What is the difference between phase noise and frequency noise?

I'm reading Audoin and Guinot's The Measurement of Time: Time, Frequency and the Atomic Clock, and ran across a confusing an interesting passage (Section 5.2.5, pp 72-73):

The physical origins of white phase and frequency noise are understood. They are associated with thermal noise in the radio region or shot noise from a particle flux.

There is also a chart that explicitly lists out phase noise and frequency noise as different components that lead to time and frequency instability.

I have always understood phase noise to be what is described in the Wikipedia article:

$$v(t) = A\cos(2\pi f_0t + \varphi(t))$$

where the $\varphi(t)$ portion is the phase noise of the sinusoidal signal.

I can't determine from the context of this book what frequency noise means, nor do any of my books mention frequency noise, just phase noise.

So, what is the difference between phase noise and frequency noise?

Phase Noise and Frequency Noise are not two different noise sources, they are artifacts of the same noise, it is just a matter of what units you want to use. Frequency and Phase are directly related as frequency is phase changing with time, so if you have one you will always have the other; frequency and phase are related by derivatives and integrals: the time derivative of phase equals frequency (as a change in phase versus time is frequency).

For a waveform in the time domain that represents the frequency changing with time as $$\omega(t)$$ (in radians/sec), the equivalent function as phase changing with time would be:

$$\phi(t) = \int \omega(t) dt$$

And likewise to go from phase (in radians) to frequency (in radians/sec):

$$\omega(t)= \frac{d\phi(t)}{dt}$$

Note this can be very confusing as we are talking about a waveform whose value represents frequency changing with time; so to be very clear this waveform I am describing is indeed in the time domain (as the independent variable is time), and we are observing how the frequency changes versus time. We can take the Fourier Transform of this waveform, and then observe the rate of change of frequency (which is the frequency of our frequency variable....even more confusing!).

However getting that straight is important to understanding the difference between Phase Noise and Frequency Noise:

Integration in the time domain is a low pass filtering response in the frequency domain, as indicated by the Laplace Transform property for integration:

$$L\left\{ \int x(t)dt\right\}=\frac{1}{s}X(s)$$

And similarly the derivative in time results in a multiplication by s in Laplace, which represents a high pass function (as you can get the frequency response (or Fourier Transform) for both by setting $$s=j\omega$$)

Therefore a Phase power spectral density and Frequency power spectral density are related as follows:

$$S_f(f) = f^2 S_{\phi}(f)$$

Where $$S_f(f)$$ is the power spectral density due to frequency fluctuations, and $$S_\phi(f)$$ is the power spectral density due to phase fluctuations. ($$f$$ and $$\omega$$ are related by $$\omega= 2\pi f$$ where $$\omega$$ is frequency in units of radians/sec and $$f$$ is frequency in units of cycles/sec or Hz.)

For example, if you had a white frequency noise process (meaning the FREQUENCY power spectral density was flat for all frequencies), the PHASE power spectral density would be going down at the rate of $$1/f^2$$ or 20 dB/decade.

Here is a plot I have showing the relationship between Phase Noise and Frequency Noise as time domain plots. Notice how the phase is smoother but wanders off relatively slowly with varying random offsets (low pass with random walk), while the frequency plot has higher frequency content but does not deviate from 0 for long durations (high pass filtered). The closest thing I'm aware of to a "rule" is the small-angle criterion, which relies on the $\sin(x) \approx x$ approximation to allow higher-order Bessel sidebands to be disregarded in phase noise measurement. If the power of the modulating noise is high enough to put significant energy into higher-order sidebands, the small-angle criterion no longer holds true. At that point you have FM noise. Outside the time/frequency community, FM noise of various slopes is commonly lumped into the term "residual FM."

Basically, the search terms you're looking for are "phase noise small angle criterion." They should lead you in a more authoritative direction.

• Although the small angle criteria is true, I do not believe this is correct as an explanation between phase noise and frequency noise. Phase modulation is the change of phase with time. If you take the derivative then you can equally measure the change of frequency with time (nothing has changed in the waveform; it is both phase and frequency modulated). The effect of the small angle criteria is when the modulated signal has only one significant sideband, and in that condition the sideband in relative amplitude will equal the phase in radians, so convenient for phase noise measurement. Jun 25 '17 at 11:06
• (You can modulate with large angles and still call in PM...such a 8PSK for example--- it is also FM it just depends on what units you want to use and that you properly translate between the two using the derivative relationship of frequency and phase--the waveform is still the same!) Jun 25 '17 at 11:09

I cannot post comments, as I created this account only to add something to this answer and I must have 50 reputation, as I found this post very helpful as a memory refresher (currently writing my PhD thesis). I hope this is not considered necro-posting.

I would like to add to Dan Boschen's answer that one must pay A LOT OF ATTENTION to units:

It is true that by having the starting statement as a function that tracks $$f$$ through time, after integration, the output has no units, or in other words, it is in normalized phase units to 1 ( 1=2$$\pi$$ rad).

From the second equation: $$f(t)=d\phi/dt$$, $$\phi$$ should instead be in radians, meaning that the output is in rad/s. This means that the formulation in concise notation (or more common notation) should be $$\omega(t)=d\phi(t)/dt$$, leading to the more common and concise notation of: $$\begin{equation} f(t)=\frac{1}{2\pi}\frac{d\phi(t)}{dt} \end{equation}$$

Which simply arises from the fact that a perfect sinusoidal at frequency $$f$$ will have the phase $$\phi$$ changing through time as $$\phi(t)=2\pi ft\equiv\omega t$$

My opinion is just that the post above should be completely redacted to concise units, as this can lead to very wrongly plotted functions if someone who is just learning does not realize this problem. Or maybe in my field in physics rad/s are important as we always use complex exponentials?

• Yes good point Jose! I updated my answer to be clearer that I was using radians/sec. Thanks for the comment. Nov 25 '21 at 15:45