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What exactly is phase noise? (Read 600 times)
Frank Wiedmann
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What exactly is phase noise?
Nov 09th, 2002, 5:59am
 
Here is a more theoretical question that I have been wondering about for some time now: If both the signal and the strobed noise of the signal have an arbitrary periodic waveform, is there a unique way to determine the phase noise of the signal?

Let me explain what I mean: If I have a sinewave signal with pure phase noise, the strobed noise will also have a sinusoidal variation and the maxima of the strobed noise will coincide with the zero crossings of the signal. If I have a sinewave signal with pure amplitude noise, the strobed noise will have a sinusoidal variation also in this case, but the maxima will coincide with the extrema of the signal. A sinewave signal with constant strobed noise can be regarded as having equal amounts of uncorrelated phase noise and amplitude noise as they will add up to a constant value.

What happens now if we have a sinewave signal for which the strobed noise has an arbitrary waveform? Is there any unique way to decompose the noise into amplitude and phase noise? What happens if the signal also has an arbitrary waveform? Will there be a different phase noise value for each harmonic? What would be the combined phase noise of the entire signal in this case?

The practical answer to my question is of course the one that is given in section 6 of Ken's paper on phase noise and jitter (http://www.designers-guide.com/Analysis/PLLnoise+jitter.pdf). Here, the phase noise is defined with respect to a threshold level and is actually nothing else than timing jitter multiplied by the angular frequency of the fundamental. The consequence of this definition is, however, that it is not possible to determine a unique phase noise value for an arbitrary periodic signal. Instead, the phase noise will generally be different depending on the threshold level one chooses.
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Ken Kundert
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Re: What exactly is phase noise?
Reply #1 - Nov 10th, 2002, 11:52am
 
Frank,
   I talk about these topics in my paper and presentation on cyclostationary noise, which can be found in the Theory section of this website. To explore these questions, I like to work with a noisy signal vn(t) that is constructed by starting with a noise-free T-periodic signal v(t) where
   vn(t) = (1 + a(t))v(t + p(t)T/(2pi))
where a represents the amplitude noise and p represents the phase noise.

Quote:
Let me explain what I mean: If I have a sinewave signal with pure phase noise, the strobed noise will also have a sinusoidal variation and the maxima of the strobed noise will coincide with the zero crossings of the signal. If I have a sinewave signal with pure amplitude noise, the strobed noise will have a sinusoidal variation also in this case, but the maxima will coincide with the extrema of the signal. A sinewave signal with constant strobed noise can be regarded as having equal amounts of uncorrelated phase noise and amplitude noise as they will add up to a constant value.

This is the case where a or p are stationary. Note that even though the amplitude and phase noise contributions are stationary, the noise in vn is cyclostationary because the noise from a or p is being modulated by v.

Quote:
What happens now if we have a sinewave signal for which the strobed noise has an arbitrary waveform? Is there any unique way to decompose the noise into amplitude and phase noise? What happens if the signal also has an arbitrary waveform? Will there be a different phase noise value for each harmonic? What would be the combined phase noise of the entire signal in this case?

In this case, a or p  become cyclostationary, meaning that the amount of amplitude and phase noise vary along the phase of v. To see this variation you will have to be "observing" vn across multiple harmonics, but this does not imply that the amplitude or phase noise of each harmonic is different. You can see this by taking the Fourier expansion of v before adding in the amplitude and phase noise.

Quote:
The practical answer to my question is of course the one that is given in section 6 of Ken's paper on phase noise and jitter (http://www.designers-guide.com/Analysis/PLLnoise+jitter.pdf). Here, the phase noise is defined with respect to a threshold level and is actually nothing else than timing jitter multiplied by the angular frequency of the fundamental. The consequence of this definition is, however, that it is not possible to determine a unique phase noise value for an arbitrary periodic signal. Instead, the phase noise will generally be different depending on the threshold level one chooses.

That particular section of the paper is discussing the phase noise of frequency dividers, which are logic circuits and so inherently have thresholds. And the thresholds play a critical role in the phase noise behavior of the divider. However, as I have shown above, one can define amplitude and phase noise without resorting to use of either thresholds or strobed noise. The same is also true with jitter. Though jitter is normally defined as the variation in the timing of discrete events, one can also talk about jitter simply being a "noise in time" in which case the jitter j = pT/(2pi).
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« Last Edit: Nov 13th, 2002, 10:04am by Ken Kundert »  
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #2 - Nov 10th, 2002, 1:21pm
 
Let me ask my question a little differently: Given a certain v(t) and vn(t), is there a unique way to determine a pair a(t) and p(t) that describe the cyclostationary amplitude and phase noise of the signal?
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Ken Kundert
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Re: What exactly is phase noise?
Reply #3 - Nov 13th, 2002, 10:01am
 
a and p represent the amplitude and phase modulation of the carrier v. To extract them from vn one would use some form of amplitude or phase detector. Many detectors employ sampling (generally at the peaks) to extract the amplitude or phase modulation, and so are only able to determine the modulation signals at the sample points. The only way I know to extract the modulation signals over all time is using synchronous detection.
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #4 - Nov 13th, 2002, 12:02pm
 
I am actually not so much interested in how one would measure a(t) and p(t). What I would really like to know is how to clearly distinguish between cyclostationary amplitude noise and phase noise.

