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  4. Carrier phase modulation

0.1 Carrier phase modulation

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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

Binary s 1 t or s 2 t

Representing the signals

An orthonormal basis to represent the signals is

ψ 1 t 1 E s A P T t 2 f c t
ψ 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,
ψ 1 t 2 T P T t 2 f c t
ψ 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 ψ 1 t
ψ 1 t 2 T P T t 2 f c t
ψ 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 φ 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 φ N t
The statistics
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
r 1 ± α A 2 t 0 T 4 f c t φ φ ^ φ 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
where η 1 α t 0 T N t ω c t φ ^ is zero mean Gaussian with variance α 2 N 0 T 4 .

Therefore,

P e Q 2 α A T 2 φ φ ^ 2 α 2 N 0 T 4 Q φ φ ^ A T N 0
which is not a function of α and depends strongly on phase accuracy.
P e Q φ φ ^ 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 τ θ ′
x t -1 i A 2 f c t 2 f c τ ′ 2 f c τ θ ′

Local oscillator should match to phase θ .
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Source:  OpenStax, Intergrated library system management. OpenStax CNX. Jul 29, 2009 Download for free at http://cnx.org/content/col10801/1.1
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