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

Attila Kinali attila at kinali.ch
Sun Sep 16 17:06:06 EDT 2018

```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.

```