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Phase noise to phase jitter for square waves

yizh

Hi,

I'm simulating a free running oscillator for jitter and I have the following question:

I have to run a "PNOISE - sources" simulation in order to recieve phase noise, since I have to filter the phase noise before integrating in to extract jitter (in order to mimic a PLL / CDR transfer function).

A few papers were written on the subject, some of them state that the integration upper limit is Fc/2 while others state that it is a few Fc. I assume that it should be a few Fc if the tested wave is a sine wave (i.e. no harmonics appear in the phase noise) and Fc/2 if it is a square wave.

As far as I understand, for square waves the jitter behavior of the first harmonic is similar to the jitter behavior of the square wave, thus it is assumed that integration up to Fc/2 takes into account only the first harmonic, otherwise the jitter will be summed more than once.

Please correct me if so far I'm wrong. Otherwise, here is a correction that I would like to do in my PNOISE simulation settings: instead of mixing the noise with many harmonics (i.e. Maximum sideband >> 1) and then integrating up to Fc/2, I might set maximum sideband to 1, thus the noise will be mixed only with the first harmonic, such that I will see a phase noise as if I had a pure sine wave at the input and not a square wave. Then, I would integrate up to a few Fc and see a more accurate jitter result.

In my simulations I see substantial difference between the two options, that's why the question is very important.

Any respose will we appreciated. I would especially like to hear Andrew Beckett's opinion on this.

Thanks!

 

  • Edouard
    Thanks Yizh  ,

    You seem to be highly confident on the workaround you have setup for your jitter calculation. If that works for you, i am happy for you too.  Unfortunately  the conclusion you have come up with may not be generalized to all designs, since they are simply drawn from on a number of simulation results observations and your theory is somehow going against math evidences upon which the simulator has been built.   

    Firstly , as you pointed out yourself ( post Feb/ 5 )  noise response is indeed a coherent addition all mixing products between the independent noise sources stimuli  and all harmonics of  the large signal pump. Hence accuracy of the simulation will recommend that you use a high enough maxsideband, especially as the content of higher harmonics is large, especially with  the  square wave you are talking about.  Your proposition (use maxsideband=1) is not likely to work in most cases, because you are just missing the important influence of  higher harmonics in shaping and transporting internal noise stimuli to the output bandwidth.  Integrating noise spectrum  you get that way, even going  from 0 to infinity, will not anyway tell you about that higher harmonics influence you have missed.  

    Secondly, Jitter is a sampled process, as such you MAY  NOT  integrate beyond F0/2 to compute total power.  If you need to, it is an indication of  a flaw in your analysis principle, so you probably need to reconsider and certainly not generalize.

    A few comments to your points

    1. How was the expression to oscillator signal as a sum of exponents/sines was developed? I'm not sure that it is so straight forward since a disturbed clock is no longer a periodic signal, thus some assumptions have to be performed in order to develop a Fourier series for it.

    The expression for oscillator signal in my notes  is a classical expression of a perturbed periodic signal. It analogous to the Q(t) what you write in your  3rd point, to the difference that  your expression of Q(t) is missing amplitude noise and basically holds for high Q oscillators only.  You might be aware that depending on the threshold, amplitude noise has non negligible  impact on threshold crossing jitter.

    2. I understand that you claim that for oscillators, frequency conversion is much smaller related to frequency modulation, which is the phase perturbation due to noise on the oscillator loop. I would like to challenge that claim, based on simulation results. Simulation leads me to believe that the significant contributor to jitter is the buffer that extracts CMOS clock from the sine wave at the input to the oscillator's loop amplifier. I simulated "multiple pnoise" with the consecutive nets, before and after the buffers, so see how jitter accumulates with each buffering stage. The results are as follows: at the input to the loop amplifier I saw 2.5ps RMS phase jitter. After the first amplifying stage I saw 4.2ps and after the second amplifying stage I saw 2.25ps (probably due to the steeper slope)……

    May be it was not clear;  my notes  did say that the noise stimuli contributing outside the oscillation feedback loop will follow a frequency conversion mechanism; so it is reasonable that  jitter you measure  in you output buffer  may be significantly different from one point to another.
    Unless you used PM jitter , steepness of the slope is probably not the reason for lower jitter on the second stage.  FM jitter filters the signal and consider only a single harmonic (you precisely mentioned this as the reason you prefer  FM to PM jitter) .  Hence  it ignores  the real slew rate of the signal and cares only of the amplitude of the first harmonic in your case.  So  jitter decrease you observe, may be simply that first harmonic amplitude has doubled in the second amplifier stage.     This also is an indication that the output buffer excessively noisy ,  it completely  swamp root oscillator loop phase noise; which also explains  why you get similar result whem you switch from the noisy oscillator signal to a noiseless sine source.