From your equation above, it seems like for a given v(t) there might be many different possible combinations of a(t) and p(t) that would result in the same vn(t). Is this true?
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Ken Kundert
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Re: What exactly is phase noise?
Reply #5 - Nov 13th, 2002, 9:53pm
 
I don't believe so. Amplitude and phase modulation are orthogonal decompositions, at least if the modulation is small signal.
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #6 - Nov 14th, 2002, 11:52am
 
Let's do some math. If I take the equation

vn(t) = (1 + a(t)) * v(t + p(t) * T / (2pi)), † † (1)

set p(t) = 0 and solve for a(t), I get

a(t) = (vn(t) - v(t)) / v(t). † † (2)

If I take equation (1) and set a(t) = 0, I get equation (28) from your paper "Predicting the Phase Noise and Jitter of PLL-Based Frequency Synthesizers" (http://www.designers-guide.com/Analysis/PLLnoise+jitter.pdf) with f0 = 1/T. For small phase noise, equation (28) can be linearized, resulting in equation (29). Solving equation (29) for p(t) gives

p(t) = (vn(t) - v(t)) * 2pi / (T * (dv(t)/dt)). † † (3)

Now, for any given pair v(t) and vn(t), it seems like one could always choose the pair a(t) and p(t) in the following ways (among many others) so that equation (1) is fulfilled:

First way: a(t) is given by equation (2) at all points in time except where v(t) = 0. At this point, a(t) = 0. p(t) is given by (3) where v(t) = 0. At all other points in time, p(t) = 0. Seen like this, the noise is almost entirely cyclostationary amplitude noise, there is hardly any phase noise.

Second way: p(t) is given by equation (3) at all points in time except where dv(t)/dt = 0. At this point, p(t) = 0. a(t) is given by (2) where dv(t)/dt = 0. At all other points in time, a(t) = 0. Seen like this, the noise is almost entirely cyclostationary phase noise, there is hardly any amplitude noise.

Is there anything wrong with the reasoning above? If so, what would be the correct equations to calculate a unique pair a(t) and p(t) from v(t) and vn(t)?
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« Last Edit: Nov 14th, 2002, 1:55pm by Frank Wiedmann »  
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Ken Kundert
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Re: What exactly is phase noise?
Reply #7 - Nov 14th, 2002, 3:22pm
 
Won't equations (2) and (3) work as long as as you assume that a and p are continuous?

In the examples you gave it seemed that a and p were unique except over a set of measure zero, and such distinctions are uninteresting in an engineering context.
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #8 - Nov 15th, 2002, 11:02am
 
Ok, let's disregard these points for a moment. This allows me to formulate my question in a simpler way:

It seems that any periodic signal with cyclostationary noise, given by a pair v(t) and vn(t), can be interpreted in two different ways:

First way: The signal has only cyclostationary amplitude noise as given by equation (2) and no phase noise.

Second way: The signal has only cyclostationary phase noise as given by equation (3) and no amplitude noise.

Is this correct?
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Ken Kundert
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Re: What exactly is phase noise?
Reply #9 - Nov 17th, 2002, 3:17pm
 
I think I am starting to see your point. I believe that you are correct. Except for the points in v where either it or its first time derivative are zero, the choice of a and p is not unique for a given vn.

I'm used to thinking of the modulation signals as being bandlimited with the bandwidth being much lower than the frequency of the carrier. In that case, the choice of a and p is unique.
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #10 - Nov 18th, 2002, 11:37am
 
Quote:
I think I am starting to see your point. I believe that you are correct. Except for the points in v where either it or its first time derivative are zero, the choice of a and p is not unique for a given vn.

Let me add another remark, even though it is probably not very significant in an engineering context:

If the signal v(t) happens to have a point t0 with both v(t0) = 0 and dv(t0)/dt = 0, but vn(t0) - v(t0) != 0 (!= meaning "not equal"), then it seems that the noise at t0 can be expressed neither as finite amplitude noise a(t0), nor as finite phase noise p(t0).

