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

CASE I - PERFECT LO PHASING AND DELAY
MATCHING
If we set
Thus
Pn = ap 2 = aq z = bp 2 = bq 2
(M11)
$ = -Wor, mod 2rr (M5)
P = 2 [2Ps+Pn] (MI2)
then equation (M4) becomes
gb(t) = [Sp(t - r) + bp(t)] cos %(t - r)
+ [Sq(t - r) + bq(t)] sin ¢o[(t - r)
The signal to noise ratio of the combined signal is thus
twice that of the individual signals. For n signals added
this way, the improvement is n fold.
CASE I1 - NO LO PHASING
Here we set _ = 0 in equation(M4) and delay the IF
signals of (M3) by an amount ra = r + Ar to obtain
(M6)
ga(t - ra) = [Sp(t - Ta) + ap(t - ra)l cos cot(t - ra)
If we now delay ga(t) by the time r and add ga(t - r) to
equation (M6) we get for the combined signal
g(t) = [2sp(t -'r) + ap(t - r) + bp(t)l cos wt(t - r)
+ [Sq(t - ra) + aq(t - ra) ] sin col(t - ra)
(MI3)
+ [2Sq(t - r) + aq(t - r) + bq(t)] sin wt(t - r)
which represents a power
e=-
1{ 12sp(t-
2
1 { 4Sp2 2
r) + ap(t - r) + bp(t)] 2
(M7)
+ [2Sq(t - r) + aq(t - r) + bq(t)] 2} (MS)
+ap 2 +bp 2 +4Sq 2 +aq 2 +bq 2 _ (M9)
If Ps is the received signal power per receiver and Pn is
the noise power per receiver, then
I
Ps = "_ (Sp2 +sq 2) = Sp =
_
Sq
(M 1O)
J
"t
!
If we now
gb(t) = [sp(t - r) + bp(t)l cos (wit - Wcr )
set
+ [Sq(t - r) + bq(t)] sin (wit - Wcr )
(MI4)
wit a = COcr (M 15)
the arguments of the trigonometric functions will agree.
We note that if w i = w c, ra = r and the delays match.
This merely says that if there is no heterodyning or if
the IF channels are remodulated up to the original
frequency before combining them (and the delay is
introduced after this remodulation), no delay error is
introduced.
Since ordinarily wi/w c > r and hence also that Ar >> r were it
not for the fact that (MI5) need only be satisfied
modulo 2n. We thus can set
Wc
2nk
Ta =-- T -- --
w i w i
, k =0,1,2... (Ml6)
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