    3.  I would like to emphasize that looking at your claim itself, I don’t necessarily agree that your conclusion from it fits my original claim, ….

    The base harmonic that I speak of is the base harmonic of the total signal, such that  tethaH(t)  contains all the phase perturbation due to frequency conversion and modulation. This will be easily extracted from pnoise simulation where maxsideband=1.  So I don’t neglect any of the noise sources ….

    Your expression of Q(t) has two important  flaws;  it is lacking an amplitude noise term and does not apply for frequency conversion noise (except for a sinusoidal drive).

     Note that frequency conversion noise is equally divided in amplitude and phase perturbation. So if you think your circuit has important frequency conversion noise, you will have to deal with an amplitude noise term.  So you should write something like  (1+deltaA(t))Q(t+tetaQ(t)/2pi).   But even with this improved expression, you will not be able to handle conversion noise for a non sinusoidal drive. The reason is that the phase perturbations due to frequency conversion  mechanisms are not coherent from one harmonic to the others, hence you cannot factor these in a single term tetaQ(t), as you tend to believe.  The convenient expression for that problem is the  perturbed Fourier expansion you find in my previous notes.  In that expression, the terms deltaVk(t) are  frequency conversion perturbations.  These are non coherent from one harmonic to the other, as opposed to frequency modulation noise terms that maintain the coherence for all harmonics; you find the latter only in an oscillator.

    4. I’m not sure that I agree with the integration claims either J. The first of them is that the higher frequency energy content is negligible: at least in my design,…

    Answer already given in the introduction.

    5. A small Matlab test we ran supports my claim. Take (1) a sine wave with phase perturbation at single frequency and (2) a square wave with zero crossings at the same time points of the sine wave. FFT both. Observe that the ratio between the sine wave and the small intermodulation next to it is similar to the ratio between the first harmonic and the small intermodulation next to it in the square wave.   That ratio is the phase noise. So, this Matlab test shows that at least for phase perturbation at one frequency, the phase noise of a sine wave is similar to the phase noise of the first harmonic of a square wave. If the phase noise is similar then clearly the jitter is similar either, because in the time domain the original signals have similar jitter. This could be generalized from one single frequency to a range of frequencies, as is the case for random noise.

    No, I am sorry this is not right.  You are hang on the idea that there is a direct correspondence between first harmonic phase noise and jitter.  This correspondence does not  exist for an arbitrary waveform; it only exists  in a limited sense (zero crossing) for a sinusoidal waveform.   Consider an hypothetical ideal square wave (zero rise and fall times)  which has same zero crossing times with a similar sine wave (i.e amplitude of sine = 1.27 times amplitude of square wave), then your experiment will give exact phase noise for both signals, which is 100% right !.  If now you extrapolate phase noise to jitter, as you think you can do (jitter = phase_noise/(2pi*F0)) , you’ll find identical zero crossing jitter for both waves, which is obviously  wrong for the square wave:   Ideal square wave has a  zero crossing jitter (infinite slew rate) .

    6. Answering the last question, the reason that I want to have phase noise (coming out of “sources” simulation) is that the PN has to be filtered prior to integration and this is not possible, as far as I understand, in PM jitter simulation. See my post dating Feb 3 2013. I also see that PM jitter gives similar results to “sources” if “sources” is integrated correctly (Feb 5 post). This discussion also has to do with post-Si measurements, when clock is observed on an SSA and we would like to filter it before extracting jitter.

