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Design >> RF Design >> Trying to make sense of Period Jitter Equation
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Message started by Jonathon on Jan 24^th, 2004, 1:09pm

Title: Trying to make sense of Period Jitter Equation
Post by Jonathon on Jan 24^th, 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.

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jitter Man on Jan 24^th, 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.

[glb]Jitter Man[/glb]

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jonathon on Jan 25^th, 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.

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jitter Man on Jan 26^th, 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.

[glb]Jitter Man[/glb]

Title: Re: Trying to make sense of Period Jitter Equation
Post by Nate on Jan 26^th, 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 :)

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jitter Man on Jan 26^th, 2004, 5:47pm

Of course, but you do not use dimensional analysis to find errors in factors of 2 pi.

[glb]Jitter Man[/glb]

Title: Re: Trying to make sense of Period Jitter Equation
Post by Paul on Jan 26^th, 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

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jonathon on Jan 27^th, 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.

Title: Re: Trying to make sense of Period Jitter Equation
Post by Ken Kundert on Jan 29^th, 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 = S_v/V₁² (see (22)). Since S_v has units of V²/Hz the ratio L must have units of 1/Hz.

-Ken

Title: Re: Trying to make sense of Period Jitter Equation
Post by Jonathon on Jan 30^th, 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.

Title: Re: Trying to make sense of Period Jitter Equation
Post by Ken Kundert on Feb 1^st, 2004, 2:58pm

Jonathan,
Your question seems to be "how do I convert from S_phi 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) S_v(f) = (A²/4)(S_phi(f - f₀) + S_phi(f + f₀))
Let f = f₀ + df and assume that the phase noise is bandlimited so that S_phi(f + f₀) is negligible. Then
(4) S_v(f₀ + df) = (A²/4)S_phi(df)
Finally,
(5) L(f₀ + df) = (A²/4)S_phi(df)/(A²/2)
(6) L(f₀ + df) = S_phi(df)/2

Clearly from (1), S_phi 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