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Trying to make sense of Period Jitter Equation (Read 9242 times)
Jonathon
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Trying to make sense of Period Jitter Equation
Jan 24th, 2004, 1:09pm
 
I have a question on the accumulating jitter equation covered on page 29-31 (section 10) in the "Predicting the Phase Noise and Jitter of PLL-Based Frequency Synthesizers" at this site.

Specifically, my concern is on units.  For equation 73,
J= sqrt(cT) = sqrt(L(f)*df^2 / (fo^3).

For J to equal a unit of time, it is easy to see that L(f) (converted to a noise value, 10^(L(f)/10) ) would have to have units of 1/Hz.  But this seems incorrect.  Phase noise can either have units of rad^2/Hz or cycles^2/Hz, typically the former.

Therefore, I'm wondering where these extra units are accounted for.  In the typical literature, when phase noise is integrated by a particular bandwidth, the result is a value in radians which must be scaled by 1/(2*pi*fo) to arrive at a value in terms of time.

Thanks in advance.
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Jitter Man
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Re: Trying to make sense of Period Jitter Equation
Reply #1 - Jan 24th, 2004, 4:50pm
 
Both radians and cycles are unitless quantities. So from a dimensional analysis perspective, rads/s, cycles/s, and /s are all equivalent.

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Jonathon
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Re: Trying to make sense of Period Jitter Equation
Reply #2 - Jan 25th, 2004, 2:45pm
 
I'm fairly confident in saying that rad^2/Hz, cycles^2/Hz
are not unitless quantities.

For example, if one were to integrate phase noise over a particular bandwidth, which one typically does when converting to jitter, the result when the phase noise has units of rad^2/Hz would be radians while the results would be cycles for phase noise with units of cycles^2/Hz.

I understand that when the phase noise is taken as a logarithmic measure the radian or cycle units fall out since you're comparing the noise power to the signal power (hence the cancellation), but when you convert back to an actual noise value for calculating jitter, there is no longer any such comparison made; the noise itself must have units.

In almost all the literature I have read, the conversion of L(f) to noise (S(f)) gives units of rad^2/Hz.  This is also fits with narrow band modulation theory which states
Vout(t) = A*sin(wt+phi(t))
= A * ( sin(wt)+phi(t)cos(wt) )
This says that the phase noise amplitude, relative to the carrier, will be phi(t).

Thanks for your responses.
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Jitter Man
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Re: Trying to make sense of Period Jitter Equation
Reply #3 - Jan 26th, 2004, 12:02am
 
Let me try again. What I should have said was the radians and cycles are dimensionless. Thus, rads2/Hz or cycles2/Hz both have dimensions of seconds. The reason I am distingushing units from dimensions, is that it is very common for dimensionless units such a radians to get lost in a derivation. Thus, dimensional analysis often simply neglects dimensionless units such as radians. If you are sufficiently determined, you can track them down and keep them balanced, but it is generally more trouble than its worth.

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Nate
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Re: Trying to make sense of Period Jitter Equation
Reply #4 - Jan 26th, 2004, 10:50am
 

Seems to me that if you do not track a pi or 2*pi, your answer will be off by a factor of 3-6.

Seems pretty imporant to me that the units do match.

Without proper units, your Rover could easily crush into Mars Smiley
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Re: Trying to make sense of Period Jitter Equation
Reply #5 - Jan 26th, 2004, 5:47pm
 
Of course, but you do not use dimensional analysis to find errors in factors of 2 pi.

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Paul
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Re: Trying to make sense of Period Jitter Equation
Reply #6 - Jan 26th, 2004, 11:43pm
 
I think that depends on your undergrad lecturers. Some of mine ideed used a lot of rad/s to Hz conversions to make 2pi appear and disappear. I guess that was Jonathon's point.

Paul
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Jonathon
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Re: Trying to make sense of Period Jitter Equation
Reply #7 - Jan 27th, 2004, 11:11am
 
Here's an example of why you need to keep track between radians and cycles.  VCO output frequency vs. input voltage gains (commonly referred to as KVCO) can either be given in units of rad/s/V or cycles/s/V.  Both are exceptable and easily interchangeable since 1 cycle=2pi radians.

Now, if you were designing a PLL that used this VCO and were working the phase transfer functions (for determining loop stability) in units of radians then you would want to be certain that the KVCO value you used were in units of rad/s/V.  If you slipped in the value in terms of cycles/s/V, your open loop gain would appear 6x (2*pi) less and you would have a much less stable PLL.

By the same reasoning, the only unit of noise I know that can be extracted from a phase noise L(f) is rad^2/Hz (as I explained in my last message) but this is not does not fit with Eq. 73.  

