Transcript Lecture 4b

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
16QAM
“0000”
s1
“1000”
s5
8PSK
 2 (t )
“0001” “0011” “0010”
s2
“1001”
s6
-3
-1
s9
s10
“1100”
“1101”
3
s3
s14
“0100”
“0101”
“011”
“001”
s2
s4
Es
“000”
s1
“110”
s5
s7
s8
“111”
1
s
3
s
 1 (t )
s
s
“100”
s6
s8
“101” s 7
 2 (t )
12
11
-1
“1111” “1110”
QPSK
s13
“010”
s4
“1011” “1010”
1
s3
 2 (t )
s 2“01”
16
15
-3
“0111” “0110”
“00”
s1
Es
 1 (t )
Lecture 8
s3
“11”3
“10”
s4
 1 (t )
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.
 1 (t )

T
r1
0
r (t )
 N (t )

T
0
rN
 r1
 

 r N
Lecture 8




r
r
Decision
Circuits
Compare z
with threshold.
5
ˆ
m
Error probability …


The matched filters output (observation vector= r ) is
the detector input and the decision variable is a z  f (r)
function of r , i.e.
 For MPAM, MQAM and MFSK with coherent detection z  r
 For MPSK with coherent detection z  r
 For non-coherent detection (M-FSK and DPSK), z | r |
We know that for calculating the average probability of
symbol error, we need to determine
Pr(
r
lies
inside
|
s
sent)
Z

Pr(z
satis
con
C
|
s
s
i
i
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  si  n



(a
,a
,...,
a
) is deterministic.
The signal vector s
i
i1
i2
iN
,n
n
The elements of the noise vector n(n
1
2,...,
N) are
i.i.d Gaussian random variables with zero-mean and
variance N 0 / 2 . The noise vector's pdf is
2


n
1

p
(
n
)
 N
exp
 
n
/2
N




N
0
0


rN)
The elements of the observed vector r(r
1,r
2,...,
are independent Gaussian random variables. Its pdf is
2


r

s
1
i


p
(
r
|s
)
 N
exp

r
i
/
2
 N

N

0
0



Lecture 8
7
Error probability …

BPSK and BFSK with coherent detection:
 2 (t )
“0”
BPSK
s


s
/2
1
2


P
Q
B
 N/2
0


“1”
s1
s2
 Eb
Eb
BFSK
 1 (t )
 1 (t )
“0”
s1
Eb
s 2“1”
s1s2 2 E
b
Eb
 E

b

P
Q
B
 N
 0
2

E
b

P
Q
B
 N
 0
Lecture 8
s1s2  2E
b
8
 2 (t )
Error probability …

Non-coherent detection of BFSK
Decision variable:
Difference of envelopes
z  z1  z2
2/Tcos(

t)
1

T

T

T
r11
 2
0
z1 r
r
11 
12
2
2/Tsin(

t)
1
r (t )
r12
0
2/Tcos(

t)
2
r21
 2
 2
2
Decision rule:
+
z
-
ˆ
ifz(T
)
0
,m
1
ˆ
ifz(T
)
0
,m
0
0
2/Tsin(

t)
2
z2  r21r22
2

T
0
r22
 2
Lecture 8
9
2
ˆ
m
Error probability – cont’d

Non-coherent detection of BFSK …
1
1
P

Pr(
z

z
|
s
)

Pr(
z

z
|
s
)
B
1
2
2
2
1
1
2
2



Pr(
z

z
|
s
)

E
Pr(
z

z
|
s
,
z
)
1
2
2
1
2
2
2
p


Pr(
z

z
|
s
,
z
)
p
(
z
|
s
)
dz

(
z
|
s
)
dz
p
(
z
|
s
)
dz
1
2
2
2
2
2
2
1
2
1
2
2
2





0
0
z
2






1  E
b

P
 
B exp


2 2
N
0

Rayleigh pdf
Rician pdf
Similarly, non-coherent detection of DBPSK

1  E
b

P
 
B exp


2  N
0
Lecture 8
10
Error probability ….

Coherent detection of M-PAM

Decision variable:
z  r1
“00”
4-PAM
s1
 3 Eg
 1 (t )
r (t )
“01”

T
0
“11”
s3
s2
 Eg
0
r1
Eg
“10”
s4
 1 (t )
3 Eg
ML detector
(Compare with M-1 thresholds)
Lecture 8
11
mˆ
Error probability ….


