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

[Murray Lang]

Thanks Robert,

I need to go away and try to get my head around it some more. I need to 
understand the relationship between the sampling resolution of I/Q and the 
resolution (ie step size) of phase adjustments that it can apply to RF. 
It's not going to be continuous. I'm starting to work on the math, but I 
have a way to go.

Cheers
Murray
VK6HL

At 12:43 AM 6/09/2006, Robert McGwier wrote:
>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>
>>>To: "Robert McGwier" <rwmcgwier at gmail.com>
>>>Cc: <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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