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Phase Shift Keying
• Transmit information by modulating the phase of a sine wave • Binary Phase Shift Keying (BPSK) - 180° Phase Shift – Cosine Channel Only
• Quaternary Phase Shift Keying (QPSK) - 180° Phase Shift – Sine and Cosine Channels.
Bit0 (T seconds) Bit1 (T seconds)
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Receiver Block diagram for a BPSK receiver: s(t) Iin(t) Iout(t) y(t) n(t) (Local Oscillator)
• Assume Coherence i.e. Local Oscillator is synchronized to s(t) • n(t) is White Gaussian Noise (WGN), ,
Power Spectral Density
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N0 (W/Hz). • At the Integrator Input:
• At the Integrator Output:
• So the Integrator Output at time T, , is + Noise. What is the
Standard Deviation ( ) for the Noise?
+ x
t=(n+1)T
+ -
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σout2 ≡Variance = E(N 2) − E 2(N) = E(N 2)
since N is zero mean.
σout2 = E n t( )cos(2πf0t)dt
0
T
∫⎡
⎣ ⎢
⎤
⎦ ⎥
2⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
= E n t( )n u( )cos(2πf0t)cos(2πf0u)dudt0
T
∫0
T
∫⎛
⎝ ⎜
⎞
⎠ ⎟
= E n t( )n u( )[ ]cos(2πf0t)cos(2πf0u)dudt0
T
∫0
T
∫
=N0
2δ(t − u)cos(2πf0t)cos(2πf0u)dudt
0
T
∫0
T
∫
=N0
2cos2(2πf0t)dt
0
T
∫
=N0
41+ cos(4πf0t)( )dt
0
T
∫
=N0T
4
σout = σout2 =
12
N0T
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Waveforms for (all examples have Fs = 1000 Hz, BW = 500 Hz):
Zoom In of Final Value of Integrator
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Waveforms for:
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f0 =10 Hz, N0 =158µW /Hz, A = +1 V, and T =1 sec
Zoom In of Final Value of Integrator
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Histogram of the final values of Iout(T)
•
o
o
o
o
•
•
•
•
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Pr(Error) =12erfc
AT2
σout 2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=12erfc
AT2
12
N0T 2
⎛
⎝
⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟
=12erfc A T
2N0
⎛
⎝ ⎜
⎞
⎠ ⎟
•
•
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Pr(Error) =12erfc Eb
N0
⎛
⎝ ⎜
⎞
⎠ ⎟
Pr(0/1) | Pr(1/0)
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The following graph is a plot of
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Pr(Error) vs Eb
N0 where is the Energy per Bit and
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N0is the Power Spectral Density of the AWGN noise of the channel. For Example,
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N0 = 0.198W /Hz, A =1, T =1, Fs =100
Eb =A2T2
10log Eb
N0
⎛
⎝ ⎜
⎞
⎠ ⎟ =10log 0.5
0.198⎛
⎝ ⎜
⎞
⎠ ⎟ = 4 dB
Pr(Error) =12erfc Eb
N0
⎛
⎝ ⎜
⎞
⎠ ⎟ =
12erfc(1.58) = 0.0125
This is a point on the curve below for Coherent PSK.
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Quaternary Phase Shift Keying
• Use the Sine and Cosine channels simultaneously.
• Transmit twice as much information in the same bandwidth! • Also uses twice the power
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Icos_in(t) Icos_out(t) s(t) (Local Oscillator) n(t) Isin_in(t) Isin_out(t) (Local Oscillator)
+
x
t=(n+1)T
+ -
t=(n+1)T
+ -
x
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Cosine Channel :
Icos_ in t( ) = s t( ) + n t( )[ ]cos(2πf0t)
Icos_ out t( ) = ±Acos(2πf0t) + ± Asin(2πf0t) + n t( )[ ]cos(2πf0t)dt0
T
∫
= ±Acos2(2πf0t)0
T
∫ dt + ±Asin(2πf0t)cos(2πf0t)dt + n t( )cos(2πf0t)dt0
T
∫0
T
∫
= ±AT2
+ 0 + N
σ cos_ out =12
N0T
Sine Channel :
Isin_ in t( ) = s t( ) + n t( )[ ]sin(2πf0t)
Isin_ out t( ) = ±Acos(2πf0t) + ± Asin(2πf0t) + n t( )[ ]sin(2πf0t)dt0
T
∫
= ±Asin2(2πf0t)0
T
∫ dt + ±Asin(2πf0t)cos(2πf0t)dt + n t( )sin(2πf0t)dt0
T
∫0
T
∫
= ±AT2
+ 0 + N
σ sin_ out =12
N0T
References: • “Principles of Communications, Systems, Modulation, and Noise”, Ziemer and Tranter, pp.
343, 455. • Dr. Morley’s EE437 Lecture Notes, Fall 2003.