The phase lock loop provides estimates of the phase of the
incoming modulated signal. A phase ambiguity of exactly
π is a common occurance in many phase
lock loop (PLL) implementations.

Therefore it is possible that,
θ
^
=θ+π
θ
^
θ
without the knowledge of the receiver. Even if there is no noise, if
b=1
b
1
then
b
^
=0
b
^
0
and if
b=0
b
0
then
b
^
=1
b
^
1
.

In the presence of noise, an incorrect decision due to noise may
results in a correct final desicion (in binary case, when there
is π phase ambiguity with the
probability:

P
e
¯
=1−Q2
E
s
N
0
P
e
¯
1
Q
2
E
s
N
0

(1)
Consider a stream of bits
a
n
∈01
a
n
0
1
and BPSK modulated signal

∑n-1
a
n
A
P
T
t−nTcos2π
f
c
t+θ
n
n
-1
a
n
A
P
T
t
n
T
2
f
c
t
θ

(2)
In differential PSK, the transmitted bits are first encoded
b
n
=
a
n
⊕
b
n
−
1
b
n
a
n
b
n
−
1
with initial symbol (e.g.
b
0
b
0
) chosen without loss of generality to be either 0 or 1.

Transmitted DPSK signals

∑n-1
b
n
A
P
T
t−nTcos2π
f
c
t+θ
n
n
-1
b
n
A
P
T
t
n
T
2
f
c
t
θ

(3)
The decoder can be constructed as

b
n
−
1
⊕
b
n
=
b
n
−
1
⊕
a
n
⊕
b
n
−
1
=0⊕
a
n
=
a
n
b
n
−
1
b
n
b
n
−
1
a
n
b
n
−
1
0
a
n
a
n

(4)
If two consecutive bits are detected correctly, if
b
^
n
=
b
n
b
^
n
b
n
and
b
^
n
−
1
=
b
n
−
1
b
^
n
−
1
b
n
−
1
then

a
^
n
=
b
^
n
⊕
b
^
n
−
1
=
b
n
⊕
b
n
−
1
=
a
n
⊕
b
n
−
1
⊕
b
n
−
1
=
a
n
a
^
n
b
^
n
b
^
n
−
1
b
n
b
n
−
1
a
n
b
n
−
1
b
n
−
1
a
n

(5)
if

b
^
n
=
b
n
⊕1
b
^
n
b
n
1
and

b
^
n
−
1
=
b
n
−
1
⊕1
b
^
n
−
1
b
n
−
1
1
. That is, two consecutive bits are detected
incorrectly. Then,

a
^
n
=
b
^
n
⊕
b
^
n
−
1
=
b
n
⊕1⊕
b
n
−
1
⊕1=
b
n
⊕
b
n
−
1
⊕1⊕1=
b
n
⊕
b
n
−
1
⊕0=
b
n
⊕
b
n
−
1
=
a
n
a
^
n
b
^
n
b
^
n
−
1
b
n
1
b
n
−
1
1
b
n
b
n
−
1
1
1
b
n
b
n
−
1
0
b
n
b
n
−
1
a
n

(6)
If

b
^
n
=
b
n
⊕1
b
^
n
b
n
1
and

b
^
n
−
1
=
b
n
−
1
b
^
n
−
1
b
n
−
1
, that is, one of two consecutive bits is detected in
error. In this case there will be an error and the probability
of that error for DPSK is

P
¯
e
=Pr
a
^
n
≠
a
n
=Pr
b
^
n
=
b
n
b
^
n
−
1
≠
b
n
−
1
+Pr
b
^
n
≠
b
n
b
^
n
−
1
=
b
n
−
1
=2Q2
E
s
N
0
1−Q2
E
s
N
0
≃2Q2
E
s
N
0
P
¯
e
a
^
n
a
n
b
^
n
b
n
b
^
n
−
1
b
n
−
1
b
^
n
b
n
b
^
n
−
1
b
n
−
1
2
Q
2
E
s
N
0
1
Q
2
E
s
N
0
2
Q
2
E
s
N
0

(7)
This approximation holds if

QQ is
small.