multiphase shift keying (mpsk) lecture 8
TRANSCRIPT
Lecture 8
Goals
� Be able to analyze MPSK modualtion
� Be able to analyze QAM modualtion
� Be able to quantify the tradeoff between data rate and energy.
VIII-1
Multiphase Shift Keying (MPSK)
si � t � � � 2Pcos
�
2π fct �
2πM
i
�
pT � t � 0 � t � T
� Ac i
2T
cos � 2π fct � pT � t ��� As i
2T
sin � 2π fct � pT � t �
� Ac iφ0 � t � � As iφ1 � t �
for i � 0 1 � � � M� 1, where φ0 � t � � �
2T cos � 2π fct � pT � t � and
φ1 � t � � �
2T sin � 2π f0t � pT � t � .
Ac i � � E cos �
2πiM �
As i � � E sin �
2πiM �
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Constellation
s0 � t �
s1 � t �
s2 � t �
s3 � t �
s4 � t �
s5 � t � s6 � t �
s7 � t �
φ0 � t �
φ1 � t �
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Decision Regions
φ0 � t �
φ1 � t �
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Notes on MPSK
For this modulation scheme we should use Gray coding to map bits into
signals.
M � 2 � BPSK M � 4 � QPSK
This type of modulation has the properties that all signals have the same
power thus the use of nonlinear amplifiers (class C amplifiers) affects each
signal in the same manner. Furthermore if we are restricted to two dimensions
and every signal must have the same power then this signal set minimizes the
error probability of all such signal sets.
QPSK and BPSK are special cases of this modulation.
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000
001011
010
110
111 100101
φ0 � t �
φ1 � t �
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Symbol Error Probability for MPSK
The error probability for MPSK can be determined as follows. Consider a
signal 0 transmitted where with the constellation shown above. The
probability of error given signal 0 transmitted is the probability that the noise
brings the received signal outside the region R0 where the decision is that
signal 0 was transmitted. This is given as
Pe 0 �� � Rc
0
fZ0 � H0 � z0 � H0 � fZ1 � H0 � z1 � H0 � dz0dz1
�� � Rc
0
12πσ2 exp ��
12σ2� � z0� � E � 2 � z2
1 �� dz0dz1
�� � Rc
0� E 0 �
12πσ2 exp ��
12σ2� z2
0 � z21 �� dz0dz1
� 2 �
π 1� 1M �
ψ 0 �
∞
r R ψ �
12π
r2σ2 exp ��
r2
2σ2� drdψ
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� 2 �
π 1� 1M �
ψ 0
12π �
∞
r R ψ �
r2σ2 exp ��
r2
2σ2� drdψ
� 2 �
π 1� 1M �
ψ 0
12π �
∞
u R2 ψ �� 2σ2exp �� u� dudψ
� 2
�
π 1� 1M �
ψ 0
12π
exp �� R2 � 2σ2� dψ
where Rc0 is the complement of R0 and R is the distance from the signal point
s0 to the line with slope π � M.
R � � E sin � π � M �
sin � π� � ψ � π � M � �
� � E sin � π � M �
sin � ψ � π � M �
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Thus
R2
2σ2
� E sin2 � π � M �
N0 sin2 � ψ � π � M �
In the derivation of the error probability the last line follows from a change of
variables where the point z0 z1 is mapped to an angle φ from the horizontal
axis with reference point s0 and a magnitude r from the point s0. The symbol
error probability is thus
Pe s � 2 �
π 1� 1M �
ψ 0
12π
exp ��E sin2 � π � M �
N0 sin2 � ψ � π � M �� dψ
Eb � E � log2 � M �
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φ0 � t �
φ1 � t �
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π � M ψR
� E
φ0 � t �
φ1 � t �
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Constellation (Eb � N0 � 0dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 2dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 4dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 6dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 8dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
VIII-16
Constellation (Eb � N0 � 10dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
VIII-17
Constellation (Eb � N0 � 12dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 14dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 16dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure
Pha
se
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Constellation (Eb � N0 � 20dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure−
Pha
se
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Constellation (Eb � N0 � 30dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure−
Pha
se
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Constellation (Eb � N0 � 50dB)
