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A description of phase shift keying in which information is conveyed through the phase of the modulated signal.

Phase shift keying (psk)

Information is impressed on the phase of the carrier. As data changes from symbol period to symbol period, the phase shifts.

m m 1 2 M s m t A P T t 2 f c t 2 m 1 M

Representing the signals

An orthonormal basis to represent the signals is

1 t 1 E s A P T t 2 f c t
2 t -1 E s A P T t 2 f c t

The signal

S m t A P T t 2 f c t 2 m 1 M
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

The signal energy

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
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
(Note that in the above equation, the integral in the last step before the aproximation is very small.)Therefore,
1 t 2 T P T t 2 f c t
2 t 2 T P T t 2 f c t

In general,

m m 1 2 M s m t A P T t 2 f c t 2 m 1 M
and 1 t
1 t 2 T P T t 2 f c t
2 t 2 T P T t 2 f c t
s m E s 2 m 1 M E s 2 m 1 M

Demodulation and detection

r t s m t N t for some m 1 2 M

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 t 2 f c t 2 m 1 M N t

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 Q A T N 0
The receiverwhere
r t A P T t 2 f c t N t
The statistics
r 1 t 0 T r t 2 f c t t 0 T A 2 f c t 2 f c t t 0 T 2 f c t N t
r 1 A 2 t 0 T 4 f c t 1
r 1 A 2 T t 0 T A 2 4 f c t 1 A T 2 1
where 1 t 0 T N t c t is zero mean Gaussian with variance 2 N 0 T 4 .

Therefore,

P e Q 2 A T 2 2 2 N 0 T 4 Q A T N 0
which is not a function of and depends strongly on phase accuracy.
P e Q 2 E s N 0
The above result implies that the amplitude of the local oscillator in the correlator structure does not play a role inthe performance of the correlation receiver. However, the accuracy of the phase does indeed play a major role. Thispoint can be seen in the following example:

x t -1 i A 2 f c t 2 f c
x t -1 i A 2 f c t 2 f c 2 f c

Local oscillator should match to phase .

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Source:  OpenStax, Digital communication systems. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10134/1.3
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