## Where Does The Noise In A Signal Come From?

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 to 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), there is a clear difference.  All the noise is in the phase component!

If we were to consider the amplitude spectrum or the auto spectrum, 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 15o.

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

When looking at the spectrum, 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 1.5o 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. 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 severely phase modulated signal (blue) and the ideal pure sine wave (red) are shown in Figure 9 below.

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

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#### Dr Colin Mercer

Founder / Chief Signal Processing Analyst (Retired) at Prosig
Dr Colin Mercer was formerly at the Institute of Sound and Vibration Research (ISVR), University of Southampton where he founded the Data Analysis Centre. He then went on to found Prosig in 1977. Colin retired as Chief Signal Processing Analyst at Prosig in December 2016. He is a Chartered Engineer and a Fellow of the British Computer Society.

### This Post Has 3 Comments

1. Vivian Silva Mizutani

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

2. Julio

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?

1. 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.

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