Converting between Sine Amplitude and PSD

If you are doing sine on random mixed mode testing it is often useful to be able to convert between PSD and sine amplitude and visa versa.

The following simple formulae can be used:

As an example consider you are generating a 2.5g sine tone and want to check its amplitude on a measurement channel that is displaying a PSD result. The frequency range for the random is set to 3200Hz with 800 line resolution hence the frequency resolution is 3200 / 800 = 4 Hz.

The expected PSD amplitude for a 2.5g sine tone would be 2.5 * 2.5 / 4 / 3 = 0.5208 g^2/Hz.

Potential errors

At least that is what the value is when the sine wave frequency is actually on an FFT line, ie 4, 8, 12 , 16 Hz etc in this example.

A signal processing phenomenon called FFT "leakage" causes errors when the sine tone is between FFT lines. This error is well defined and illustrated in the following graph:

Sine amplitude curve black

This graph shows the resulting sine amplitude value as the sine frequency is varied between FFT lines. Half way between lines there is an apparent reduction in peak amplitude of about 15% (-1.4dB). Converting this to the PSD error results in a worst case 28% reduction, ie our 0.5208g^2/Hz example could read as low as 0.375 g^2/Hz.

This error is the reason that m+p vibration controllers analyse and display sine tones separately from the random signal background so that an accurate measurement, control and display of the sine levels can be achieved.

Another possible source of error when measuring sine amplitudes using the broadband display is when the sine tone is sweeping. The averaging process tends to cause the tone to lag behind on the display (you will see a "tail" on the sweeping tone) and this also suppresses the PSD value. This effect is not easily calculated and it's best just to say that the slower the sweep the better the result. A fixed frequency tone is best!

Another method

You may now be thinking that perhaps the sine amplitude can not be accurately measured from the PSD however there is a better and simple way to do this calculation:

With a fixed frequency signal put the cursor on the maximum PSD value, note this, move the cursor to the line below and above this and note those values. Add the three PSD values, multiply by the frequency resolution and take the square root. The result is the rms amplitude of the sine tone and can be converted to the normal peak value by multiplying by 1.414 (ie root 2).

Using this method also reduces the leakage error to below 2% (-0.17dB).

Taking our example above and assuming the sine tone is on an FFT line then the peak PSD will measure 0.5208 and the two adjacent lines will be 0.1302 so:

add them, 0.5208 + 0.1302 + 0.1302 = 0.7812 multiply by the resolution, 0.7812 * 4 = 3.125 take the square root and multiply by root 2, 1.768 * 1.414 = 2.5 g peak

As an exercise try this same calculation using the worst case PSD reading you would see from the first example above. You would see two lines at 0.375 and the third line would be very small so could be ignored.

By adding a few more lines this technique will also produce a good result even if the sine tone is sweeping !

This technique does assume that the sine tone amplitude is well above the random background level.

Final note

The conversion formulae apply to the m+p VibControl range of systems and all others that use a Hanning window in the spectrum analysis.