Phase Shift Keying
•
•
Transmit information by modulating the phase of a sine wave
Binary Phase Shift Keying (BPSK) - 180° Phase Shift – Cosine Channel Only
Bit0 (T seconds)
•
Bit1 (T seconds)
Quaternary Phase Shift Keying (QPSK) - 180° Phase Shift – Sine and Cosine
Channels.
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Receiver
Block diagram for a BPSK receiver:
s(t)
x
+
n(t)
•
•
•
Iin(t)
Iout(t)
t=(n+1)T
+
-
y(t)
(Local Oscillator)
Assume Coherence i.e. Local Oscillator is synchronized to s(t)
n(t) is White Gaussian Noise (WGN),
Power Spectral Density N 0 (W/Hz).
At the Integrator Input:
€
•
At the Integrator Output:
•
So the Integrator Output at time T,
Standard Deviation (
, is
) for the Noise?
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+ Noise. What is the
,
2
σout
≡ Variance = E(N 2 ) − E 2 (N) = E(N 2 )
since N is zero mean.
⎛ ⎡ T
⎤ 2 ⎞
2
σout = E ⎜⎜ ⎢ ∫ n ( t ) cos(2πf 0 t)dt ⎥ ⎟⎟
⎦ ⎠
⎝ ⎣ 0
⎛ T T
⎞
= E ⎜ ∫ ∫ n ( t ) n ( u) cos(2πf 0 t)cos(2πf 0 u)dudt ⎟
⎝ 0 0
⎠
T T
=
∫ ∫ E [n(t)n(u)] cos(2πf t)cos(2πf u)dudt
0
0
T T
=
=
∫∫
0
0
N0
cos2 (2πf 0 t)dt
2
0
T
=
∫
0
=
N0
δ (t − u)cos(2πf 0 t)cos(2πf 0 u)dudt
2
T
∫
0
0
N0
(1+ cos(4πf 0t))dt
4
N 0T
4
2
σout = σout
=
1
N 0T
2
€
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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:
f 0 = 10 Hz, N 0 = 158 µW /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)
Pr(0/1) | Pr(1/0)
•
o
o
o
o
•
•
•
•
⎛ AT ⎞
⎛
⎞
AT
⎛
⎜ 2 ⎟ 1
⎜
⎟ 1
1
T ⎞
2
Pr (Error) = erfc⎜
=
erfc
=
erfc
A
⎜
⎟
⎟
⎜ 1
⎟
2
2
2
2N
σ
2
⎝
⎠
0
out
⎜
⎟
⎜
N 0T 2 ⎟
⎝
⎠
⎝ 2
⎠
•
€
•
⎛ E ⎞
1
b
Pr (Error) = erfc⎜
⎟
2
N
⎝
0 ⎠
€
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Eb
where
is the Energy per Bit and
N0
N 0 is the Power Spectral Density of the AWGN noise of the channel.
The following graph is a plot of Pr (Error) vs
For Example,
€
€
N 0 = 0.198 W /Hz, A = 1, T = 1, Fs = 100
A 2T
2
⎛ E ⎞
⎛ 0.5 ⎞
10log⎜ b ⎟ = 10log⎜
⎟ = 4 dB
⎝ 0.198 ⎠
⎝ N 0 ⎠
Eb =
⎛ E ⎞ 1
1
b
Pr (Error) = erfc⎜
⎟ = erfc(1.58) = 0.0125
2
⎝ N 0 ⎠ 2
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
o
x
s(t)
+
Icos_in(t)
Icos_out(t)
t=(n+1)T
+
-
(Local Oscillator)
n(t)
x
Isin_in(t)
Isin_out(t)
t=(n+1)T
(Local Oscillator)
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+
-
Cosine Channel :
Icos_ in ( t ) = [ s( t ) + n ( t )] cos(2πf 0 t)
T
Icos_ out ( t ) =
∫ [± Acos(2πf t) +
0
± Asin(2πf 0 t) + n ( t )] cos(2πf 0 t)dt
0
T
=
T
T
∫ ± Acos (2πf t)dt + ∫ ± Asin(2πf t)cos(2πf t)dt + ∫ n(t) cos(2πf t)dt
2
0
0
0
=±
0
0
0
0
AT
+0+N
2
1
N 0T
2
Sine Channel :
σ cos_ out =
Isin_ in ( t ) = [ s( t ) + n ( t )] sin(2πf 0 t)
T
Isin_ out ( t ) =
∫ [± Acos(2πf t) +
0
± Asin(2πf 0 t) + n ( t )] sin(2πf 0 t)dt
0
T
=
T
2
0
0
=±
σ sin_ out =
€
T
∫ ± Asin (2πf t)dt + ∫ ± Asin(2πf t)cos(2πf t)dt + ∫ n(t ) sin(2πf t)dt
0
0
0
0
0
AT
+0+N
2
1
N 0T
2
References:
• “Principles of Communications, Systems, Modulation, and Noise”, Ziemer and Tranter, pp.
343, 455.
• Dr. Morley’s EE437 Lecture Notes, Fall 2003.
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