[time-nuts] Phase Detectors/Mixers for DMTD and PN measurements

[Attila Kinali]

On Thu, 16 Jan 2020 12:24:30 -0500
Bob kb8tq <kb8tq at n1k.org> wrote:

> > Another trick you can employ is to pass the LO through a limiting
> > amplifier. The idea is that the main contributor to noise in a
> > diode ring mixer is the noise (or uncertainty) of switching delay
> > of the diodes (which is not symmetric, btw). By having an LO with
> > a high slope, this uncertainty is reduced. But high slope means
> > high power, which might damage the mixer. Using a clamped signal
> > instead limits the power into the mixer and thus prevents it from
> > frying while giving the advantage of having fast rise/fall times
> > for the diodes to switch with (note that the swtiching time is
> > limited by how fast the space charge zone can be built up and removed).
> > Big disadvantage of this is, that now the LO has lots of harmonics
> > which will lead to noise down-folding. As 1/f noise is the limiting
> > factor for DMTD, especially even harmonics have to be kept as low as
> > possible, ie the duty cycle (or DC offset) of the LO signal should
> > as close to 50% (or 0, respectively) as possible. A nice side effect
> > of this is that the switch-on-switch-off asymmetry of the diodes is
> > reduced by having a higher slope, thus reducing even harmonics.
> 
> If you look at the response of a normal diode ring mixer, it already 
> *has* responses at the odd harmonics of the LO signal. The reason is
> pretty simple - the diodes have turned the drive into a square wave, regardless
> of it being a sine wave to start out with …..

Yes, and that's exactly the problem.
To expand a little bit: Noise down-folding comes to be, when
a signal with harmonic content passes through a non-linear element.
Then, this element acts as kind of a mixer on the signal with itself,
thus the spikes, that the harmonics are, mix down the noise around
those spikes down to the fundamental frequency. (For a more thorough
mathematical explanation, see [1])

Now in the mixer case, we have the mixer as non-linear element
and we feed it with a signal that is high in harmonic content.
Those harmonics will down mix the noise down to the signal 
frequency and DC. If there are only odd harmonics, then there
will be only white noise mixed down to the signal frequency
(spacing of odd harmonics is twice the signal frequency, thus
there is only white noise in f_0 distance of an odd harmonic).
This leads to a additive increase of noise power (the 3dB per
doubling type increase) per harmonic (white noise is uncorrelated
unless harmonics have the same power as the signal).

If there are even harmonics as well, then 1/f noise will be
mixed down to the signal frequency as well. But now we have
correlated noise (the 1/f noise around the harmonics has
the same origin at the fundamental of the signal) thus we
get additive increase of noise voltage (the 6dB per doubling
type increase).

Unlike in [1] where only the fundamental frequency was important,
in a mixer for DMTD application we care about the noise around DC.
In this case we get down-folding of 1/f noise even if there are
no even harmonics present, as the noise around each harmonic gets
mixed by its harmonic down to DC.

So, how do we combat this? Easiest is to use a pure sinosoidal signal
with no harmonics. If the signal slope is to be increased and power
needs to be limited, then it's best to add a filter between the
squaring circuit and the mixer to limit the number of harmonics.
Even a first or second order filter is enough and you don't need
to go crazy[2]. The corner frequency should be choosen such that
the increase in slope vs increase in noise reaches an optimum.

			Attila Kinali

[1] "A Physical Sine-to-Square Converter Noise Model",
by Attila Kinali, 2018
http://people.mpi-inf.mpg.de/~adogan/pubs/IFCS2018_comparator_noise.pdf

[2] The harmonics of a square wave decrease with 1/i, i being the
number of the harmonic. Passing it through a first order filter
adds a 1/f term, ie the harmonics then decay with 1/i^2.
Every increase in filter order adds another 1/f or 1/i term.
The sum of all 1/i for i=1,2,3,...infinity diverges (ie goes to infinity). 
The sum of all 1/i^2 for i=1,2,3,...infinity is ~1.645.
The sum of all 1/i^3 for i=1,2,3,...infinity is ~1.202.
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