Lecture 8 2
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
Lecture 8 3
Example of two dim. modulation
)(1 tψ
2s 1s
sE
“00”
“11”
)(2 tψ
3s 4s“10”
“01”QPSK
3s
7s
“110”)(1 tψ
4s 2ssE
“000”
)(2 tψ
6s 8s
1s5s
“001”“011”
“010”
“101”
“111” “100”
8PSK
)(1 tψ
)(2 tψ
2s1s 3s 4s“0000” “0001” “0011” “0010”
6s5s 7s 8s
10s9s 11s 12s
14s13s 15s 16s
1 3-1-3
“1000” “1001” “1011” “1010”
“1100” “1101” “1111” “1110”
“0100” “0101” “0111” “0110”
1
3
-1
-3
16QAM
Lecture 8 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 5
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.
Nr
r1
r=∫T
0
)(1 tψ
∫T
0
)(tNψ)(tr
1r
Nr
rDecision CircuitsCompare z
with threshold.
m̂
Lecture 8 6
The matched filters output (observation vector= ) is the detector input and the decision variable is a function of , i.e.
For MPAM, MQAM and MFSK with coherent detection For MPSK with coherent detection For non-coherent detection (M-FSK and DPSK),
We know that for calculating the average probability of symbol error, we need to determine
Hence, we need to know the statistics of z, which depends on the modulation scheme and the detection type.
Error probability …
)(rfz =r
rr=z
r∠=z|| r=z
sent) |C condition satisfies Pr(zsent) | Zinside lies Pr( ii ii ssr ≡
Lecture 8 7
Error probability …
AWGN channel model: The signal vector is deterministic. The elements of the noise vector are
i.i.d Gaussian random variables with zero-mean and variance . The noise vector's pdf is
The elements of the observed vector are independent Gaussian random variables. Its pdf is
),...,,( 21 iNiii aaa=s
),...,,( 21 Nrrr=r
),...,,( 21 Nnnn=n
nsr += i
2/0N
( )
−=
0
2
2/0
exp1)(NN
p N
nnn π
( )
−−=
0
2
2/0
exp1)|(NN
p iNi
srsrr π
Lecture 8 8
Error probability …
BPSK and BFSK with coherent detection:
)(1 tψ
2s1sbE
“0” “1”
bE−
)(2 tψ
bE221 =− ss
)(1 tψ
2s
1s
)(2 tψbE
bEbE221 =− ss“0”
“1”
BPSK BFSK
2
0
=
NEQP b
B
2/
2/
0
21
−=
NQPB
ss
0
=
NEQP b
B
Lecture 8 9
Error probability …
Non-coherent detection of BFSK
∫T
0
)cos(/2 1tT ω
∫T
0)(tr
11r
12r
∫T
0
∫T
0
21r
22r
Decision rule:
)cos(/2 2tT ω
)sin(/2 2tT ω
)sin(/2 1tT ω
( ) 2
( ) 2
( ) 2
( ) 2 +
-
z
0ˆ ,0)( if1ˆ ,0)( if
=<=>
mTzmTz
m̂
212
2111 rrz +=
222
2212 rrz +=
21 zzz −=
Decision variable:Difference of envelopes
Lecture 8 10
Error probability – cont’d
Non-coherent detection of BFSK …
Similarly, non-coherent detection of DBPSK
[ ]
∫ ∫∫∞ ∞∞
=>=
>=>=
>+>=
0 2221210 2222221
2221221
112221
)|()|()|(),|Pr(
),|Pr()|Pr(
)|Pr(21)|Pr(
21
2
dzzpdzzpdzzpzzz
zzzEzz
zzzzP
z
B
ssss
ss
ss
−=
02exp
21
NEP b
B
−=
0
exp21
NEP b
B
Rayleigh pdf Rician pdf
Lecture 8 11
Coherent detection of M-PAM Decision variable:
∫T
0
)(1 tψ
ML detector(Compare with M-1 thresholds))(tr
1rm̂
Error probability ….
)(1 tψ2s1s0gE3−
“00” “01”4s3s
“11” “10”
gE− gE gE34-PAM
1rz =
Lecture 8 12
Error probability ….
