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

[Magnus Danielson]

Hi,

On 2020-01-17 10:32, Attila Kinali wrote:
> 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])

The main benefit of diodes is that they are non-linear. The McLauren
expansion of their PN-junction property provides x, x^2, x^3 and higher
terms and in that you have multiplications of the signals visible at the
PN-junction, the one or two signals, their noise, the noise at DC, the
noise contribution of the diode itself. It is those x^2 and higher order
terms that provides mixing as the signals under mix is added linearly
together. The double-balanced arrangement is there to create
cancellation of odd terms. Then this can be sliced and diced in many
ways, misunderstood in many ways etc.

For modern mobile base-stations, they have great problem with the
multi-carrier setup that can produce Passive InterModulation (PIM),
which comes from all kinds of metal-metal interfaces with a bit of oxide
or so in them, as it will acts as a diode which will mix the signals and
re-transmit them in-band. Typically it's the third degree mixing which
is troublesome, that is x^3 as you have twice one carrier minus another
or similar. They need to measure that and then go around and tighten or
replace bolts and nuts etc. Nonlinear responses is both a blessing and a
curse.

>
> 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).
There is NIST reports about dividers doing odd divisions having
increased noise, but dividing by 2 removes that. This is where it is
very useful to understand how a PWM waveform spread it's energy into
even and odd harmonics, as 50% has the even harmonics go to zero, and as
a consequence not contributing to 1/f noise mix 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.
The odd/even balance does not go to full zero as you get off axis.
>
> 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.
No harmonics is overstating it, sufficiently low harmonics is the only
thing you can wish for. As for filter frequency, see my note below,
which is extra relevant as for since it needs some care to ensure low
AM, and AM often is shaved down by convert some of the energy to overtones.
>
> 			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.

You make an assumption about the first order lowpass filter frequency
for those numbers. You would prefer to keep the bandwidth somewhat
higher than the frequency of the square, since that will provide too
strong AM-to-PM conversion, but with that the converged numbers will
become somewhat larger.

Cheers,
Magnus