Quote:
I'm used to thinking of the modulation signals as being bandlimited with the bandwidth being much lower than the frequency of the carrier. In that case, the choice of a and p is unique.

With this in mind, would you like to clarify the exact meaning of statements like: "The noise of a free-running oscillator close to the carrier is predominantly phase noise."?
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Ken Kundert
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Re: What exactly is phase noise?
Reply #11 - Nov 18th, 2002, 1:31pm
 
Your first point indicates that the noise model I proposed is not complete. It needs to have an additive noise term.
   vn(t) = (1 + a(t))v(t + p(t)T/(2pi)) + n(t)
where n is the additive noise term. Of course, adding this new term gives you even more ways of specifying the same noise. But it also makes it more likely that the noise model will better fit a variety of realistic situations and be predictive without requiring a great deal of mathematical gymnastics. As an example, consider modeling the noise produced by a signal generator. The signal is initially produced by a free running oscillator, and so contains substantial phase noise. It has a controlled amplitude, and so may have an AGC loop that adds amplitude noise. Finally, it has a 50 Ohms output impedance and so includes additive noise. Modeling the output noise using the above expression would be relatively straight forward, but if any term were missing it would be much more complicated.

Allow me to respond to your second point in a subsequent post.
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Ken Kundert
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Re: What exactly is phase noise?
Reply #12 - Nov 21st, 2002, 5:36pm
 
Quote:
With this in mind, would you like to clarify the exact meaning of statements like: "The noise of a free-running oscillator close to the carrier is predominantly phase noise."?

Regardless of how you write the model, the underlying physical noise processes are acting to modulate the phase of the oscillator. And so I think the statement is still an accurate portrayal of the situation. Your point is that the model I have given that allows me to separate the amplitude and phase noise becomes ambiguous when the modulating signals are allowed to become cyclostationary. But I donít think that is important because the knowledge of the modulating signals in this model is rarely used directly. Instead we use the model to produce the correct modulated waveform vn(t) and then use our knowledge of the larger system to determine what the effect of noise will be. For example, when trying to compute the jitter of a frequency divider, we donít use either the amplitude or the phase noise terms from this model directly, we always have to use them to determine what the noise will be at the threshold crossing and then use that to determine the effect of the noise on the larger system. In this case, whether we model the noise as being either completely in the phase or completely in the amplitude does not matter as long as we properly determine vn(t) and use that information to predict the effect of the noise.
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Frank Wiedmann
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Re: What exactly is phase noise?
Reply #13 - Nov 22nd, 2002, 11:56am
 
My point was indeed that, contrary to what one might think, it seems to be generally impossible to unambiguously separate the noise of a signal into amplitude noise and phase noise. On the other hand, I agree with you that the statement I quoted does have a meaning.

A somewhat more precise formulation of this statement might be: "The noise of a free-running oscillator close to the carrier is predominantly stationary phase noise." This is supposed to mean that if you look at the noise of a free-running oscillator close to the carrier and calculate its phase noise p(t) according to equation (3), this phase noise will be almost constant. Another way to express this fact would be to say that the strobed noise vn(t) - v(t) of a free-running oscillator close to the carrier is almost exactly proportional to the time derivative dv(t)/dt of the signal. You explain the reason for this behavior in section 3.1 of your paper on phase noise and jitter (http://www.designers-guide.com/Analysis/PLLnoise+jitter.pdf).

The insight expressed by this statement is important because spectrum analyzers and some simulation software only give you information about the power of vn(t) over frequency, which only allows you to determine the time-average of the noise. What is missing is information about the correlation between the different sidebands of vn(t) that would allow you to determine how the strobed noise vn(t) - v(t) varies over the period of the signal. For a sinewave signal, stationary amplitude noise a, stationary phase noise p, and stationary additive noise n all look exactly the same on a spectrum analyzer and in some simulation software. So, additional insight as expressed by the quoted statement is important in order to interpret the results of measurements and simulations correctly.
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Asad Abidi
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Re: What exactly is phase noise? (Additive noise)
Reply #14 - Apr 1st, 2003, 4:50pm
 
I don't think it is necessary to add Ken's third term n(t):

Quote:
Your first point indicates that the noise model I proposed is not complete. It needs to have an additive noise term.
   vn(t) = (1 + a(t))v(t + p(t)T/(2pi)) + n(t)
where n is the additive noise term.

This is because one can decompose each spectral component of added noise into equal AM and PM sidebands around the oscillation frequency, and absorb the two components into a(t) and p(t).

I should point out that the spectral density of the sidebands of p(t) arising from additive noise do not grow as they approach the carrier, because the noise adds outside the autonomous oscillator circuit. Nonetheless, it is phase noise.
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