−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
In−phase
Qua
drat
ure−
Pha
se
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Symbol Error Probability
2420161284010 -5
10 -4
10 -3
10 -2
10 -1
10 0
Perform ance of M PSK M odulation
Eb/N 0 (dB)
Symbol Error Probability
M = 32M = 2 M = 4 M = 8 M = 16
Figure 39: Symbol Error Probability for MPSK Signaling
VIII-24
Bit Error Probability
2420161284010 -5
10 -4
10 -3
10 -2
10 -1
10 0
Bit Error Rate for M PSK
Eb/N 0 (dB)
Bit Error Probability
M = 2,4 M = 8 M = 16 M = 32
Figure 40: Bit Error Probability for MPSK Signaling (with Gray coding)
VIII-25
M-ary Pulse Amplitude Modulation (PAM)
si � t � � Ai φ0 � t � 0 � t � T
whereAi � � 2i � 1� M � A i � 0 1 � � � M� 1
Ei � A2i
φ0 � t �
φ1 � t �
� 3A � A A 3A
s0 � t � s1 � t � s2 � t � s3 � t �
E � 1M
M� 1
∑i 0
Ei �
A2
M
M� 1
∑i 0
� 2i � 1� M � 2 � M2� 13
A2
VIII-26
Error Probability
The error probability (for 4-ary) is
Pe 1 � Pe 2 � 2Q �
A
� N0 � 2
�
Pe 0 � Pe 3 � Q �
A
� N0 � 2
�
The average error probability (for 4-ary PAM) is
Pe � 14
Pe 0 �
14
Pe 1 �
14
Pe 2 �
14
Pe 3
� 32
Q �
A
� N0 � 2
� �
32
Q � �
2E5N0
�
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In general the error probability is
Pe s �
�
2 � M� 1 �
M � Q�� �
6E
� M2� 1 � N0
��
�
�
2 � M� 1 �
M � Q�� �
6Eb log2 � M �
� M2� 1 � N0
��
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Symbol Error Probability
0 5 10 15 20 25 3010
−6
10−5
10−4
10−3
10−2
10−1
100
Eb/N
0 (dB)
Pe,s
M=2 M=4 M=8 M=32M=16
Figure 41: Symbol Error Probability for MPAM Signaling
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10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
1
0 5 10 15 20 25
Eb/N0 (dB)
Pe,b
M=4 M=8 M=16
Figure 42: Bit Error Probability for MPAM Signaling
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Quadrature Amplitude Modulation
For i � 0 � � � M� 1
si � t � � Ai cos2π fct � Bi sin2π fct 0 � t � T
si � t � � Ai
T2
φ0 � t � � Bi
T2
φ1 � t � 0 � t � T
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Constellation
φ0 � t �
φ1 � t �VIII-32
Decision Regions
φ0 � t �
φ1 � t �
VIII-33
Error Probabilities for QAM
Since this is two PAM systems in quadrature. Pe 2 � 1� � 1� Pe 1 � 2 for PAM
with � M signals
The application of QAM is to bandwidth constrained channels. We can
consider as a baseline a two dimensional modulation system transmitting a
2400 symbols per second. If each symbol represents 4 bits of information then
the data rate is 9600 bits per second. So we would like to have more signals
per dimension in order to increase the data rate. However, we must try to keep
the signals as far apart from each other as possible (in oder to keep the error
rate low). So an increase of the size of the signal constellation for fixed
minimum distance would likely increase the total signal energy transmitted.
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32-ary signal sets
Consider a 32-ary QAM signal set shown below. The average energy is 20.
The minimum distance is 2 and the rate is 5 bits/dimension.
�
�
� � � �
� � � � � �
� � � � � �
� � � � � �
� � � � � �
� � � �
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
VIII-35
64-ary signal sets
Consider a 64-ary QAM signal set shown below. The average energy is 42.
�
�
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
VIII-36
Modified QAM (used in Paradyne 14.4kbit modem). This has average energy
of 40.9375.
�
�
� � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � � � �
� � � � � �
� �
��
-9 -7 -5 -3 -1 1 3 5 7 9
-9
-7
-5
-3
-1
1
3
5
7
9
VIII-37
The following hexagonal constellation has energy 35.25 but each interiorpoint now has 6 neighbors compared to the four neighbors for the rectangularstructures.
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
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�
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�
�
-7 -5 -3 -1 1 3 5 7
-7
-5
-3
-1
1
3
5
7
VIII-38
Figure 43: Constellation used in v34 28.8kbps modems (960 points)VIII-39
Capacity
Eb � N0 (dB)
R � W
-2 2 4 6 8 10 12 14 16 18 20 22
1
2
3
4
5
6
7
BPSK
QPSK
8PSK
16PSK
32PSK
VIII-40
Capacity
R � W
Eb
� N0
(dB
)
10� 3 10� 2 10� 1 100 101-202468
1012141618202224
BPSKQPSK
8PSK
16PSK
32PSK
VIII-41 VIII-42