Transcript Modulation, Demodulation and Coding Course
EE 3220: Digital Communication
Lec-8: Error performance of bandpass modulation Dr. Hassan Yousif Ahmed Department of Electrical Engineering College of Engineering at Wadi Aldwasser Slman bin Abdulaziz University 1 Dr Hassan Yousif
Last time we talked about:
• Some bandpass modulation schemes – M-PAM, M-PSK, M-FSK, M-QAM • How to perform coherent and non-coherent detection Dr Hassan Yousif 2
Example of two dim. modulation
16QAM 8PSK s
3 2 (
t
)
“011”
2 (
t
)
“0000” “0001” “0011” “0010” s
1
s
2 3
s
3
s
4
“010” s
4
E s
“001” s
2
“110” s
5
s “000”
1 1 (
t
)
“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
s
13
s
14 -3
s
15
s “0100” “0101” “0111” “0110”
16 1 (
t
)
QPSK “111” s
6
s
2
“01”
2 (
t
)
s
7
E s
s
8
“100” “00” s
1 1 (
t
) Dr Hassan Yousif
s
3
“11” “10” s
4 3
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?
Dr Hassan Yousif 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 non coherent) 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 r N
1
r r Decision Circuits
Compare z with threshold. Dr Hassan Yousif 5
Error probability …
– – The matched filters output (observation vector= ) is the detector input and the decision variable is a function of , i.e. •
r
For MPAM, MQAM and MFSK with coherent detection
z
r
• For MPSK with coherent detection
z
r
(
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 • inside i
i
| sent) i
i Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type.
Dr Hassan Yousif 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 i.i.d 1 ,
n
2
N
) Gaussian random variables with zero-mean and variance – (
n
) 1 0
n
0
r
r r
independent Gaussian random variables. Its pdf is
p
r
|
i
1 0
s
i
0
r N
) Dr Hassan Yousif 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
P B
s s
b N
0
P
N
0
s
2
“1”
E b
2 (
t
) Dr Hassan Yousif 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
T
0
r
12 +
r
(
t
)
T
2
t
) 2
z
Decision rule:
( ) ( )
m m
T
0
r
21 2 -
T
2
t
)
T
0
r
22 2
z
2 2 2 Dr Hassan Yousif 9
Error probability – cont’d
• Non-coherent detection of BFSK …
B
1 2 1 2
z
) 2 1 2
z
2 | ) 1 1 ) 2
z z
s
2 2 0
z
| ,
p
2 2 2
s
2 2 | 1 2 | 2 ) 2 2
P B
1
E
exp 2
b N
0 Rayleigh pdf Rician pdf • Similarly, non-coherent detection of DBPSK
P B
2
E b N
0 Dr Hassan Yousif 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) Dr Hassan Yousif 11
Error probability ….
• • Coherent detection of M-PAM ….
n
1 1
s
m
the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence:
P
r E m g
g
g
( )
M
M
M s
M
M
2
b
3
E g
(
E
Q M
log 2
b
0 M M
g
( 2 0 Gaussian pdf with zero mean and variance
N
0 / 2 Dr Hassan Yousif 12
r
(
t
)
Error probability …
• Coherent detection of M-QAM
s
1
s
2 2 (
t
)
s
3 1 (
t
)
T
0
r
1
16-QAM ML detector s
4
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
Dr Hassan Yousif 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
b
0 Dr Hassan Yousif Average probability of
M
14
Error probability …
•
r
(
t
) Coherent detection of MPSK 1 (
t
) 2 (
t
) 0
T
8-PSK
r
1
r
arctan
r
1 2 ˆ
“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 | Choose smallest
T
0
r
2 Decision variable
z
r
Dr Hassan Yousif 15
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:
P E P C
M
where
s
N
0 0 2 It can be shown that (
E
) 2 2
s
sin
N
0
M
or (
E
) 2 2 0
b N
sin Dr Hassan Yousif 16
Error probability …
• Coherent detection of M-FSK
r
(
t
) 1 (
t
)
M
(
t
)
T
0
T
0
r
1
r M
r M r
1
r r ML detector:
Choose the largest element in the observed vector Dr Hassan Yousif 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 Dr Hassan Yousif 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
P k P
Dr Hassan Yousif 19
P E
Probability of symbol error for binary modulation
•
Note!
“The same average symbol energy for different sizes of signal space”
E b
/
N
0 dB Dr Hassan Yousif 20
P E
Probability of symbol error for M-PSK
•
Note!
“The same average symbol energy for different sizes of signal space”
E b
/
N
0 dB Dr Hassan Yousif 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 Dr Hassan Yousif 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 Dr Hassan Yousif 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 Dr Hassan Yousif 24
Example of samples of matched filter output for some bandpass modulation schemes Dr Hassan Yousif 25