Digital Communications I:
Modulation and Coding Course
Spring - 2013
Jeffrey N. Denenberg
Lecture 4b: Detection of M-ary Bandpass Signals

Last time we talked about:

Some bandpass modulation schemes

M-PAM, M-PSK, M-FSK, M-QAM

How to perform coherent and noncoherent detection
Lecture 8 2

Example of two dim. modulation s
3

2
( t )
“011”
16QAM

2
( t )
“0000” “0001” “0011” “0010” s
1 s
2
3 s
3 s
4
8PSK
“010” s
4
E s
“1000” “1001” “1011” “1010” s
5 s
6 s
7 s
1
8
-3 -1 1 3 s
9 s
10
-1 s
11 s
“1100” “1101” “1111” “1110”
12

1
( t )
QPSK
“110” s
5
“111” s
6 s
2
“01”
2
( t s
)
7
“00” s
1 s
13 s
14
-3 s
15 s
“0100” “0101” “0111” “0110”
16
“001” s
2 s
“000”
1

1
( t ) s
8
“100”
E s

1
( t )
Lecture 8 s
3 “11”
3
“10” s
4

Today, we are going to talk about:

How to calculate the average probability of symbol error for different modulation schemes that we studied?

How to compare different modulation schemes based on their error performances?
Lecture 8 4

Error probability of bandpass modulation

Before evaluating the error probability, it is important to remember that:


The type of modulation and detection ( coherent or noncoherent) determines the structure of the decision circuits and hence the decision variable, denoted by z
.
The decision variable, z
, is compared with M-1 thresholds, corresponding to M decision regions for detection purposes. r ( t )

1
( t )

N
( t )
 T
0
 T
0 r
1 r
N




 r
N
 r
1





 r r
Decision
Circuits
Compare z with threshold.
Lecture 8 5

Error probability …


The matched filters output (observation vector= ) is the detector input and the decision variable is a function of , i.e. r z
 f ( r )



For MPAM, MQAM and MFSK with coherent detection z
For MPSK with coherent detection z
  r
For non-coherent detection (M-FSK and DPSK), z

| r |
 r
We know that for calculating the average probability of symbol error, we need to determine
 inside i i s i i
Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type.
Lecture 8 6

Error probability …

AWGN channel model:

 r
 s i
 n s i
 a a i 2 a ) iN
The elements of the noise vector are
1 2 N i.i.d Gaussian random variables with zero-mean and

( n
)
1

0

 n
N
0
 r

( r
1
, r
2
,...,
N
) are independent Gaussian random variables. Its pdf is p r
| i
1

0

 s i
0


Lecture 8 7

Error probability …

BPSK and BFSK with coherent detection:
BPSK s
1
“0”

E b

2
( t )
“1”
B
  s s
N
0
/ 2
2

1
( t )
E b s
2

1
( t )
BFSK
“0” s
1
E b s s
2
 s s s
2
“1”
E b

2
( t )
P
B
 b

N
0

P

N
0


Lecture 8 8

Error probability …

Non-coherent detection of BFSK
/ T

)
/ T

)
 T
0 r
11
 
2
Decision variable:
Difference of envelopes z
 z
1
 z
2 z
1
 r
2 
11
2 r
12 r ( t )
T

2 t )
 T
0 r
12  
2
+ z
Decision rule: if ( )
 if ( )

 T
0 r
21
 
2
-
T

2 t )
 T
0 r
22
 
2 z
2
 2  2
Lecture 8 9

Error probability – cont’d

Non-coherent detection of BFSK …
1
P
B z
1
2

2
)
2
1
2 z
2
| )
1 1 z )
2
 z z s
2 2

 
0 z | , p
2 2
|
2 s
2 2
 
|
1 2


|
2
)
2 2
P
B
1
2


E exp
2 b
N
0


Rayleigh pdf Rician pdf

Similarly, non-coherent detection of DBPSK
P
B
1 exp
2


E b
N
0


Lecture 8 10

Error probability ….