Coherent detection of M-PAM ….
Error happens if the noise, n1 r1 sm , exceeds in amplitude
one-half of the distance between adjacent symbols. For symbols
on the border, error can happen only in one direction. Hence:



and


P
(
s
)

Pr
n

r

s

E
P
(
s
)

Pr
n

r

s


E
P
(
s
)

Pr
|
n
|

|
r

s
|

E
for
2

m

M

1
;
e
m
1
1
m
g
e
1
1
1
1 g
e
M1
1
Mg

 
 

M
1
M

2
1
1
P
(
M
)

P
(
s
)

Pr
|
n
|

E

Pr
n

E

Pr
n


E

E
em
1
g
1
g
1
g
M
M
M
M
m

1


2
E
2
(
M

1
)
2
(
M

1
)
2
(
M

1
)
g


Pr
n

E

p
(
n
)
dn

Q
1
g
n

1
E
N

g
M
M
M
0

2


(
M

1
)
E

(log
M
)
E

E
s
2
b
g
3

E
2
(
M

1
)
6
log
M
b
2


P
(
M
)

Q
E
2


M

1
N
0
M

Lecture 8
Gaussian pdf with
zero mean and variance
12
N0 / 2
Error probability …

Coherent detection
of M-QAM
 2 (t )
“0000”
“0001”
s2
s 3“0011”s 4 “0010”
“1000”
s
“1001”
s
s 7“1011”s8 “1010”
s1
5
6
16-QAM
s9
“1100”
s13
 1 (t )

T
r1
ML detector
“0100”
r (t )
Parallel-to-serial
converter

T
0
s11
“1101”
“0101”
r2
mˆ
ML detector
(Compare
with
M

1
threshold
s)
Lecture 8
13
s12
“1111” “1110”
s15
s14
(Compare
with
M

1
threshold
s)
0
 2 (t )
s10
 1 (t )
s16
“0111” “0110”
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:
P
(
M
)

1

P
(
M
)

1

Pr(
no
error
detecte
on
I
and
Q
bran
E
C
Pr(
no
error
detected
on
I
and
Q
branch

Pr(no
error
on
I)Pr
err
on
Q



2
2

Pr(no
error
on
I)

1

P
M
E


E
3
log
M
 1

b
2


P
(
M
)

4
1

Q

E



1
N
 M

0
M

Lecture 8
Average probability of
symbol error for MPAM
14
Error probability …

Coherent detection
of MPSK
 2 (t )
s 3 “011”
“010”
s4
s“001”
2
Es
8-PSK
“110”
 1 (t )
s5
“111”
 1 (t )
s8“100”
s6

T
0
r (t )
s“000”
1
 2 (t )

r1
r1 ˆ
arctan
r2
“101”s 7
Compute
Choose
smallest
| i  ˆ |
T
0
r2
Lecture 8
Decision variable
z ˆ r
15
mˆ
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:




M
/
M
1
P
(
M
)

1

P
(
M
)

1

P
(
s
)

1

P
(
s
)

1

p
(
)
d
ˆ

E
C
c
m
c
1


/
M
M
m

1
where
 




E
E
2
s
s 2


p
(
)

cos(
)
exp

sin
;||

ˆ



N
N
0
0

 2

It can be shown that
2

E



s

or
P
(
M
)

2
Q
sin

E
N 

M

 0 

Lecture 8





2
log
M
E


2
b


P
(
M
)

2
Q

E
 N sin

M


0


16
Error probability …

Coherent detection of M-FSK
 1 (t )

T
r1
0
r (t )
 M (t )

T
0
rM
 r1


 r M
Lecture 8




ML detector:
r
r
Choose
the largest element
in the observed vector
17
mˆ
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:
E

s



P
(
M
)

M

1
Q
E
N

 0
or, equivalently



log
M
E
2
b




P
(
M
)

M

1
Q
E
 N
0


Lecture 8
18
Bit error probability versus symbol error
probability
 Number of bits per symbol k log
2M
 For orthogonal M-ary signaling (M-FSK)
PB 2k1 M/ 2
 k

PE 2 1 M1
PB 1
lim 
k P
2
E

For M-PSK, M-PAM and M-QAM
P
E
P
for
P
1
B
E
k
Lecture 8
19
Probability of symbol error for binary
modulation
Note!
•
“The same average symbol
energy for different sizes of
signal space”
PE
Eb / N0 dB
Lecture 8
20
Probability of symbol error for M-PSK
Note!
•
“The same average symbol
energy for different sizes of
signal space”
PE
Eb / N0 dB
Lecture 8
21
Probability of symbol error for M-FSK
Note!
•
“The same average symbol
energy for different sizes of
signal space”
PE
Eb / N0 dB
Lecture 8
22
Probability of symbol error for M-PAM
Note!
•
“The same average symbol
energy for different sizes of
signal space”
PE
Eb / N0 dB
Lecture 8
23
Probability of symbol error for MQAM
Note!
•
“The same average symbol
energy for different sizes of
signal space”
PE
Eb / N0 dB
Lecture 8
24
Example of samples of matched filter output
for some bandpass modulation schemes
Lecture 8
25