Phase noise to phase jitter for square waves

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

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