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 is usually written in the form
where is the amplitude, is the frequency and is the phase in radians. If the phase is time-varying then it is better to write this in the form
We may of course write it in the form
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.