Phase noise to phase jitter for square waves 

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!

 

I think there are some assumptions you are making here which aren't necessarily correct. In general you should use the "jitter" noisetype if you want to measure jitter. There are two approaches - "FM" jitter (which uses a conversion from the PM part of the noise, as computed with the "modulated" noise type to a time metric), and "PM" jitter (which samples the noise at the specified threshold crossing, and then uses this in conjunction with the slope of the signal at the threshold crossing to compute the equivalent time variation).

The FM approach (which is not disimilar to what you are trying to do) is best suited to sinusoidal (or near-sinusoidal signals) rather than square-ish waveforms; for those (particularly when used in decision making circuits, or clock recovery circuits), the PM jitter approach is better.

In addition, integrating the noise has to be done carefully - for example, making sure that you integrate the area under the curve properly when you have a log axis - particularly for handling any flicker noise. A lot of thought and attention has gone into the integration in our jitter direct plot form.

I don't think it makes sense to set maxsideband to 1 and then integrate over multiples of the carrier. For a start, the noise will be infinite at multiples of the carrier (1/f is infinite at f=0 - of course, noise is not really that large at low frequencies, because non-linear effects start to come into play), but more importantly if you are only including noise contributions from up to the first side band, high frequency noise is being excluded! In many oscillators the noise in sidebands 0 and +/-1 are the biggest, but of course it depends...

Whilst the phase variation of the fundamental harmonic of a square wave will be similar to that of a sine wave, it won't be precisely the same, and I don't think that means that jitter is the same either - the PM jitter approach will be better.

Note that it doesn't make sense to sweep past half the fundamental frequency when using PM jitter, because it adds an ideal sampler at the fund frequency rate to the output of the circuit, which naturally folds the noise from higher frequencies - so it is sufficient to sweep to half the fundamental. Sweeping beyond that will mean you double count the noise. Maybe that's what you're read?

Regards,

Andrew.

Quoting:

I think there are some assumptions you are making here which aren't necessarily correct. In general you should use the "jitter" noisetype if you want to measure jitter. There are two approaches - "FM" jitter (which uses a conversion from the PM part of the noise, as computed with the "modulated" noise type to a time metric), and "PM" jitter (which samples the noise at the specified threshold crossing, and then uses this in conjunction with the slope of the signal at the threshold crossing to compute the equivalent time variation).

The FM approach (which is not disimilar to what you are trying to do) is best suited to sinusoidal (or near-sinusoidal signals) rather than square-ish waveforms; for those (particularly when used in decision making circuits, or clock recovery circuits), the PM jitter approach is better.

Thanks for the answer.

First, a word about my motivation or - why pnoise-jitter does not suit my needs. In my application, jitter has to be filtered prior to integration with a physical (i.e. not a "brick wall") filter. Take for example IEEE KX spec (annex 70.7.1.9) which states: "For the purpose of jitter measurement, the effect of a single-pole high-pass filter with a 3 dB point at 750 kHz is applied to the jitter". Pnoise-jitter does not have this capability as far as I've seen, thus I must run jitter-sources, extrace phase noises, multiply it with the filter and then integrate.So, even though the wave I measure is a square-ish wave, I don't see an option to run the PM approach.Please correct me if I'm wrong.

Quoting:

I don't think it makes sense to set maxsideband to 1 and then integrate over multiples of the carrier. For a start, the noise will be infinite at multiples of the carrier (1/f is infinite at f=0 - of course, noise is not really that large at low frequencies, because non-linear effects start to come into play), but more importantly if you are only including noise contributions from up to the first side band, high frequency noise is being excluded! In many oscillators the noise in sidebands 0 and +/-1 are the biggest, but of course it depends...

 I think that the noise will not be infinite at multiples of the carrier since maxsideband=1, so no harmonic is viewed at the phase noise plot. The plot is smooth at the harmonics.

High frequence noise, as my understanding goes, is not excluded since integration goes up to a high frequency. This is equivalent to folding the noise around half the carrier frequency. Only in my proposition noise is folded only "implicitly" since I don't actually fold but instead integrate up to a high frequency.

 It is true that noise mixing with high harmonics is not taken into account, but this is due to the assumption that I described at the first post of this thread.

Quoting:

Whilst the phase variation of the fundamental harmonic of a square wave will be similar to that of a sine wave, it won't be precisely the same, and I don't think that means that jitter is the same either - the PM jitter approach will be better.

Note that it doesn't make sense to sweep past half the fundamental frequency when using PM jitter, because it adds an ideal sampler at the fund frequency rate to the output of the circuit, which naturally folds the noise from higher frequencies - so it is sufficient to sweep to half the fundamental. Sweeping beyond that will mean you double count the noise. Maybe that's what you're read?

No, I'm speaking of phase noise integration, i.e. FM jitter. See for example here:

http://www.maximintegrated.com/app-notes/index.mvp/id/3359

Or in the Cadence app note titled "Jitter Measurements Using SpectreRF", eq. 1-17.

Pnoise jitter analysis also gives you the spectrum of the sampled noise as a result. You can multiply this with any function you like before doing the integration. This is actually being done to calculate the accumulated (k-cycle) jitter Jc, where the spectrum is multiplied with a sine function before integration (see http://www.designers-guide.org/Forum/YaBB.pl?num=1224609785/9#9).

Frank made exactly the same suggestion that I was going to make - you could use an svcvs source from analogLib to describe the filter in the s-domain. 