Therefore, the unit conversion was either handled in some way that I didn't see, 10^(L(f)/10) does not have units of rad^2/Hz, or Eq. 73 has an error.  This may seem like mathematical minutea, but we're talking about a potential 6x (2*pi) error in a jitter calculation.

Thanks again for your opinions.
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Ken Kundert
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Re: Trying to make sense of Period Jitter Equation
Reply #8 - Jan 29th, 2004, 9:32pm
 
Actually the solution to the puzzle is relatively simple. Jonathon asked how the phase noise density L can have units of 1/Hz. The answer is ...
L is not phase noise density. It is the normalized single sideband noise density, L = Sv/V12 (see (22)). Since Sv has units of V2/Hz the ratio L must have units of 1/Hz.

-Ken
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Jonathon
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Re: Trying to make sense of Period Jitter Equation
Reply #9 - Jan 30th, 2004, 12:40pm
 
Ken,

I entirely agree that L(f) literally has units of 1/Hz which is arrived at from the ratio of the noise voltage power (V^2/Hz) to the signal voltage power (V^2).

What is still a little unclear is how L(f) is being converted to a single side band phase noise density (Sphi(f)).  As it is, L(f) is a dimensionless density.  In order to relate it to phase, we need a power density that carries units of rad^2/Hz or cycles^2/Hz.  When that power density is integrated over a particular bandwidth, it need to indicate phase in either radians or cycles.  Both radians and cycles are technically dimensionless since they both describe a portion of a circle, but obviously with a 2pi scaling difference.

From narrow band modulation theory,
Vout(t) = A * sin(2pi*fout*t+phi(t)) is approximately equal to
Vout(t) = A*(sin(2pi*fout*t)+phi(t)*cos(2pi*fout*t))

L(f) = [Sphi/Hz]/1^2 = Sphi/Hz

From this basic relationship, the phase noise induced voltage magnitude should be phi(t) times the signal amplitude.  Therefore, L(f) = Sphi/1^2.  What I draw from this is that L(f), while still technically 1/Hz, should equal the single sideband phase noise power density and that the units that one gets with this are in rad^2/Hz.  This tracks with most the literature (Hajimiri for example) that computes period jitter as:

sigma^2 = 1/2pi/fo*sqrt(2 * integrate(Sphi(f)*sin^2(2pi*f*1/fout), from 0 to infinity)

The key feature to that equation is that the quantity generated by the integration is rad^2.

From Eq. 19 (page 12), Sphi is being carried through in cycles/Hz and from Eq. 27 (page 15) it has the same dimensions as L(f).

So, the big question is: How are you moving from a dimensionless power spectral density L(f) to Sphi(f) with dimensions of phase^2/Hz?

Thanks.
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Ken Kundert
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Re: Trying to make sense of Period Jitter Equation
Reply #10 - Feb 1st, 2004, 2:58pm
 
Jonathan,
Your question seems to be "how do I convert from Sphi to L?". Let me show you how its done assuming a sinusoidal signal. Let,
(1)    v(t) = A sin(2 pi f0 t + phi(t))
Assume phi is small, then
(2)    v(t) = A sin(2 pi f0 t) + A cos(2 pi f0 t) phi(t)
(3)    Sv(f) = (A2/4)(Sphi(f - f0) + Sphi(f + f0))
Let f = f0 + df and assume that the phase noise is bandlimited so that Sphi(f + f0) is negligible. Then
(4)    Sv(f0 + df) = (A2/4)Sphi(df)
Finally,
(5)    L(f0 + df) = (A2/4)Sphi(df)/(A2/2)
(6)    L(f0 + df) = Sphi(df)/2

Clearly from (1), Sphi has units of rads2/Hz (as it does everywhere in my paper, including eqn 19 on page 12). But as you said, L has units of 1/Hz.  So, from all appearances, the units on (6) do not balance. That is because factor of 1/2 appears to be a unitless number. In fact, it has units of 1/rads2. It got those units when the Taylor series was applied to (1).  To see how, let
   x(phi) = 2 pi f0 t + phi
   y(x) = A sin(x)
Then
   dy/d phi = dy/dx dx/d phi
Notice that dy/dx has units of V/rad and dx/d phi has units of rads/rad. So dy/d phi has units of V/rad. And so
    y(x) = A sin(2 pi f0 t + phi)    [V]
    dy/d phi = A cos(2 pi f0 t)      [V/rad]
Thus, the last term in (2) appears to have units of [V*rads], but it really has units of [V/rad][rads] = [V].

I believe it was these subtleties in the units that Jitter Man was trying to warn you about.

-Ken
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