r
t
=
s
m
t+
N
t
,
for some
m∈12…M
r
t
s
m
t
N
t
, for some
m
1
2
…
M

(14)
We must note that due to phase offset of the oscillator at the
transmitter, phase jitter or phase
changes occur because of propagation delay.

r
t
=A
P
T
tcos2π
f
c
t+2π(m−1)M+φ+
N
t
r
t
A
P
T
t
2
f
c
t
2
m
1
M
φ
N
t

(15)
For binary PSK, the modulation is antipodal, and the optimum
receiver in AWGN has average bit-error probability

P
e
=Q2
E
s
N
0
=QAT
N
0
P
e
Q
2
E
s
N
0
Q
A
T
N
0

(16)
The receiver
where

r
t
=±A
P
T
tcos2π
f
c
t+φ+
N
t
r
t
±
A
P
T
t
2
f
c
t
φ
N
t

(17)
The statistics

r
1
=∫0T
r
t
αcos2π
f
c
t+
φ
^
dt=±∫0TαAcos2π
f
c
t+φcos2π
f
c
t+
φ
^
dt+∫0Tαcos2π
f
c
t+
φ
^
N
t
dt
r
1
t
0
T
r
t
α
2
f
c
t
φ
^
±
t
0
T
α
A
2
f
c
t
φ
2
f
c
t
φ
^
t
0
T
α
2
f
c
t
φ
^
N
t

(18)
r
1
=±αA2∫0Tcos4π
f
c
t+φ+
φ
^
+cosφ−
φ
^
dt+
η
1
r
1
±
α
A
2
t
0
T
4
f
c
t
φ
φ
^
φ
φ
^
η
1

(19)
r
1
=±αA2Tcosφ−
φ
^
+∫0T±αA2cos4π
f
c
t+φ+
φ
^
dt+
η
1
±αAT2cosφ−
φ
^
+
η
1
r
1
±
α
A
2
T
φ
φ
^
t
0
T
±
α
A
2
4
f
c
t
φ
φ
^
η
1
±
α
A
T
2
φ
φ
^
η
1

(20)
where

η
1
=α∫0T
N
t
cos
ω
c
t+
φ
^
dt
η
1
α
t
0
T
N
t
ω
c
t
φ
^
is zero mean Gaussian with

variance≃α2
N
0
T4
variance
α
2
N
0
T
4
.

Therefore,

P
e
-=Q2αAT2cosφ−
φ
^
2α2
N
0
T4=Qcosφ−
φ
^
AT
N
0
P
e
Q
2
α
A
T
2
φ
φ
^
2
α
2
N
0
T
4
Q
φ
φ
^
A
T
N
0

(21)
which is not a function of

αα and depends strongly on
phase accuracy.

P
e
=Qcosφ−
φ
^
2
E
s
N
0
P
e
Q
φ
φ
^
2
E
s
N
0

(22)
The above result implies that the amplitude of the local
oscillator in the correlator structure does not play a role in
the performance of the correlation receiver. However, the
accuracy of the phase does indeed play a major role. This
point can be seen in the following example: