Comparator: Noisy sine to square wave, how much phase noise?

Depending on how the spectral density is provided, it is essentially asin

Determine the phase error due to the hysteresis:

\$ \Theta_{low} = sin^{-1}(-0.3) \$

\$ \Theta_{high} = sin^{-1}(0.3) \$

This is the phase error purely due to the hysteresis if a pure sinewave was applied.

Assuming you have or have converted your spectral density into magnitude & equally assuming it is normally distributed. generate the MEAN and 1 standard deviation.

LOW:

\$ \Theta_{low_error\_mean} = sin^{-1}(-0.3) - sin^{-1}(-0.3 + mean) \$

\$ \Theta_{low\_error\_+\sigma} = sin^{-1}(-0.3) -sin^{-1}(-0.3 + \sigma) \$

HIGH:

\$ \Theta_{high\_error\_mean} = sin^{-1}(0.3) - sin^{-1}(0.3 + mean) \$

\$ \Theta_{high\_error\_+\sigma} = sin^{-1}(0.3) -sin^{-1}(0.3 + \sigma) \$

With the mean and the standard deviation "phase error" you can reconstruct a phase error distribution curve.

However... if the spectral density isn't normally distributed you will need to derive errors at a number of specific points to reconstruct a phase error curve specific to the information you have

This answer ----- The amplitude of the phase noise (in µs) is simply the noise voltage divided by the slope ----- is from Dave Tweed. Or TimeJitter = Vnoise / SlewRate

is the form I've used for over 2 decades.

In ONNN Semi PECL, using Bandwidth of 10GegaHertz and Rnoise of 60 Ohm (1nV/rtHz), with Slewrate of 0.8v/40picoseconds, the TimeJitter is Vnoise = 1nV * sqrt(10^10) = 1nV * 10^5 = 100 microVolts RMS. SlewRate is 20 volts/nanosecond. The TimeJitter is 100uV RMS / (20v/nS) = 5 * 10^-6 * 10^-9 = 5 * 10^-15 seconds RMS.

What is the spectral density of the jitter? We simply scale down by the sqrt(BW) which is 10^5, yielding 5 * 10^-20 seconds/rtHz.

For your question: 1MHz, 1voltPeak, 20dB SNR and Tj = Vnoise/SR, we have Vnoise = 1V/10 = 0.1vRMS (ignoring any sin-peak-rms ratios) SlewRate = 6.3 Million volts/second, therefor TimeJitter = 0.1v/6.3Mega v/Sec = 0.1 * 0.16e-6 = 0.016e-6 = 16 nanoSeconds RMS.

As an advice, you could reduce the noise by adding a low-pass filter to your design before going into the comparator. This would cut-off the higher frequencies of your signal which is the noise in this case.

To calculate the frequency of the phase noise, you can use FFT or perform a spectrum analysis of the signal. A frequency spectrum would give you the frequency of your signal plus the frequency of the un-wanted noise.

The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency.

Derive an equation for the signal you are getting, and perform a Fourier transform to get the amplitude and phase plotted against frequency.

Given the spectral density of the noise on the input signal, how can I calculate the phase noise of the square wave?

This is just a thought on how to possibly get to a value...

I think I'd be tempted to use a PLL (phase locked loop) to generate a squarewave from its VCO that tracks the basic fundamental signal. Your schmitt comparator is a good start and could feed a PLL nicely. The output from the PLL's phase comparator would need to be highly low-pass filtered so that the control voltage to the PLL's VCO would be very smooth and cause minimal jitter on the VCO.

The raw output from the phase comparator would be a very good measure of the phase noise. If there were no phase noise, that output would be very regular.

Anyway, it's just a thought.

For a random noise signal of Npp around 10% with a signal Vpp comparing peak-peak ratio it can be seen that if the signal is a triangle waveform that the amplitude noise is converted to phase noise in a linear equation where is S/N=1 each edge has T/2 jitter p-p.

However the amplitude of the sine fundamental component is 81% of a Vpp triangle waveform and thus it's slope is 1/81% or 1.23 steeper thus phase noise is reduced to 81% of the ratio with hysteresis set to just higher than the peak noise level.

Thus the jitter on each edge is 81% of the Vpp/Npp ratio. It could be shown that slope matches the triangle wave when the Npp reaches 75% of the Vpp or a Vpp/Npp ratio of 1.33.

Normally jitter errors are measure in RMS noise power and energy per bit and statistical probability of error, but this was shown from the perspective of the question for time jitter over any measurement time period.

This ignores any asymmetry error which may be caused by a DC offset or the comparator positive output feedback not biased properly. The phase shift and the edge jitter is also proportional to 81% of the % Npp/Vpp inverse SNR ratio for levels below the 20% range roughly.

e.g. Consider Noise is 10% in pp ratios then each edge will have jitter of 8.1% of T/2