ends of the time signal wrap around should add high frequency noise and spreads the bins. If you do not multiply your time data by a window function before the FFT you will have deep notch filters between each df bin. There are implications when you use an IFFT to generate a stimulus which has a particular power spectrum. A reverse window could be applied to spread the peaks and fill the notches.

Interesting. I never knew windows were used to intentionally create spectral leakage; I've only ever seen them used to try and prevent it.