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Overall rms Variations when Converting Waveforms to Spectra


QUESTION

When I convert a time waveform to a spectrum I notice that the overall rms value changes a little. Why is this?

ANSWER

The overall rms for a time waveform is calculated digitally (by taking the square root of the sum of the square of every sample).

The overall rms for a spectrum is also calculated digitally (by taking the square root of the sum of the square of every FFT bin). If you were to work through the math (integrals and cosine functions - very messy) you would find that this amounts to exactly the same calculation as for a time waveform.

(You can tell that there must be a but coming.)

But in practice you DO see a slight difference. This is caused by windowing.

The difference between the mathematical world and the real world is that a vibration signal is continuous in time, but we only record a short period of it with an abrupt beginning and end. We must use window functions to get around this issue and avoid bad things like spectral leakage.

By windowing the waveform we are reducing the amplitude of the samples near the ends. From here on in we have to use a window factor to scale up any rms calculations so that we compensate for the power lost by zeroing out the ends of the waveforms.

When we calculate the rms value of the spectrum by adding together the frequency bins, we are actually calculating the rms of the windowed waveform then scaling it up by a window factor. But this scaling factor is a theoretical value that will only work for - you guessed it - stochastic signals i.e. signals where one bit of the waveform is just as good as another. In reality, our waveforms aren't uniform from start to finish.

To illustrate: You have a waveform of 1024 points. The very last sample is one thousand times higher than anything else in the waveform. When we calculate the rms of the waveform it comes out to be a huge number because of the effect of that sample near the end. Now we take the same waveform, apply the Hanning window and take the FFT. The Hanning window actually comes down to zero at the ends and it zeroes out that last sample, so the spectrum doesn't have any hint that such a huge amplitude value ever existed. The rms value calculated by adding together the frequency bins of the spectrum will be much lower than that of the waveform.

An easy way to think about it is that the windowed FFT mainly uses the 'middle' part of the time waveform. So if your time waveform happens to have a burst of noise in the middle the spectrum will be high. Alternatively, if there is a low amplitude period in the middle (relative to the ends) the spectrum will be low.

When recording spectra we use averaging and overlap to fix this problem caused by windowing. It is typical to take four averages with an overlap of 50%, which will more than compensate for any error introduced by the windowing.