The principal figure of merit for a clock is that at various times in the future it is correct. This requires in turn a high degree of frequency stability. So long as the phase noise is zero mean over the shortest period you wish to measure with the clock and this holds true over the full range from the shortest to the longest, the phase noise is irrelevant to time keeping.
That makes sense.
The claim in bold is false.
In a clock, the oscillator's phase noise will accumulate over time as an imprecision in time keeping, even if the average value of the phase noise is zero.
I did not make the statement in bold. My statement was a sentence with an important constraint which you wish to ellide and then claim I am wrong. I included that constraint precisely because the statement in bold is false.
Would you please show mathematically why a clock meeting the constraint I imposed would lead to an accumulation of errors? I shall be very interested in your argument (aka mathematical proof).
By clock, I will understand a time keeping device.
As an example, let's consider a clock made from an oscillator followed by a counter. The counter will count the number of oscillations. By reading the counter, we can measure time.
To simplify this thought experiment, let's consider a square wave oscillator followed by a digital counter that counts the number of rising edges.
By phase noise, will understand the next rising edge of the oscillator arrives to the counter slightly faster, or slightly delayed, than expected from an ideal oscillator (ideal as in constant frequency and no phase noise).
For the first case, let's consider the clock has an ideal oscillator: constant known frequency, and no phase noise. We can measure the time by reading the counter, and compare the number with another reference clock. The maximum error will be +/-1 count. This error will never increase with time, because we assumed our oscillator is ideal, and the reference clock is also ideal.
Now, let's add some phase noise. Let's say we have a true random numbers generator that generates only -1 and +1, with 50/50 chances. The average of many random numbers will converge to zero. We will generate a random number for each oscillator's period. If the random number is +1, we artificially move the rising edge of the oscillator to the right (let's say +1us). If the random number is -1, we move the edge to the left (-1us).
On average, the period of the oscillator stays the same, because our random +/-1 has exactly 50/50 chances.
At the first look, we will be tempted to say that our clock will not be affected, because the average frequency of the oscillator stays unchanged, but this is NOT true: the clock will be affected. The later we read our clock (counter) the more erroneous readings we get. That phase noise accumulates with time.
To understand why, we need to consider the worst possible scenarios. Let's say we read the counter after 100 rising edges. The worst possible error will be +100 or -100us. If we read the counter after 1000 edges, then the worst possible result will be +1000us or -1000us.
For an oscillator with white phase noise, the errors will have a Gaussian distributtion. The bell of errors is always centered on zero, but the shape of the bell goes wider and wider in time, and that's because of that zero mean phase noise.
In conclusion, measurement errors caused by phase noise increases with time.
This will affect the time keeping (for either long or short time), as well as any phase measurement of the oscillator. The later we measure, the bigger the errors.
Here is an experiment to check the above conclusion:
- We have a DDS generator (Rigol DG4102) and an oscilloscope (Rigol DG1054Z), each with their own internal oscillator, and their own phase noise
- the DDS generates pulses of 10ns at each 500ms
- the oscilloscope visualizes the pulses in 3 situations:
1. First pulse, at the trigger moment (video between minute 00:07 and 00:10)
2. The second pulse, at 500ms after the trigger moment (video between minute 01:57 and 03:10)
3. Third pulse, at 1s after the trigger moment (video between minute 00:35 and 01:43)
The blue, orange and red spikes are only some fixed markers. They are just as references. Ideally they should be only one spike in the center of the grid, but they have different positions because of some small difference in frequency between the DDS and the oscilloscope's oscillators.
The useful signal (the 10ns pulse) is the green trace.
The video is unedited, so please look only at the specified moments, and ignore the periods where I was changing the oscilloscope's settings.
In case 1, the pulse is stable, in case 2, the pulse WIGGLES around the 500ms mark, in case 3, the pulse wiggles EVEN MORE around the 1 second mark. The errors increases with time.
Of course this error accumulation caused by the phase noise can be alleviated buy averaging repeated measurements, but repeating the measurement is not always possible. Even if it were, the effect of phase noise is still important in order to decide how many measurements we need to average.
Now, all these are nothing more than my own intuitive explanation, using only time domain and common sense about probabilities. I still didn't provide a mathematical demonstration. Probably that would be to demonstrate the Central Limit Theorem, but then I will just copy/paste, and it will still make no sense without all the above.