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When one thinks of noise in a signal it is generally associated with having been added in some way to the amplitude of a signal. This is not always the case.

The two sinewaves shown in Figure 1 appear similar with the same frequency and amplitude except that the top one appears to have some noise or distortion present.

If one is overlaid on the other (Figure 2) then the noise is more apparent.

The amplitude (modulus) spectrum for both is identical as shown in Figure 3.

However, if we look at both the amplitude and the phase spectra (Figure 4) then there is a clear difference. All the noise is in the phase component!

If we were just to consider the amplitude spectrum, or the autospectrum, then both signals would agree and we would not detect any difference. In this example the added phase noise was Gaussian with a standard deviation of 15^{o}.

If the phase noise is increased by a factor of 10 to a standard deviation of 150^{o} then the distortion to the signal is readily observed (Figure 5). But it looks just like it was added to the amplitude.

As previously, when looking at the spectrum, then all the noise is in the phase (Figure 6).

If the phase noise is decreased by a factor of 10 to a standard deviation of 1.5^{o} then there is no apparent distortion and the signals appear identical (Figure 7).

But, as we see in figure 8, the noise is still present in the phase spectrum. As before the amplitude spectra appear identical.

This is not totally true as there are no observable differences in the amplitude spectra. If the amplitude spectra are plotted on a dB scale then the noisy signal has a lower signal to noise ratio. However, the amplitude noise floor is nearly 100dB down so if only the amplitude is being examined, it is clearly possible to erroneously conclude that the signal we know is noisy is a ‘clean’ signal.

A change in phase in the frequency domain is of course a time delay in the time domain. The general equation of a sinewave $y(t)$ is usually written in the form

\[

y(t) = Asin(2{\pi}ft + {\phi})

\]

where $A$ is the amplitude, $f$ is the frequency and $\phi$ is the phase in radians. If the phase is time varying then it is better to write this in the form

\[

y(t) = Asin(2{\pi}ft + {\phi}(t))

\]

We may of course write it in the form

\[

y(t) = Asin(2{\pi}f\{t+{\phi}(t)/2{\pi}f\})

\]

Or as

\[

y(t) = Asin(2{\pi}f\{t+ {\tau}(t)\})

\]

where it is now easily recognised as a time delay.

What would cause such a signal? One possibility is that in a sampled data system the sample rate clock was behaving erratically. This would not have been uncommon in the last century, but is very unlikely now. A more likely possibility is if the sampling rate is controlled by say an optical tachometer and the tachometer is itself mounted on a vibrating structure, then this will impose the equivalent of a time delay. The nature of the delay is of course dependent on how the tachometer is vibrating. Rather than being random it is probably more likely to introduce phase modulation. A quite severly phase modulated signal (blue) and the ideal pure sinewave (red) are shown in Figure 9 below.

The result looks similar to the random phase effect. However, the frequency spectrum of the phase modulated signal is very different to that from a random phase signal. In the example shown here the main carrier is at 64Hz and the phase modulation is occuring at 170Hz.

Figure 11 shows that the phase is very clean and we have three significant frequencies in the amplitude spectrum at 64Hz, 106 Hz and 234Hz. The 234Hz signal is simply the sum of 64 and 170, and the 106Hz peak is the difference. There are no phase changes at 106Hz and 234Hz but there are phase changes with no significant amplitude change at 276, 404, 616 and 744 Hz. These latter frequencies may be recognised as

276 = (2*170-64), 404 = (2*170 + 64), 616 = (4*170-64) and 744 = (4*170 + 64).

Another possible cause is that the medium through which the signal travelled was subject to some rapid random variations. For example the speed of sound changes as the absolute temperature so if the temperature was fluctuating very rapidly then phase distortion could occur, perhaps a sound wave travelled through the hot vortex of an exhaust.

#### Dr Colin Mercer

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Thank you for your article, it was very usefull and interesting.

What significant changes (in amplitud, in phase, in frequency?) can one expect should we measure a cracked damaged structure where the vibration modulation due to nonlinear effects play a role? What kind of measurement equipment would be best to use?

The real answer here is “it depends” – on the structure and crack size. Clearly with a very large crack then everything will have changed, but usually we are interested in the situation where the crack is “starting”. If we are talking about a rotating shaft with a such a crack then the phase usually changes more significantly than the amplitude. Frequency shifts tend to only occur as things get extreme. Amplitude changes are initially small. Changes in the rms level (total power) are also generally undetectable initially in real situations.