[time-nuts] Noise of digital frequency circuits (was: Programmable clock for BFO use....noise)

[Bob kb8tq]

Hi

It’s pretty easy to demonstrate that squaring up a sine wave, even with fairly simple
circuits does not create crazy phase noise issues. People have been doing it successfully 
for a lot of years. In general faster saturated logic produces lower noise floors than slower
logic. 

Bob

> On Sep 16, 2018, at 4:33 PM, Dana Whitlow <k8yumdoober at gmail.com> wrote:
> 
> I'd been thinking, in an admittedly non-rigorous sort of way, about this
> matter for some years.
> 
> As I see it, it is certainly true that the phase of an oscillator's output
> is a continuous funciton
> of time.  It could be described as a continuous ramp, whose slope
> corresponds to the frequency,
> and with a little bit of non-flat random noise superimposed on it.
> 
> Now if you square up the waveform and do digital things with it (such as
> freq dividing, digital
> phase detection, etc), you are really only glimpsing the phase noise at
> transition times, and
> are blind in between.  Thus the very process amounts to sampling the phase
> noise waveform
> with a sampling phase detector.  This view suggests that all the phase
> noise power is aliased
> and folded  back into the band ranging from DC to Fsamp / 2, where Fsamp is
> the frequency
> of the waveform after frequency division.  This is why the time domain
> jitter of the oscillator's
> waveform is unchanged by "perfect" frequency division (or multiplication).
> 
> It is why I wonder about the wisdom of doing phase comparison at
> unnecessarily low frequency-
> all that noise would seem to be scrunched down into a bandwidth of half the
> comparison frequency.
> 
> Does this explanation help, and how does it sit with those of  you who have
> more expertise
> than I?
> 
> Dana
> 
> 
> 
> 
> On Sun, Sep 16, 2018 at 4:06 PM, Attila Kinali <attila at kinali.ch> wrote:
> 
>> Moin,
>> 
>> On Sat, 15 Sep 2018 08:38:55 -0700
>> "Richard (Rick) Karlquist" <richard at karlquist.com> wrote:
>> 
>>> On 9/15/2018 3:26 AM, Attila Kinali wrote:
>>> 
>>>> possible logic family for the task. Otherwise the harmonics of the
>>>> switching of the FF will down-mix high frequency white noise down
>>>> to the signal band (this is the reason for the 10*log(N) noise scaling
>>>> of digital divider that Egan[1] and Calosso/Rubiola[2] and a few others
>>>> mentioned).
>>> 
>>> Wow, I never knew this in 45 years of designing synthesizers!
>>> I do remember that some of the frequency counter engineers at HP
>>> talked about noise aliasing.  I think this is another way of
>>> describing the same problem.
>> 
>> Yes. This effect has been known for a few decades at least.
>> What kind of puzzles me is, that I have not seen a mathematically
>> sound explanation of it, so far. People talk of aliasing and sampling,
>> but do not describe where the sampling happens in the first place.
>> After all, it's a time-continuous system and as such, there is no
>> sampling. One could look at it as a (sub-harmonic) mixing system,
>> but even that analogy falls short, as there is no second input.
>> It also fails at describing why there is not infinite energy being
>> down-mixed, as the resulting harmonic sum does not converge.
>> 
>> If someone knows of a description that goes beyond handwavy arguments,
>> I would very much appreciate hearing of them.
>> 
>> The only way to explain the effect in a rigorous way, that I could
>> figure out, is to apply Hajimiri and Lee's Impulse Sensitivity Function[1],
>> and adapt from the oscillators they discribed to general periodic systems.
>> (The step, as one can guess, is small, but hic sunt dracones)
>> Doing this, it becomes obvious that the down-mixing is an inherent
>> property of all systems that use or generate non-sinusoidal waveforms.
>> It is this ISF that is the source of the down-mixing/aliasing effect,
>> as it has a periodic waveform of sharp spikes.
>> 
>> As the ISF is probably (this is my intuition and I have, unfortunately,
>> no proof of this) related to the derivative of the produced output
>> waveform,
>> it becomes important to limit the slew rate of the output, to introduce
>> a second pole in the ISF and thus limit the number of harmonics.
>> Yet, it is also important to keep the input slew rate high, in order to
>> keep the width/height of the ISF pulses low.
>> 
>> A partial discussion of this can be found in the paper I presented
>> at IFCS earlier this year[2]. Unfortunately, the write-up is not
>> nice and I only realized after the deadline that I should have
>> all written it using a different approach. Sorry for that.
>> If something is not clear, do not hesitate to send me an email.
>> 
>>> About 10 years ago, the frequency synthesizer chip vendors started
>>> talking about a Figure of Merit (FOM) that predicted phase noise floor,
>>> and it also included the 10 LOG N noise scaling.  An application