Coherent detection of M-PAM

Decision variable:
4-PAM
“00” s
1

3 E g
“01” s
2

E g
0 z
 r
1
“11” s
3
E g
“10” s
4 
1
( t )
3 E g

1
( t ) r ( t )
 T
0 r
1
ML detector
(Compare with M-1 thresholds)
Lecture 8 11

Error probability ….


Coherent detection of M-PAM ….
Error happens if the noise, , exceeds in amplitude
1 1 m one-half of the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence:
P

 r
 m g  g
 
M g
( )
M 
M

M
 
M
  g
M

E s

M

M
2
 b

3
E g

P
E
M 
M log
2
 b
0
Lecture 8
 p g n
E

M
E
Q
N
0
Gaussian pdf with zero mean and variance
N
0
/ 2
12

r ( t )
Error probability …

Coherent detection of M-QAM s
1

2
( t )
“0000” “0001” “0011” “0010” s
2 s
3 s
4

1
( t )
 T
0 r
1
16-QAM
ML detector s
5 s
6 s
7 s
8 s
9

1
( t ) s
10 s
11 s
12
“1100” “1101” “1111” “1110” s
13 s
14 s
15 s
16
“0100” “0101” “0111” “0110”
Parallel-to-serial converter

2
( t )
 T
0 r
2
ML detector
Lecture 8 13

Error probability …




Coherent detection of M-QAM …
M-QAM can be viewed as the combination of two
M

PAM modulations on I and Q branches, respectively.
No error occurs if no error is detected on either the I or the Q branch.
Considering the symmetry of the signal space and the orthogonality of the I and Q branches:
( 1

Q and I)Pr(no

 
E
)


1 log
Q
M
2
E
 b
0
Lecture 8
Average probability of
M

14

Error probability …

Coherent detection of MPSK r ( t )

1
( t )

2
( t )
8-PSK

0
T r
1 arctan r
1 r
2
 ˆ
 T
0 r
2
Lecture 8
“110” s
5 s
4
6 s
3

2
( t )
“011” s
“001”
2
E s s
“000”
1

1
( t ) s
8
“100”
“101” s
7
|
Compute
 i
  ˆ
|
Decision variable z
    r
15
Choose smallest

Error probability …




Coherent detection of MPSK …
The detector compares the phase of observation vector to M-1 thresholds.
Due to the circular symmetry of the signal space, we have:
)
E C

M
  where
 cos(
0

N

0



2
It can be shown that
(
E
) 2
2 s
N
0
 sin or (
E
) 2

2

N
0




Lecture 8 16

Error probability …

Coherent detection of M-FSK r ( t )

1
( t )

M
( t )
 T
0
 T
0 r
1 r
M




 r
M

1 r





 r r
ML detector:
Choose the largest element in the observed vector
Lecture 8 17

Error probability …

Coherent detection of M-FSK …
 The dimension of the signal space is M . An upper bound for the average symbol error probability can be obtained by using the union bound. Hence:
  or, equivalently
( )
E
  
2
 b
N
0
Lecture 8 18



Bit error probability versus symbol error probability
Number of bits per symbol k
 log
2
M
For orthogonal M-ary signaling (M-FSK)
P
B
P
E

2
2 k k

1

1

M / 2
M

1 k lim
P
B
P
E

1
2

For M-PSK, M-PAM and M-QAM
P
 k
P 1
Lecture 8 19

Probability of symbol error for binary modulation
P
E
Note!
• “The same average symbol energy for different sizes of signal space”
E b
/ N
0 dB
Lecture 8 20

Probability of symbol error for M-PSK
Note!
• “The same average symbol energy for different sizes of signal space” P
E
E b
/ N
0 dB
Lecture 8 21

Probability of symbol error for M-FSK
P
E
Note!
• “The same average symbol energy for different sizes of signal space”
E b
/ N
0 dB
Lecture 8 22

P
E
Probability of symbol error for M-PAM
Note!
• “The same average symbol energy for different sizes of signal space”
E b
/ N
0 dB
Lecture 8 23

P
E
Probability of symbol error for M-
QAM
Note!
• “The same average symbol energy for different sizes of signal space”
E b
/ N
0 dB
Lecture 8 24

Example of samples of matched filter output for some bandpass modulation schemes
Lecture 8 25