[time-nuts] Re: uncertainty/SNR of IQ measurements

[Dana Whitlow]

Hi Jim,

I think the best way is to view the signal as a phasor, with any
noise present adding a
random trajectory (a fuzzball) to the tip of the signal vector.
Conceptually speaking,
this eliminates needing to worry about the distribution of power between I
& Q, etc.
It lets you view the whole thing without regard for choice of axes,
coordinate system,
and all that.

If the S/N is good, the fuzzball is small in size compared to the length of
the phasor,
and you can immediately see that neither the length nor the angle of the
sum vector
is much affected.

But as the SNR is reduced, you eventually reach the point where some of the
noise
peaks almost reach down to the origin, and as the vector tip swings near
the origin
the phase angle changes very rapidly by nearly 180 deg, but the effect is a
temporary
glitch of zero area.

But if the noise peak is a little bit bigger, the vector tip swings all the
way around the
origin, yielding an eventual effect of an added 360 deg (a whole extra
cycle) in phase
shift.  This tends to have a far more deleterious effect on a signal.  I
had a text given
to me by a friend in which the author used this kind of explanation to
explain, for
example, the "threshold effect" of noise in FM demodulation.  I just
looked, but could
not find the book, else I'd have given you the title and author information.

This mode of thought also leads towards an understanding of the "FM capture
effect",
which spec was always highlighted in datasheets of HiFi FM tuners.  But one
hears
little of it nowadays, I suspect because the advent of fast ICs has made it
so easy
to very- closely approach the "theoretical limit" that everybody is about
the same.
BTW, said "theoretical limit" is not fixed until one specifies other
parameters, and
at one time there was a standard test definition so that such a limit could
be defined
and measured against.

In IQ demodulation I find the ATAN2 function a good deal more useful than
the old
arctan function, which needs a lot of help in order to work usefully.  The
ATAN2
function takes two arguments (I & Q values) and automatically places the
angular
result in the correct quadrant and is not bothered by either of the
arguments being
zero.  The only place it gets in trouble is if *both* arguments are zero,
which is an
infinitely-tough nut to crack in any case.

As with all the inverse trig functions, ATAN2 has a limited angular range,
from
-180 deg through zero to +180 deg, then snaps back to -180 deg again.
But it's not too difficult to "unwrap" the results so that a continuous
rotation of a
phasor leads to a nice smooth phase ramp with no discontinuities at all.
In many
cases this presentation makes the picture much clearer, although overly high
noise peaks can create what some will call a false transition.  If you're
really
interested in the signal alone, yes you have a problem then.  But if you
consider
the "signal" to be the composite vector sum of some signal and added noise,
the unwrap process works correctly.

Whew!

Dana   K8YUM



On Thu, Aug 26, 2021 at 3:52 PM Poul-Henning Kamp <phk at phk.freebsd.dk>
wrote:

> --------
> Lux, Jim writes:
>
> >I'm looking for a simplified treatment of the uncertainty of I/Q
> >measurements.  Say you've got some input signal with a given SNR and you
> >run it into a I/Q demodulator - you get a series of I and Q measurements
> >(which might, later, be turned into mag and phase).
> >
> >[...]
> >
> >I'm looking for a sort of not super quantitative and analytical
> >treatment that I can point folks to.
>
> Good luck with that :-)
>
> Some of the noise processes will be along the "vector" and distributed
> between I & Q components depending on the phase, while other noise
> processes affect the components individually.
>
> To make matters worse, both kinds of noise processes may depend on the
> phase, usually because of cross-talk and/or insufficient isolation.
>
> Low-resolution ADC's are a particular nasty problem, because they add
> +/-1 count jitter independent of the phase, and that causes very
> large arctangent errors.
>
> Counterintuitive as it may sound, it is easier to process the bits from
> ADC's where the low two bits are pure noise, than ADC's where all bits
> are good...
>
> --
> Poul-Henning Kamp       | UNIX since Zilog Zeus 3.20
> phk at FreeBSD.ORG         | TCP/IP since RFC 956
> FreeBSD committer       | BSD since 4.3-tahoe
> Never attribute to malice what can adequately be explained by incompetence.
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