Information is impressed on the phase of the carrier. As data
changes from symbol period to symbol period, the phase shifts.
∀m,m∈12…M:
s
m
t=A
P
T
tcos2π
f
c
t+2πm-1M
m
m
1
2
…
M
s
m
t
A
P
T
t
2
f
c
t
2
m
1
M
(1)
Binary
s
1
t
s
1
t
or
s
2
t
s
2
t
An orthonormal basis to represent the signals is
ψ
1
t=1
E
s
A
P
T
tcos2π
f
c
t
ψ
1
t
1
E
s
A
P
T
t
2
f
c
t
(2)
ψ
2
t=-1
E
s
A
P
T
tsin2π
f
c
t
ψ
2
t
-1
E
s
A
P
T
t
2
f
c
t
(3)
The signal
S
m
t=A
P
T
tcos2π
f
c
t+2πm-1M
S
m
t
A
P
T
t
2
f
c
t
2
m
1
M
(4)
S
m
t=Acos2πm-1M
P
T
tcos2π
f
c
t-Asin2πm-1M
P
T
tsin2π
f
c
t
S
m
t
A
2
m
1
M
P
T
t
2
f
c
t
A
2
m
1
M
P
T
t
2
f
c
t
(5)
The signal energy
E
s
=∫-∞∞A2
P
T
2tcos22π
f
c
t+2πm-1Mdt=∫0TA212+12cos4π
f
c
t+4πm-1Mdt
E
s
t
A
2
P
T
t
2
2
f
c
t
2
m
1
M
2
t
0
T
A
2
1
2
1
2
4
f
c
t
4
m
1
M
(6)
E
s
=A2T2+12A2∫0Tcos4π
f
c
t+4πm-1Mdt≈A2T2
E
s
A
2
T
2
1
2
A
2
t
0
T
4
f
c
t
4
m
1
M
A
2
T
2
(7)
(Note that in the above equation, the integral in the last
step before the aproximation is very small.)
Therefore,
ψ
1
t=2T
P
T
tcos2π
f
c
t
ψ
1
t
2
T
P
T
t
2
f
c
t
(8)
ψ
2
t=-2T
P
T
tsin2π
f
c
t
ψ
2
t
2
T
P
T
t
2
f
c
t
(9)
In general,
∀m,m∈12…M:
s
m
t=A
P
T
tcos2π
f
c
t+2πm-1M
m
m
1
2
…
M
s
m
t
A
P
T
t
2
f
c
t
2
m
1
M
(10)
and
ψ
1
t
ψ
1
t
ψ
1
t=2T
P
T
tcos2π
f
c
t
ψ
1
t
2
T
P
T
t
2
f
c
t
(11)
ψ
2
t=2T
P
T
tsin2π
f
c
t
ψ
2
t
2
T
P
T
t
2
f
c
t
(12)
s
m
=
E
s
cos2πm-1M
E
s
sin2πm-1M
s
m
E
s
2
m
1
M
E
s
2
m
1
M
(13)
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+2πm-1M+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
=Q
2
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:
x
t
′
=-1iAcos-2π
f
c
t
′
+2π
f
c
τ
x
t
′
-1
i
A
2
f
c
t
′
2
f
c
τ
θ
′
(23)
x
t
=-1iAcos2π
f
c
t-2π
f
c
τ
′
-2π
f
c
τ+
θ
′
x
t
-1
i
A
2
f
c
t
2
f
c
τ
′
2
f
c
τ
θ
′
(24)
Local oscillator should match to phase
θθ.
