Project Cyclops, A Design... - Department of Earth and Planetary ...

andis the (discrete) two-dimensional Fourier transform
of A(x i_vj). Because y is constant during the summation
over i (or, alternatively, x is constant during the
summation over /), equation (47) can be written as a
two-step
process
e
-ikuxi
a(u,v) = _ -ikvy] _. A(xi'Yi)e (48)
Radiative
Imaging
1 t
One obvious way to implement equation (47) is to
radiate the IF signals from the signal plane and to receive
this radiation in the image plane, in exact analogy to the
optical spectrum analyzer. The waves radiated may be
acoustic, as suggested by Oliver (ref. 11) and McLean
and Wild (ref. 12), or electromagnetic. If the signal and
image planes are separated a distance _, several alternatives
exist for bringing the waves to focus in the image
plane:
1. We may interpose a lens of focal length _/2
midway between the planes.
2. We may interpose two lenses one of focal length
(the focussing lens) in front of the signal plane and
the other of focal length _/2 (the field flattene0 in
front of the image plane.
3. We may curve the signal and image planes into
appropriate spherical surfaces.
4. We may delay the signals to the central elements
so as to radiate a spherical wavefront from a plane
array.
With any of these alternatives, the phase shift produced
by the propagation delay between points on the two
surfaces will be a constant (-2rr_/?_ i for alternative 3)
plus a term
27r
¢i = - _ (ux' + vy')
_i
2fro
= - _ (ux + vy) (49)
where _'i is the wavelength of the radiation used.
Comparing equations (49) with (38) we see that
2ff(l
- (50)
so that the effective focal length equation (50) can now
be written
frXfl hi _ frv
F ......... (51)
oc _r o rico
where v is the velocity of propagation of the waves used.
We now wish to point out a fundamental limitation
of all radiative imaging processes. If fi is obtained by
heterodyning fr, then fi = fr - fo where fo is a constant
frequency. Then equation (51) becomes
F = (52)
teA-re
As fr varies from one end of the IF band to the other,
the fractional variation in fr-re is greater and the
effective focal length F changes with frequency• Low IF
frequencies are imaged with more magnification than
high IF frequencies. Thus, a wide-band source will be
imaged not as a point but as a radial line whose intensity
prof'fle is the power spectrum (versus wavelength)of the
source. This may have its uses, but it does not produce
good images• By analogy with a similar defect in lenses,
we shall call this phenomenon lateral chromatic aberration.
There appears to be no way to avoid lateral
chromatic aberration other than to make .to zero. This
means doing the radiative imaging at the original RF
frequency.
We then have yet another problem: Unless adequate
shielding is provided, the electrical signals generated for
imaging could be picked up by the antennas, thereby
producing serious feedback. Unless we are careful we
might end up with the world's most expensive oscillator.
Because the antennas need not be aimed at the imager
and because the array is completely dephased for any
nearby signal, we are really concerned only with the far
out side lobe response of the nearest elements. Hopefully,
this can be 20 dB or more below that of an
isotropic antenna. Nevertheless, careful shielding is
essential.
If we are forced by the feedback problem to use a
frequency offset, some chromatic aberration will remain.
To be effective in suppressing feedback, fo must be at
least equal to the system bandwidth B. lff c is the center
frequency of the RF band and .to = -B (that is, we
actually use an upward offset), and if we let x = Bffo
then from equation (52) the fractional change in F is
144