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 �

VIII-2

Constellation

s0 � t �

s1 � t �

s2 � t �

s3 � t �

s4 � t �

s5 � t � s6 � t �

s7 � t �

φ0 � t �

φ1 � t �

VIII-3

Decision Regions

φ0 � t �

φ1 � t �

VIII-4

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.

VIII-5

000

001011

010

110

111 100101

φ0 � t �

φ1 � t �

VIII-6

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

VIII-7

� 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 �

VIII-8

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 �

VIII-9

φ0 � t �

φ1 � t �

VIII-10

π � M ψR

� E

φ0 � t �

φ1 � t �

VIII-11

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

VIII-12

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

VIII-13

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

VIII-14

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

VIII-15

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

VIII-18

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

VIII-19

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

VIII-20

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

VIII-21

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

VIII-22

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

VIII-23

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

�

VIII-27

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

��

VIII-28

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

VIII-29

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

VIII-30

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

VIII-31

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.

VIII-34

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.

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

-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