    I agree, PNoise noise you’ll  get from  SSA equipment  fits PNoise from FM jitter.  In both cases first harmonic is bandpass filtered and phase noise is sensed from the resulting perturbed sinusoid.  Jitter you will extrapolate from this measurement will for sure give you some valuable estimate of the actual jitter  of your oscillator,  but the error you make for non sinusoidal oscillator  is unknown, since you are definitely missing the phase noise picture of higher harmonics.  Note however that equivalence between the two is obtained only when you set maxsideband=full spectrum.  For squared wave oscillator you should probably prefer direct jitter measurement equipment, whch will then fit spectre PM jitter.

    Edouard
  • yizh

     Edouard,

    I read your post once and again (and again). You seem very confident, and besides it is much easier to walk in a beaten path than build a new one, but I couldn't get myself conviced that you are right.

    However, at this point it I feel that it will not be appropriate to continue the discussion as it is since it seems that we begin to walk in circles. Some points I would still like you to clarify, if you could:

    1.  You write "your theory is somehow going against math evidences upon which the simulator has been built". Could you ellaborate?

    2. Andrew forwarded to you a while ago what I think might be a proof to my theory. Could you review it?

    3. I didn't understand your response to my proposed Matlab example.  You wrote "you’ll find identical zero crossing jitter for both waves, which is obviously  wrong for the square wave:   Ideal square wave has a  zero crossing jitter (infinite slew rate)". Since the square wave was created from the sine wave (say, by defining: if the sine wave is >0 then the square wave is =1, otherwise it is =-1), then clearly both waves have similar jitter. What do the infinite slew rate has to do here?

    4. How would YOU simulate jitter, given that you have a square wave and want to run the FM jitter ("sources"). Please relate to the following:

    a. Number of harmonics

    b. Integration limits

    c. Noise folding (aliasing of noise from higher frequencies due to the sampling of the signal edge)

     

    Thanks,

    Yizhak

  • Edouard

     Thanks Yizh for your reply,  I appreciate your interaction,  I wish we don't walk in circles but converge as we progress.  Below a quick reply to your questions.

    1. Yes this is because the basic principle behind periodic noise analysis is that all harmonic components in the large signal simultaneously cooperate to shape noise in any bandwidth. So you may not accurately reproduce this behavior saying I neglect the harmonics cooperation and  consider just the fundamental one.

    2.  Are you referring to the maxim application note ?  it basically says that for an oscillator, jitter measured from  filtered fundamental harmonic is practically the same as  jitter of the overall square wave output.  As I already said this is true, provided your oscillator is pretty clean, i.e. no significant frequency conversion noise (clean output buffer).

    3. For a driven circuit, zero crossing jitter for a square is zero, because jitter is simply j = noise/slew_rate  ; and slew_rate = dv/dt is infinite at zero crossing for an ideal square.

    4. Two cases for square wave:

         - Driven circuit: do not use FM jitter ("sources") to compute jitter, it is likely to give you a wrong answer, use PM, all harmonics, integration limit Fo/2  and specify the crossings.

         - Oscillator circuit: if you are pretty sure that your output buffer is clean, you can use FM jitter ("sources"), full maxsideband, integration limit Fo/2; otherwise use PM as above.

     

  • Edouard

     Hizh,

    Andrew just forwarded to me the proof of your theory, which i have reviewed carefully; unfortunately i am sorry to say that it has some flaws, especially in the section proof/induction basis. In fact when stacking  up perturbed sine harmonic terms to form a square wave as you do, the only possibility to have the resulting signal to have identical zero crossings with the fundamental harmonic sine signal is that phase perturbations of the subsequent harmonics are integer multiples of the fundamental harmonic perturbation. This is quite obvious: tetha_n(t), the perturbation of the nth harmonic must be n times tetha_h(t) and not tetha_n(t)=tetha_h(t) as in your proof.  Such a situation is hardly achievable in a driven circuit, only cumulative phase noise in an oscillator circuit provide these conditions.

    I hope this helps.

  • Fazil

    Dear Edouard, 

    I would highy appreciate if you can elaborate one point to me. 

    As per above point :4, if there are square wave buffers after oscillator. I should use PM jitter analysis to get correct number.