Coherent detection of M-PAM …. Error happens if the noise, , exceeds in amplitude
one-half of the distance between adjacent symbols. For symbols on the border, error can happen only in one direction. Hence:
( )( ) ( )gMMege
gmme
ErnPErnP
MmErnP
−<−==>−==
−<<>−==
ssss
ss
111111
11
Pr)( and Pr)(
;12for ||||Pr)(
−
−=0
22
1log6)1(2)(
NE
MMQ
MMMP b
E
gbs EMEME3
)1()(log2
2−==
mrn s−= 11
Gaussian pdf with zero mean and variance 2/0N
( ) ( ) ( )
( )
−=−=>−=
−<+>+>−==
∫
∑∞
=
01
1111
2)1(2)()1(2Pr)1(2
PrM1 Pr
M1||Pr2)(1)(
1 NE
QMMdnnp
MMEn
MM
EnEnEnMMP
MMP
g
E ng
ggg
M
mmeE
g
s
Lecture 8 13
Error probability …
Coherent detection of M-QAM
∫T
0
)(1 tψML detector1r
∫T
0
)(2 tψML detector
)(tr
2r
m̂Parallel-to-serialconverter
s) threshold1 with (Compare −M
s) threshold1 with (Compare −M
)(1 tψ
)(2 tψ
2s1s 3s 4s“0000” “0001” “0011” “0010”
6s5s 7s 8s
10s9s 11s 12s
14s13s 15s 16s
“1000” “1001” “1011” “1010”
“1100” “1101” “1111” “1110”
“0100” “0101” “0111” “0110”
16-QAM
Lecture 8 14
Error probability …
Coherent detection of M-QAM … M-QAM can be viewed as the combination of two 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:
PAM−M
branches) Q and Ion detectederror noPr(1)(1)( −=−= MPMP CE
( )( )22 1I)on error Pr(no
Q)on error I)Pr(noon error Pr(nobranches) Q and Ion detectederror noPr(
MPE−==
=
−
−=
0
2
1log3114)(
NE
MMQ
MMP b
E Average probability of symbol error for PAM−M
Lecture 8 15
Error probability …
Coherent detection of MPSK
Compute Choose smallest2
1arctanrr φˆ
|ˆ| φφ −i
∫T
0
)(1 tψ
∫T
0
)(2 tψ)(tr
1r
2r
m̂
3s
7s
“110”)(1 tψ
4s 2ssE
“000”
)(2 tψ
6s 8s
1s5s
“001”“011”“010”
“101”
“111” “100”
8-PSK
Decision variable r∠== φˆz
Lecture 8 16
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:
where
It can be shown that
φφπ
π φ dpPPM
MPMPM
Mc
M
mmcCE )(1)(1)(11)(1)(
/
/ ˆ11
∫∑ −=
−=−=−=−= ss
≈MN
EQMP sE
πsin22)(0
( )
≈MN
EMQMP bE
πsinlog22)(0
2or
2|| ;sinexp)cos(2)( 2
00ˆ
πφφφπ
φφ ≤
−≈NE
NEp ss
Lecture 8 17
Error probability …
Coherent detection of M-FSK
Mr
r1
r=∫T
0
)(1 tψ
∫T
0
)(tMψ)(tr
1r
Mr
rML detector:
Choose the largest element
in the observed vector
m̂
Lecture 8 18
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
( )
−≤
0
1)(NEQMMP s
E
( ) ( )
−≤
0
2log1)(N
EMQMMP bE
Lecture 8 19
Bit error probability versus symbol error probability
Number of bits per symbol For orthogonal M-ary signaling (M-FSK)
For M-PSK, M-PAM and M-QAM
21lim
12/
122 1
=
−=
−=
∞→
−
E
Bk
k
k
E
B
PP
MM
PP
Mk 2log=
1for < <≈ EE
B PkPP
Lecture 8 20
Probability of symbol error for binary modulation
EP
dB / 0NEb
Note!• “The same average symbol
energy for different sizes of signal space”
Lecture 8 21
Probability of symbol error for M-PSK
EP
dB / 0NEb
Note!• “The same average symbol
energy for different sizes of signal space”
Lecture 8 22
Probability of symbol error for M-FSK
EP
dB / 0NEb
Note!• “The same average symbol
energy for different sizes of signal space”
Lecture 8 23
Probability of symbol error for M-PAM
EP
dB / 0NEb
Note!• “The same average symbol
energy for different sizes of signal space”
Lecture 8 24
Probability of symbol error for M-QAM
EP
dB / 0NEb
Note!• “The same average symbol
energy for different sizes of signal space”
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