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Technology

Carrier Phase Modulation

Module by: Behnaam Aazhang

Summary: 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∈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)

Example 1

Binary s 1 t

s 1 t

or s 2 t

s 2 t

Representing the Signals

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)

Demodulation and Detection

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 α

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:

Example 2

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 θ

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This work is licensed by Behnaam Aazhang. See the Creative Commons License about permission to reuse this material.

Last edited by Elizabeth Gregory on Mar 17, 2004 5:25 pm US/Central.

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