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

Bob kb8tq kb8tq at n1k.org
Mon Sep 17 10:24:16 EDT 2018


Hi

For moderate division ratios ( like 100 MHz down to 1 MHz ), the 20 log N stuff holds 
pretty well ….

Bob

> On Sep 17, 2018, at 12:37 AM, Dana Whitlow <k8yumdoober at gmail.com> wrote:
> 
> The act of squaring up the waveform alone might not do much harm, depending
> on the extent
> to which the phase noise on said waveform has already been filtered off.
> But it's mainly when
> the signal gets divided down by large ratios that the difference would
> become really noticeable.
> 
> For example, take the case of 10 MHz starting frequency; the phase noise
> several MHz out
> is likely to be nil.  But divide the 10 MHz down to, say, 1 Hz; then
> there's likely to be quite a
> lot of phase noise within "folding range" of many Nyquist bands about 1 Hz.
> 
> This, again, is why I wonder so much about our efforts in re-synthesizing
> higher frequencies from
> the 1PPS from GPS receivers.  I don't know much the architecture of GPS
> receivers, but it seems
> to me it would sure be nice if there were some convenient way to extract a
> clean signal at the
> chipping rate, for use in generating standard reference frequencies.
> 
> Dana
> 
> 
> 
> 
> On Sun, Sep 16, 2018 at 9:15 PM, Bob kb8tq <kb8tq at n1k.org> wrote:
> 
>> 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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