>>> engineer at ADI told me this was a characteristic of the sampling phase
>>> detector that all these chips used.  But I always wondered if the
>>> frequency divider could come into play.  The way FOM is defined,
>>> it doesn't distinguish between phase detector and divider noise.
>> 
>> The 10*log(N) also applies to the phase detector in PLL chips,
>> where N becomes the ratio of the phase detector bandwidth divided
>> by the phase detector input frequency.
>> 
>> Given that the phase noise is dominated by the input source' phase
>> noise, there will be no appreciatable difference in whether the
>> down-mixing happens in the divider or the phase detector, as long
>> as the bandwidth of all components is the same. If the bandwidth
>> is different, we get into something akin Collins' zero crossing
>> detector[3] where appropriately designed stages with different
>> input bandwidths limit the energy that is down-mixed.
>> 
>>> At Agilent, we used to make a lot of lab demos using a Centellax
>>> (now Microsemi AKA Microchip) frequency divider that could divide by any
>>> number between 8 and 511 up to 10 GHz.  It was absolutely fabulous for
>>> dividing 10 GHz down to 2.5 GHz.  But 20 LOG N quit working if I tried
>>> to divide down to 50 MHz.  Now you have explained it.
>> 
>> Hmm? Are you implying those chips somehow were able to give
>> a 20*log(N) phase noise behaviour? If so, do you know how
>> they achieved such a feat?
>> 
>> 
>>>> If you divide by something that is not a power of 2, then it is
>> important
>>>> that each stage produces an output waveform with a 50% duty cycle.
>> Otherwise
>>>> flicker noise which has been up-mixed by a previous stage, will be
>> down-mixed
>>>> into the signal band, increasing the close-in phase-noise.
>>> 
>>> Wow, another thing I never knew.
>> 
>> I do not think that anyone was aware of this. A least I do not remember
>> seeing this being mentioned in any of the papers I have read. I, myself,
>> stumbled over it by accident. I was trying to design a sine-to-square
>> wave converter and wanted to understand what happend to the noise.
>> Especially the AM to PM conversion that a few people here have mentioned
>> a few times. I was looking at Claudio's measurement [4, page 28] and,
>> after applying Hajimir and Lee's ISF, I could (mathematically) explain
>> everything but what Enrico so nicely labled as "bump". None of the
>> explanations that I exchanged with Enrico, Claudio, Magnus and a few
>> other people made sense with the complete data. An external influence
>> didn't make sense as the flicker noise went from a straight ~6dB/oct line
>> to a straight ~3db/oct line below 25MHz. This hunch got stronger when
>> Claudio shared the complete circuit they used with me(see figure 3 in [2]).
>> The feedback circuit, which stabilizes duty cycle, has a -3dB frequency
>> of 0.28Hz, which is exactly the frequency where the bump is. And below
>> it, the flicker noise behavior seems to go back to approximately 6dB/oct.
>> For a complete explanation, see my paper[2] section 5.D "Scaling in a
>> Multi-Stage Sine-to-Square Converter."
>> 
>> 
>>> The conventional wisdom was to
>>> divide by any number (even or odd) and then follow that divider
>>> with a divide by 2 flip flop to get 50%.  Now, that is in question.
>>> The now correct answer is to us a variable modulus prescaler to
>>> divide by P and P+1, controlled by a toggle flip flop to make
>>> half the divisions at P and half at P+1.
>> 
>> I don't think the modulus prescaler is a good approach.
>> It will help reduce flicker noise, at the price of incrased
>> white noise, as the two division values will generate two
>> frequency spikes in the ISF that are close to each other.
>> There is probably some residual even harmonic content due to
>> the switching betwen the two scaler values, which will increase
>> flicker noise, not as much as having non-50% duty cycle, but still.
>> 
>> The right way to do it is to use both edges in case of odd division
>> factors (as some of the divider circuits by Linear/Analog seem to do).
>> Alternatively generate a ramp/sine output, ie use a Λ-divider
>> or a DDS, as both have much lower harmonics content in the ISF
>> and thus do not suffer from the down-mixing as much. If a square
>> waveform is required afterwards, a square-to-sine converter with
>> approriate bandwidth for the output frequency will solve that.
>> 
>> 
>> 
>>                        Attila Kinali
>> 
>> 
>> [1] "A General Theory of Phase Noise in Electrical Oscillators,"
>> by Hajimir and Lee, 1998
>> 
>> [2] "A Physical Sine-to-Square Converter Noise Model,"
>> by Kinali, 2018
>> 
>> [3] "The Design of Low Jitter Hard Limiters," by Collins, 1996
>> 
>> [4] http://rubiola.org/pdf-slides/2016T-EFTF--Noise-in-digital-
>> electronics.pdf
>> --
>> <JaberWorky>    The bad part of Zurich is where the degenerates
>>                throw DARK chocolate at you.
>> 
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