If maxsideband is 1, you are only including the noise contributions from sidebands +/-1 and 0 (so you will get very high up-converted flicker noise if you sweep near to the carrier frequency). You won't get noise contributions from higher frequencies "folded" by the harmonics of the carrier because you simply are not including them. Sweeping over a wide frequency range is NOT the same as including lots of sidebands - you are sweeping the output frequency yes, but you've removed the simulator's ability to compute all the transfer functions from the noise sources to the output - you've only included three noise transfer functions.

I think the approach you're suggesting is flawed - best is to include the filter in your circuit (it can be ideal), and then use the PMjitter approach.

Andrew.

 Andrew, Frank,

Let me get back to my original claim before I answer your two latest comments. Let's leave the filtration aside for now.

My claim was that it is enough  to look at the first harmonic's jitter when analyzing a square wave's jitter.

Using what I just learnt on ponise -> jitter -> pm, I thought of the following experiment: Let us take the "Jee" result of PM jitter simulation as a reference point, with integration at the range [1K, Fc/2] and maxsideband=fullspectrum. Next, let us see if my claim holds, by running a "sources" simulation with maxsideband=1 at the range [1K,300*Fc], then integrate phase noise and extract the phase jitter. If results are close, that would be an indication for the correctness of my claim (I will speak about theory later).

So, I ran the simulation on the crystal oscillator that I currently design (fullspectrum resulted in 3).

For the PM option, jitter was 1.9ps for one of the edges and  1.96ps for the other.

For the "sources" option, jitter was 1.94ps.

I also tried a third option: "sources",  fullspectrum, integration [1K, Fc/2]. Result was 1.2ps.

So I think that this is a strong indication to my claim's correctness.

 

Back to thoery, qouting:

If maxsideband is 1, you are only including the noise contributions from sidebands +/-1 and 0 (so you will get very high up-converted flicker noise if you sweep near to the carrier frequency). You won't get noise contributions from higher frequencies "folded" by the harmonics of the carrier because you simply are not including them. Sweeping over a wide frequency range is NOT the same as including lots of sidebands - you are sweeping the output frequency yes, but you've removed the simulator's ability to compute all the transfer functions from the noise sources to the output - you've only included three noise transfer functions.

I definitely agree that folding the high frequency content of the base harmonic noise is not the same as integrating the high frequency content of the high order harmonics, that were folded by the higher harmonics into the frequency range of interest (say [0,Fc/2]). BUT this is done on purpose, assuming that the claim is correct. A nice illustration of this is below, taken from spectreRF user guide.

So the point is that I believe that (1) only one skirt should be taken and (2) that skirt is folded by the sampling nature of the signal's edges hence its high frequency content should be integrated.

 

 

I'm still not convinced that what  you're claiming is correct (I'm going to consult with R&D to try to come up with a more convincing explanation). For a start, in your circuit the noise around the first sideband of the oscillator may be dominant, and so neglecting the higher sidebands may not make a significant difference - so it's hard to confirm anything for sure in the general case from the numbers you've given.

BTW, the plots you are showing from the SpectreRF manual are actually explaining the effect of maxsidebands in how inclusion of noise contributions works in the simulator.

Andrew.

I don't think that your simulation results can be generalized. For your circuit, oscillator phase noise is probably the dominant noise source. In this case, the jitter of every sideband is the same. This is generally not the case for jitter due to other noise sources.

If I have to simulate jitter, I always use pnoise PM jitter analysis. It's simply the most direct way to do it, which reduces the risk of making an error in the simulation setup.

 I don't have any clear argument why it could be generalized, it just makes sense to me to say so.

Anyway, I can say that (1 - Frank) the dominant noise source is not the oscillator but the stages that extract CMOS clock from the sine wave at the oscillator loop and (2 - Andrew) when I'm not neglecting the higher sidebands and integrate with fullspectrum, range [1K,300*Fc] I see jitter of 23ps so the higher sidebands do make a significant difference.

 

 

 

Yizhak,

I've attached an explanation from my colleague Edouard Ngoya from our R&D team. Edouard was having trouble replying on the forum himself, so I'm doing it on his behalf. 

I've attached it as a PDF because it has some equations in which are not rendered well in the forum. I think you'll find it an interesting explanation - please note that there's also a question at the end:

A question Yizh – since you have the possibility of computing an effective threshold crossing jitter figure using
Pnoise/noisetyp=PM jitter what is the primary reason why you prefer the FM one despite of the possible limitations
? I didn’t clearly get it.

Kind Regards,

Andrew.

Edouard and Andrew,

Thanks for the detailed response.

A few comments:

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.

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).

Moreover, I once replaced the input to that first inverting stage with an ideal sine source for debug reasons (of another oscillator design), and still saw almost the same jitter at the output of the circuit.

So I think that the frequency conversion's contribution is not negligible

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, that is not the case and the results when integrating only the low frequency contents are far from PM jitter result (see my post dating 02-05-2013). The second is that I shouldn’t integrate past F0/2: this is essentially correct since the phase perturbation is being sampled by the rising (or falling) edge, at a frequency of F0, so Nyquist is F0/2. However the higher frequency contents are folded into the band of interest so integration up to a high frequency is mathematically similar to the two steps process, (1) folding and (2) integrating up to F0/2.

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.

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.

Yizhak

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

 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

 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.

 

 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.