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design1

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Phase noise to jitter conversion equation
Sep 27^th, 2005, 8:56pm

Hello,

I am trying to calculate long term jitter of my PLL by using Ken's phase domain model and then integrating the resulting noise power spectrum. However, it seems there is a factor of sqrt(2) discrepancy in the literature regarding the correct constants to apply to this integral.

In Appendix A of Hajimiri's paper, he states
RMS clock jitter ^ 2 = 8 / (w0)^2 * integ(Sphi(f) * (sin (pi*f*tau))^2)

Taking the square root, letting Sphi(f)=L(f) (since the integral is from 0 to infinity) and letting tau (the measurement interval) go to infinity, the equation for RMS jitter simplifies to:

sqrt ( 4 / (w0)^2 * integ(L(f)) )
which can be re-written as
1/w0 * sqrt(4*integ(L(f)) )

However, I have found through online searches that many sources state the correct conversion to be

1/ w0 * sqrt(2*integ(L(f))) - a factor of sqrt(2) lower than Hajimiri's equation.

My time domain jitter model seems to correlate with the lower number. I'm certain I am missing something obvious, but if anyone would be kind enough to point it out I would greatly appreciate it. Thanks in advance for your response.

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design1

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Re: Phase noise to jitter conversion equation
Reply #1 - Oct 7^th, 2005, 7:19am

Hello Again,

I have another sqrt(2) question concerning the conversion from phase noise to jitter. I am now working on characterizing the synchronous jitter in my divider using equations 55, 56, and 35 from Ken's paper.

In my last post, I was trying to understand the difference between Hajimiri's equation and a more common one found in many app notes [i.e. Jitter = 1/ w0 * sqrt(4*integ(L(f))) or 1/ w0 * sqrt(2*integ(L(f))) ]. Equations 55 and 56 seem to give me a result that is again a factor of sqrt(2) lower, 1/ w0 * sqrt(1*integ(L(f)))

I started by substituting equation 56 into 55 to get

JEE = [1 / (dv(tc)/dt)] * sqrt(integ(Sn(f)))

Then using equation 35 to get Sphi, setting L(f)=Sphi, and pluging into the second equation from my prevous post, I get

JEE = 1/w0 * sqrt(2*integ([w0/(dv(tc)/dt)]**2 * Sn(f)))

Cancelling the w0 factors and pulling the dv(tc)/dt out in front of the square root leaves:

JEE = [1 / (dv(tc)/dt)] * sqrt(2*integ(Sn(f))) - a factor of sqrt(2) higher than Ken's equation.

I guess my question based on this is if Sphi from equation 35 is a double sided spectrum because that would seem to resolve this issue. I assumed it was single sided since it was derived from the single sided voltage noise spectrum Sn. Maybe it is single sided and my starting equation was just wrong?

Thanks again in advance for any assistance.

design1

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neoflash

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Re: Phase noise to jitter conversion equation
Reply #2 - Oct 29^th, 2005, 10:22pm

Hi, design1:

would you mind post the paper title of Hajimiri you mentioned here?

[quote author=design1 link=1127879810/0#0 date=1127879810]Hello,

I am trying to calculate long term jitter of my PLL by using Ken's phase domain model and then integrating the resulting noise power spectrum. �However, it seems there is a factor of sqrt(2) discrepancy in the literature regarding the correct constants to apply to this integral.

In Appendix A of Hajimiri's paper, he states
RMS clock jitter ^ 2 �= �8 / (w0)^2 �* �integ(Sphi(f) * (sin (pi*f*tau))^2)

Taking the square root, letting Sphi(f)=L(f) (since the integral is from 0 to infinity) and letting tau (the measurement interval) go to infinity, the equation for RMS jitter simplifies to:

sqrt ( 4 / (w0)^2 * integ(L(f)) �) �
which can be re-written as
1/w0 * sqrt(4*integ(L(f)) )

However, I have found through online searches that many sources state the correct conversion to be

1/ w0 * sqrt(2*integ(L(f))) �- � a factor of sqrt(2) lower than Hajimiri's equation. �

My time domain jitter model seems to correlate with the lower number. �I'm certain I am missing something obvious, but if anyone would be kind enough to point it out I would greatly appreciate it. �Thanks in advance for your response.

[/quote]

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design1 New Member Offline Posts: 6	Re: Phase noise to jitter conversion equation Reply #3 - Oct 30^th, 2005, 1:50am You can find the equation I refer to in "Jitter and Phase Noise in Ring Oscillators" but I believe it was originally published in Appendix A of his thesis paper. www.ece.wpi.edu/Research/Analog/Resources/00766813.pdf See eq. 49 on p. 14. Design1
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neoflash

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Re: Phase noise to jitter conversion equation
Reply #4 - Oct 30^th, 2005, 1:17am

I got read the paper.

I found that you might be talking about two different concepts.

hajimiri is talking about more like "Cycle Jitter", and other sources might be talking about "Phase Jitter" or "Absolute Jitter".

In general, if the auto-correlation function of delta_phi(t) is a impulse function, which means that the absolute phase jitter is white, you could assume that:
Cycle jitter rms is about sqrt(2)*absolulte jitter rms.

It could be easily derived from equation:
Cycle Jitter = phi(t+T)-phi(t)

thanks,
Neoflash

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design1

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Re: Phase noise to jitter conversion equation
Reply #5 - Oct 30^th, 2005, 9:48am

Hi Neoflash,

Thank you for your response.

I think that Hajimiri's equation describes phase noise to jitter conversion for any measurement interval (i.e. in the sin^2(pi*f*tau) factor, replace tau with the interval of the corresponding jitter type you are interested in calculating).

Therefore, for period (or n=1) jitter, substitute 1/fout for tau. For uncorrelated white noise, cycle-to-cycle jitter can be shown to be equivalent to n=2 jitter which is what Hajimiri shows in equation 51 of his paper. This is where the sqrt(2) factor comes from for cycle-to-cycle jitter.

I am interested in long term (or n=inf) jitter. As I let tau go to infinity and take the square root to get rms jitter, his equation simplifies to
1/w0 * sqrt(4*integ(L(f)) )

It is this factor of 4 which I am trying to understand, because I believe it should be a factor of 2.

-Design1

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design1

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Re: Phase noise to jitter conversion equation
Reply #6 - Nov 30^th, 2005, 8:57pm

Hello,

I found that the sqrt(2) discrepancy occurs because Hajimiri is referring to "long term" (i.e. interval or n-cycle) jitter, while the other equation refers to what is termed "wideband jitter". Wideband jitter is what we get when we integrate the phase noise spectrum after passing it through a high-pass filter to eliminate the effects of wander.

I still don't have an intuitive feel for the cause of the sqrt(2) factor, but it is a step forward to know that both equations are correct.

I found that the paper "Specifying the Jitter Performance of Audio Components" by Travis and Lesso is a good reference for this.

Design1

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