[hpsdr] My doubts about I/Q beam-forming

[Robert McGwier]

Let's take almost the simplest possible case.   I have two isotropic
radiators spaced one half wavelength apart, X and Y.   I am going to look at
two emitters.  One is collinear with my two radiators and one is
perpendicular to the line joining my two radiators


X---wavelength/2---Y


Let us that the emitter carrier frequency is F and that its total bandwidth
is B and that B is tiny compared to F.


wavelength in the antenna ascii art is determined by F.  So let us suppose
we have a receiver system that digitizes the incoming signal from X and Y
with LO's that are provided by a single oscillator and furthermore, we will
assign that the same oscillator is divided down to provide the sampling
clock for our digitizers (A/D's) and that the sample rate >  2B.  This is
bigger than Nyquist for the bandwidth of my signal.  All signals are in the
far field and so far that we may consider wavefronts arriving as parallel.
The perpendicular source is "up".   There is no delay between X and Y,  so I
simply add the signals together.   Signals arriving from different angles
than perpendicular to the line joining X and Y are attenuated with the
greatest attenuation in the collinear directions,  to the right of Y and to
the left of X respectively.

Now consider the source that is collinear with X and Y.  Assume the
collinear source is to the "right".    The signal arrives at antenna X
delayed by

wavelength /  2c    seconds.

At the frequency F this is a phase delay of pi/2 or 90 degrees since the LO
for both X and Y as well as the A/D clocks are all common or tied to the
same source.

So the X signal is delayed by wavelength/2c seconds and the phase has
advanced by pi/2.

After you have gone through the receive process, and are down to digital
samples,   AND because the bandwidth of the signal B is tiny compared to the
carrier frequency F,   you cannot distinguish the time delay wavelength/2c
because we are so grossly undersampled at the digitized samples in most
cases,  but you can easily rotate the digital samples for X by pi/2.     In
fact,  it doesn't matter what the delay is between these antennas up
rotational ambiguity and not even there if F is hugely larger than B.   All
that matters is the phase differences,  which you compute,  reverse, and
then add.  If you had N antennas,  you simply compute the signal delays for
each of the N antennas given the direction you want to aim, and then compute
what phase difference this will induce because of the continuous phase
advance of the LO (shared common amongst all antennas).   You turn these N
delays DIRECTLY into phase rotations that cancel the one at frequency F
caused by the delay.   Again,  at digitally sampled "base band",   you
cannot distinguish this delay because it is TINY compared to the sample
rate.   You ignore it, do the phase rotation, and add all N signals up.

This "ignore it"  for the delay is why you make the assumption that F  >>
B.   The phase shift across the entire bandwidth due to the delay and the
offset of the signal of interest from zero causing the phase variance is
assumed to be negligible compared to the rotation due to LO advance because
of the delay.   When the B bandwidth of the signal becomes large enough
compared to the carrier F,  so that Nyquist sampling of the bandwidth B can
"almost see" the delay between elements,  then you must compute a frequency
dependent correction.  But again,  you can do this in software with a fast
enough computer by polyphase filtering,  doing it in the frequency domain
where delays are PRECISELY phase rotations.  If you do it using an FFT,
then the bins will each have different rotations.  This is not a narrow band
phased array since compensation must be done in a tapped delay line.



Did this help at all or is it only more confusing?

Bob
N4HY


Murray Lang wrote:

***** High Performance Software Defined Radio Discussion List *****

I would have thought the accuracy required would be orders of magnitude
greater than for sideband suppression.
We're talking about phase shifts of small fractions of a wave length at 10s,
100s or 1000s of MHz.

Murray
VK6HL

At 02:46 PM 5/09/2006, John B. Stephensen wrote:


Fortunately, mixers are linear for amplitude and phase. The accuracy
required isn't any more than for sideband suppression.

73,

John
KD6OZH

----- Original Message -----
From: "Murray Lang"
<murray.lang at metoceanengineers.com><murray.lang at metoceanengineers.com>
To: "Robert McGwier" <rwmcgwier at gmail.com> <rwmcgwier at gmail.com>
Cc: <hpsdr at hpsdr.org> <hpsdr at hpsdr.org>
Sent: Tuesday, September 05, 2006 03:10 UTC
Subject: Re: [hpsdr] My doubts about I/Q beam-forming






-- 
AMSAT VP Engineering. Member: ARRL, AMSAT-DL, TAPR, Packrats,
NJQRP/AMQRP, QRP ARCI, QCWA, FRC. ARRL SDR Wrk Grp Chairman
"You see, wire telegraph is a kind of a very, very long cat.
You pull his tail in New York and his head is meowing in Los
Angeles. Do you understand this? And radio operates exactly
the same way: you send signals here, they receive them there.
The only difference is that there is no cat." - Einstein
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