RMS Jitter From Phase Noise

Dear Grant,

I've tried to answer your questions as best I can below.

Shawn

> To get RMS Jitter, in radians, from Phase Noise you must integrate the Phase

> Noise.  What are the units of integrated Phase Noise and how do they cancel.

> The equation I am currently using for this is A = Phase Noise (L(f)) + 10*

> log10(frequency2- frequency1) and to generate the RMS Jitter value in radians

> I am using sqrt(2*10^(A/10)).

If one is given the phase noise characteristic, as defined by L(f) in dBc/Hz, then over specific frequency range [f2,f1], the rms jitter in radians and in units of time are found as follows:

1. Integrate the function L(f) between f2 and f1

2. The rms jitter expressed in radians is (2*[Integral(L(f) between f2 and f1]))^0.50

3. The rms jitter expressed in units of time for the carrier frequency fc is [(2*[Integral(L(f) between f2 and f1)])^0.50]/(2pi*fc)

> Additionally, what are the units for 10*log10(frequency2- frequency1) and if L

> (f) is the ratio of Pcarrier and Poffset in dBm, and that has units of dBc/Hz

> , what would be the units if it were converted to a linear value.

The integration of L(f) is done in the linear frequency domain. The units of the integral are radian^2 as seen from [2] above. The units of L(f) are 1/Hz. The units remain the same when converted to a linear value since the quantity still represents a ratio.

> Follow up question. Since L(f) is the ratio of Poffset and Pcarrier(L(f) = Po/

> Pc), it follows that if you increase Pcarrier you decrease L(f), which in turn

> decreases Jitter. If this is true, then one could remove Jitter by increasing

> the power of the carrier. Is this true?

The general answer to your comment is yes - with some caveats. If the random or deterministic noise of concern does not increase proportionally with the carrier amplitude, then increasing the waveform amplitude is a common method used to reduce the magnitude of the jitter over a specific frequency range. However, this method quickly can reach a point of diminishing returns as the added power can make the design non-competitive.

I hope this provides some help!

Thank you smlogan.

My question about integrating L(f) over a bandwidth is more of a process based one.  The equation I'm using to get "area" is:

A = L(f) + 10*log(frequncy2-frequncy1)

Will this give me the value I'm looking for?

The next two steps to get to seconds are very straight forward, however I have found a few documents that mention the need to root-sum-square the individual jitter values in order to get the total jitter value for a given data set.  Is this true?

Thank you again for your help.

Dear Grant,

> ...question about integrating L(f) over a bandwidth is more of a process based one.  The equation I'm using to get "area" is:

> A = L(f) + 10*log(frequncy2-frequncy1)

> Will this give me the value I'm looking for?

Unless I am not understanding your question, Grant, I do not believe this will (in the general case) provide the area representative of L(f) between frequency2 and frequency1. Why do I suggest this?

L(f), in general, is not a constant value with frequency. It is expressed in units of dBc/Hz due to its rather large dynamic range. Its definition relates the noise power in a 1 Hz bandwidth to the entire carrier noise power. Hence, to estimate the rms value over an arbitrary frequency interval,  you are interested in the power, expressed in radian^2, represented by the area under the curve between frequency2 and frequency1. If L(f) happens to be a constant value between L(frequency2) and L(frequency1), say, Lo dBc/Hz,  then the area under the curve between frequency2 and frequency1 is:

(frequency2 - frequency1)*[10^(Lo/10)]  and is multiplied by 2 to get the total spectral power between frequency2 and frequency1 by the definition of L(f). 

This is similar to your expression but the area, A in your expression, must be expressed in radian^2 - not in logarithmic units.

For example, suppose for a very good 100 MHz clock the phase noise is constant at -150 dBc/Hz between 100 kHz and 1 MHz, then the area under the curve is 1.80e-09 radian^2 or the rms jitter is 0.0675 ps rms.

In general, L(f) is not constant with frequency and hence the area computation must be performed in the linear frequency domain.

I hope I  understood your question Grant!

Shawn

Hi Sean, I'm back with more questions.

So I think I understand what you mean.  Here's an small sample of the data I have...

(This goes to an offset frequency of 5Mhz.)

