# RE: variable phase delay / phased arrays

```John wrote:

> I see what you are saying. A 10 degree phase shift at the RF
> frequency would be
> translated to a 10 degree phase shift at the IF frequency by
> the mixing process.

Yes.   For a simple carrier, s(t) = A*e^(j*(2*PI*Fo*t + phi))
= A*e^(j*2*PI*Fo*t) * e^(j*phi)
^             ^
carrier freq     phase

> The actual time delays may not be correct, but will always be
> off by multiples
> of 360 degrees so the antenna pattern does not change. Any
> extra time delays
> will cause some dispersion in the received signal but that
> will not matter for
> narrowband communications.

How I look at it is that a correct time delay will line up phase exactly for
every frequency of interest.  Let's say that to beamform for an extremely
broadband pulse (from 1 to 5 GHz), a time delay of 1/10 wavelength was
needed at 1 GHz for one channel.  This works out to 100ps, or 36 degrees at
1 GHz.  At 2 GHz, the same 100 ps delay would represent 72 degrees of phase
shift.  At 3 Ghz ... and so on.  At 5 GHz, it would be 180 degrees of shift.

So if you decided to beamform with time delays, one delay per channel would
be all you need.  If you decided to beamform with phase shifters, you would
need to break up this large band into a number of smaller subbands, apply a
phase shift to each subband, and then add the subbands back up again.  The
potential problem with this 'frequency domain' approach to (broadband)
beamforming is that the size of your aperture now affects the bandwidth of
the beamformer (out of scope for now ... but this is where multiples of 360
degrees DOES matter).

Fortunately, we are interested in signals that don't deviate much from
carrier, such as 5 GHz +/- 0.001 GHz.  Using the previous example, our phase
shift values for this 'band' deviate from 180 degrees by +/- 0.036 degrees
(ie the error at the band edges would be very small if you simply used a
shift of 180 degrees after isolating just that band of interest).  Since we
are now beamforming a narrowband signal, a shift of 180 degrees or 540
degrees (ie 1/2 vs 3/2 wavelengths path difference) would have negligible
difference on a MHz wide baseband signal (several 100th's of a degree) when
the carrier is at 5 GHz.  Since we are processing digitally, we will always
used phase shifts limited to a 360 degree range (ie +/- 180, -PI to PI, 0 to
2PI, etc) to make our lives easier. ;-)

Basically, time delay is best, phase shift is cheapest, and the math shows
us when it is appropriate to use one or the other (or a hybrid).

Hope that makes things a bit clearer.

Cheers,

George
VE3LQN

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