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

The quantity gtPt is often called the effective radiated
power Perf" Thus an alternate form of equation (4) is
gtPtAr
PeffAr
Pr = 4rrR 2 4rrR 2 (5)
Making use of equation (1), we may write equation (4)
either in the form given by Friis (ref. 1):
NOISE IN COHERENT RECEIVERS
The noise level of any ideal coherent amplifier of
electromagnetic waves is:
where
¢ = hv + (8)
T- 1
Pr
Pt
AtAr
X2R2
(6)
= noise power per Hertz
h = Planck's constant
k = Boltzmarm's constant
or in terms of the gains of both
antennas
v = frequency
T = temperature of source or field of view
The appearance of X2 in the denominator of equation
(6), shows that for J_txed antenna areas, the transmission
increases as the square of the operating frequency.
However, larger antenna areas are more readily realized
at lower frequencies. As one scales all linear dimensions
of a given structure by a factor k, the deflections of the
structure under its own weight vary as k 2. Since we
can tolerate deflections proportional to ?_,we can let k2
vary with 3., which means that both A t and A r can be
proportional to X, and the 7,2 disappears. But this is only
part of the story. Both in the optical region and by the
use of phased arrays in the microwave region, we can
build antennas having the same gain-that is, the same
diameter measured in wavelengths. In both regions the
maximum practical gain is limited by the directivity
becoming so high that atmospheric turbulence or pointing
errors prevent us from reliably keeping the beam on
target. Typically this difficulty occurs in both spectral
ranges when 20,/2 is on the order of 1 arc sec and
go _ 101 1. Thus equation (7)is a more appropriate form
of the transmission equation to use when we are considering
the ultimate realizable performance. The factor _
now appears in the numerator and makes /ow frequencies
appear more attractive. However, we are paying
for the increased performance in the cost of the
increased antenna areas needed to realize a given gain.
When the transmitting system is not under our
control, as is the case when we are searching for radio
leakage or beacons radiated by extraterrestrial life,
equations (4) and (5) are most appropriate and we shall
use this form of the transmission law in developing our
range
equations.
(7)
The first term in the parentheses is black-body
radiation in a single propagation mode while the second
term arises from spontaneous emission in the amplifier.
At high frequencies such that hv/kT >> 1 the blackbody
radiation or thermal noise term disappears and
_ hr. At low frequencies where hv/kT