Phase noise to phase jitter for square waves

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

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