The formula A = L(f) + 10*log(frequncy2-frequncy1) is being applied per datapoint, so in the first case it would be done for 1Hz and 1.0180098Hz with L(f) = -38.1251812dBc/Hz.

From the literature I've read, I should be able to do this using a trapezoidal or Simpson's rule approximation and get a value that I can then convert to radians with the formula sqrt(2* 10^(A/10)).

Am I trying for too fine of detail?  I did notice in the same literature that the same calculations were being done on decade Hz scales.

Hi Grant,

The basic issue we are wrestling with, I believe, is the accuracy of using a logarithmic scale to approximate a linear scale to estimate the value of a definite integral.

In your specific example, it appears you have 129 points per decade - which is quite a few. When you use the trapezoidal rule to perform an integration, you are approximating the area of each segment by taking the average y value formed by (x1,y1) and (x2,y2) - i.e. (y1+y2)/2 - and multiplying it by the difference in x (x2-x1) to estimate the area. If you are using a logarithmic y scale, such as your formula is using, then you are assuming the logarithmic value of y1 is average y value (-38.1251812 dBc/Hz in your example). If the linear difference between y2 and y1 is very small, then the log of the average y value is approximately log y1 or log y2 since they are quite close. However, if the linear difference in the magnitude of y2 and y1 is significant, then the average value is not well approximated by log y1 or log y2. For example, if y1 >> y2, then the average linear value is about (y1)/2 - or about 6 dB less than just using log y1. 

In my opinion, if you do not want to be concerned with the possible approximation of using a logarithmic scale for linear scale in numerical integration, I would convert the phase noise to a linear scale and then perform the integration. This is the methodology I always use.

> Am I trying for too fine of detail?  

My thoughts....It is very unusual to measure phase noise at an offset frequency of 1 Hz unless you are designing a rather specialized type of oscillator - maybe you are! In fact, technically, the term describing the impact of phase noise below a 10 Hz offset is "wander" and not "jitter" and the phase noise measured at this offset frequency will be very noisy in most common oscillators. Wander measurements are rather specialized with names such as MTIE and TDEV. More conventional offset frequencies might start at 1 or 10 kHz or, perhaps, 100 Hz. A more typical number of points per decade to provide a reasonable estimate of jitter due to a phase noise characteristic is 20. The use of more points per decade might be warranted if there are a lot of spurs in your phase noise due to supply noise or deterministic jitter (for example, reference clock spurs in a frequency synthesizer). In this case, the finer frequency bins provide a more accurate estimate the impact of spurs on the jitter. With a coarse frequency interval, the spurs will artificially inflate the resulting jitter computation.

Let me know your thoughts Grant. I hope my explanation made some sense!

Shawn

In addition to Shawn's valuable comments, there is a "jitter" mode of pnoise analysis which takes care of all of this for you. For an oscillator, there is both "FM" jitter which transforms the phase noise into a jitter and also "PM" jitter which uses a time-domain approach. In both cases care has been taken on how the integration is performed when the x-axis may be logarithmic, as well as  taking into account other things which are hard to get correct from the simplistic integration approaches normally taken (for example, K-cycle jitter can't really include noise effects from frequencies much lower than 1/(K*period)).

So you should consider that. There's an application note on this in the MMSIM installation under <MMSIMinstDir>/tools/spectre/examples/SpectreRF_workshop

Regards,

Andrew.

Hi Shawn,

I follow your thinking about using a logarithmic scale when trying to do a trapezoidal approximation.  I was already working on converting the phase noise measurement to a linear value and crunching the numbers that way when I read your reply.  :) 

Right now I'm doing research and not design work.  I don't have spurs within the first 3 decades of my measurements and very few with significance beyond that.  What I'm trying to determine is what the overall impact of additional noise, at varying frequencies, is on my oscillator and if it impacts its performance.  At this point the specific type of noise and jitter aren't super important to differentiate.  I just need to be certain the the values for jitter that I'm calculating are valid, and that I understand the process of they're calculation.  I think I'm at the point where I understand the calculations and am starting to see some relation between the phase noise I'm measuring and the jitter I'm calculating.

Thank you for your help,

Grant

Great!

Good luck with your research Grant.

Shawn