Understanding Power Spectral Density and the Power Spectrum 

Compare pwelch, pspectrum and periodogram. Also comment on spectral mask and windowing for standard specific waveforms

@@srikondapa11e This video doesn't cover pwelch or any averaging of the power spectrum. It's mostly about how to scale an FFT to get the visualization you're looking for and why you may choose to look at power instead of power density and vice versa. There is a short section on windowing, but only to explain how the scaling we are doing in the video is only valid for rectangular windows. Maybe I can cover your other concerns in a different video. Thanks for the comment!

Hi Brian! Thank you so much for all the educational videos about signal processing and control engineering! I was wondering if you could make some videos about wavelet transforms? That would be amazing!

A periodic signal is a Fourier series (i.e. a countable sum of sharp frequencies). I understand that for this a Power Spectrum is more suitable/meaningful than than a PSD. Is it so? Can you elaborate?

I really love to watch your video! Brian. Very good lectures!

Thanks!

Great question. I do answer that in the video so check back on Wednesday when it's live :)

Thanks for this information, I'm interested in this video

Hi, Computation of Power Spectral Density (PSD) is reliant on various factors, including the frequency bin width parameter determined by the number of points in the Fast Fourier Transform (FFT) and the signal bandwidth. This dependency underscores the importance of selecting an appropriate frequency bin width to accurately represent the power distribution across the frequency spectrum. If the bin width is too coarse, the PSD may lack resolution and fail to capture important frequency components, despite a quality input signal. Conversely, overly fine bin widths may introduce noise and artifacts, particularly in regions of low signal-to-noise ratio. Thus, achieving an optimal balance in frequency bin width is essential to ensure that the PSD effectively characterizes the underlying signal's power spectrum. How to reach Optimal balance ? // thanks in advance

Thanks for the comment. 'lt' is the length of the FFT. If 'lt' is shorter than the signal length then increasing it will improve frequency resolution - since you're adding more signal to the FFT. If 'lt' is longer than the signal itself (like I have in my script) then the pscpectrum function zero-pads the data to the length. When you zero-pad, that doesn't change the amplitude of the returned power or the frequency resolution, it just fills in the frequencies that were previously skipped over. So, it shrinks the sampling frequency but it's not actually giving you more resolution. If those frequencies have higher or lower power values then it'll look like the amplitude is changing. But if you keep increasing 'lt' until the frequency spectrum is really dense you'll see the true shape of the spectrum (given the finite data you're looking at). For example, if the signal has a dominant frequency at 10 Hz, but your data length is such that the sampled frequencies are at 9 Hz and 11 Hz, then you'll jump over the peak and the returned power will look small. But if you her-pad the data such that you sample also at 10 Hz, then you've hit the peak and the returned power will look like it increases. But all you're doing is actually sampling at a frequency that has higher power - not changing it. So, as long your computer can perform the FFT in a reasonable amount of time, then you could zero-pad enough to fully capture the shape of the power spectrum - but just remember that you're not improving resolution, just visualization of the spectrum. I explain it a bit in my video called "Understanding the Discrete Fourier Transform and the FFT" starting at 14:40. I'd put a link here but my comments get rejected with links. Good luck!

If you have a non-periodic signal then time-frequency plots are usually the way to go. For example, you could take an FFT of a small subset of the data at the beginning, then move to the next set of data and take and FFT again, and move it again and so on. In this way, you can see how the amplitude/power/power density changes over time.

When a time vector with the sampling times t is available I sometimes have used fs = 1/median(diff(t))

@@GabrieleBunkheila thanks, I'll explore this more. 👍

I'm not sure of the question exactly, but if you are looking at normalized audio data then haven't you removed any notion of absolute power? You've already manipulated your audio signal and so a PSD would return the power density of that normalized signal and there wouldn't be a way to compare that to the recommended power density levels. Could you give more details?

Hi Brian, thanks for your wonderful videos and replying to my concern. So the audio file is in wav format. I am reading it in matlab and the converted variable is between (-1,1). Here i am unable to find the normalising constant. However when I play it using sound() command, I can hear the audio fine. How is the actual audio power conversion happening in these functions? Now, On finding the PSD of the audio wave, the values of power are in negative dB (as it is normalised) hence I am unable to find the actual power of the audio signal. Will the power change if I assume the normalising constant based on recording device depth (like 16bit/24bit)?