    I want to convert this jitter number "JEE" in plot option to phase noise. should I multiply by zero crossing slope to get phase noise from this. I tried but numbers do not make sense. 

    what is the correct way of converting JEE ( jitter type) to phase noise ( source type)

    Thanks

    Fazil

  • Edouard

    Hi Fazil,

    Sorry for late responding, I missed your email

    For some reason  the plot tool behavior is not precise,  you should not be provided JEE for an oscillator circuit; JEE is infinite for a free running oscillator since it lacks an absolute reference.  It is not always possible to convert between jitter and phase noise;   to be sure of the figures you get, you need to set simulation settings specifically, either  for jitter or for phase noise.    For oscillator you should only ask only  for self-referred jitter figures,  Jc(K) and Jcc(k),  but not JEE

    Edouard

  • ShawnLogan

    Dear yizh,

    I have a decent amount of experience with this subject and think I may be able to help. However, I will let you be the judge of that!!

    > I have to run a "PNOISE - sources" simulation in order to recieve
    > phase noise, since I have to filter the phase noise before
    > integrating in to extract jitter (in order to mimic a PLL / CDR
    > transfer function).

    Yes - this is a common way of determining the output phase noise of a VCO based PLL.

    > A few papers were written on the subject, some of them state that
    > the integration upper limit is Fc/2 while others state that it is a
    > few Fc. I assume that it should be a few Fc if the tested wave is a
    > sine wave (i.e. no harmonics appear in the phase noise) and Fc/2 if
    > it is a square wave. Please correct me if so far I'm wrong.

    The random portion of the VCO output phase noise is determined by the random time variation of its zero-crossings relative to its average period. As such, it is not significantly impacted by the shape of its output waveform as long its output buffers are reasonably well designed and not significant sources of random jitter. This is confirmed in my experience measuring the random phase noise of many different types of VCO (quartz based, LC based and ring). the open loop phase noise is not a strong function of the shape of the output waveform. For example, I have measured the random phase noise of a high frequency VCO with differential CML outputs using a differential to single ended balun (Picosecond Pulse Labs 5320B) as well as their low frequency outputs after a digital CMOS divider chain with negligible difference in their phase noise after accounting for the frequency difference between the two outputs. The integration limit on the phase noise extends to fc/2 where fc is the carrier frequency of your output waveform. The zero-crossings are effectively sampling the noise every 1/fc. Hence, the spectrum is a sampled waveform and the integration limit may be set to fc/2 to capture the entire spectrum - independent of the specific output waveform shape.


    However, I might suggest you consider the specific application of your PLL. Many telecommunication and data standards specify the frequency limits for the jitter of a PLL frequency synthesizer used as the basis of a transmitter. These limits may be more appropriate than limits of essentially 0 Hz to fc/2 Hz. For example, a number of standards specify the random jitter over the limited frequency range of fc/1667 to fc/2. Others specify a fixed frequency range of, for example, 10 kHz to 20 MHz. Obviously, these different frequency ranges will reduce the impact of the low frequency random phase noise contribution of your VCO relative to frequency limits of essentially 0 Hz to fc/2.

    > Otherwise, here is a correction that I would like to do in my
    > PNOISE simulation settings: instead of mixing the noise with many
    > harmonics (i.e. Maximum sideband >> 1) and then integrating up to
    > Fc/2, I might set maximum sideband to 1, thus the noise will be
    > mixed only with the first harmonic, such that I will see a phase
    > noise as if I had a pure sine wave at the input and not a square
    > wave. Then, I would integrate up to a few Fc and see a more
    > accurate jitter result.

    My understanding is that the parameter maxsideband is used to set the maximum frequency for noise folding. Hence, if you set it to 1, you will not accurately capture the folded noise of your VCO into the fc/2 frequency band due to the inherent sampling process of the VCO random noise. In fact, I believe it is recommended to verify that an increase in the value you choose for parameter maxsideband does not significantly impact the simulated phase noise. I will let the experts comment on this as they have far greater knowledge than I on the specific Cadence pss and pnoise algorithms.

    > In my simulations I see substantial difference between the two
    > options, that's why the question is very important.

    However, I think the latter explains why your simulation of the open loop phase noise will vary significantly as your vary maxsideband from a value of 1 to N. Your phase noise is, in the case of 1, only capturing a very small portion of the VCO open loop random noise.

    I hope this provides some help - or at least something to consider!

    Shawn

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