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

andobtain
V = 2 (Ps [! + _(Ar)] +Pn) (M19)
ff
_i
ff
_< Ar _< -- (M 17)
c°i
If n channels
are added,
n(n - 1)
as shown in Figure M-2. (The negative delay shown can
be avoided by adding delay to receiver B.)
/// ojc ///
1/ -_--i-i r ////
, l,I/
t/t _ //
/ ,/J//"
, // / //1 -W
P = " Ps +-- 1 _(Ari + Pn
n
i=1
(M20)
since there are n(n - 1)/2 cross-product terms between
pairs of antennas, each having (possibly) different delay
errors.
Let us now define ¢s as the average value of qJ(Ari).
Then equation (M20) becomes
n-I -
P = niPs(1 +-- _)+Pnl (M21)
2
With no delay errors
_ = 1 and
T
/
Figure M-2. Delay required to phase array with no
local oscillator phase shift.
If we assume the arguments of the trigonometric
functions have been set equal in this fashion, and the
signals have been added, the power will be
P- Po = n -- Ps + (M22)
The loss in signal power (and SIN ratio) caused by the
delay errors is therefore
2 +(n - I)_-
L = 10log (M23)
n+l
P =-_ { [Sp(t - ra) + Sp(t - r) + ap(t - ra) + bp(t)] 2
which for large n becomes simply
+ [Sq(t - ra) + Sq(t - r) + aq(t - ra) + bq(t)] 2}
L _ 10log _- (M24)
where
=-- 2
2
+ 2Rp(Ar) + ap 2 + bp 2
-]
+ 2Sq 2 + 2Rq(Ar) +a q 2 + bq 2 (M18)
The delay errors of the individual channels will tend
to be uniformly distributed over the range -(l/2fi) to
(1/2fi). Thus, the delay differences between pairs of
channels will have the triangular distribution
p(Ar) = fi(l -filArl), IArl < -
Īi
1
R(At) -= s(t)s(t + At)
1
= 0 )Arl_> -- (M25)
Ji
If we let _b(Ar) -= R(Ar)/R(O) be the normalized
autocorrelation of the received signal, then for the
addition of two channels, we find
If the source is "white," _ will be entirely determined
by the receiver selectivity characteristic. If F(_)
223