Atilla wrote: "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. If someone knows of a
description that goes beyond handwavy arguments, I would very much
appreciate hearing of them."
I can only offer a handwavy suggestion, or food for thought, regarding
digital dividers of all sorts.
Regardless of the type of divider or process used, the devices within
have finite gain, so imperfect isolation between the output activity and
input. Whatever is happening downstream in a divider chain can provide a
delayed, topology- and pattern-dependent signal back to the input,
containing the associated frequency content. The issue of course, is how
big the effect may be.
I don't know if this sort of thing is trivial or has already been
included somehow in the rigorous and theoretical studies, but I know
it's there, having observed such anomalies over the years. I've never
had a situation where the effect was big enough to prevent something
from working right, just casual observations that made me think about
what's going on.
You can probably observe it easily with enough dynamic range. Say, set
up an ECL FF to divide by two, and AC couple everything for ground
reference. Put in an RF clock signal - sine, square, doesn't matter -
and look at this input signal with a spectrum analyzer (don't worry too
much about impedance matching - just get the clock signal big enough to
toggle). You should see the strong clock and its harmonics, as expected,
and if you dig deep enough, should be able to see the one-half frequency
that shouldn't exist with a perfect FF. Now, how it gets there may be
due to a number of reasons like ground loops or power supply coupling,
but some of it is going right through the device from out to in.
Ed
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