Transcript
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Outline
Transmitters (Chapters 3 and 4, Source Coding andModulation) (week 1 and 2)
Receivers (Chapter 5) (week 3 and 4) Received Signal Synchronization (Chapter 6) (week 5)
Channel Capacity (Chapter 7) (week 6)
Error Correction Codes (Chapter 8) (week 7 and 8)
Equalization (Bandwidth Constrained Channels) (Chapter10) (week 9)
Adaptive Equalization (Chapter 11) (week 10 and 11)
Spread Spectrum (Chapter 13) (week 12)
Fading and multi path (Chapter 14) (week 12)
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Digital Communication System:
Transmitter
Receiver
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Receivers (Chapter 5) (week 3 and 4)
Optimal Receivers
Probability of Error
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Optimal Receivers
Demodulators
Optimum Detection
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Demodulators
Correlation Demodulator
Matched filter
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Correlation Demodulator
Decomposes the
signal into
orthonormal
basis vectorcorrelation terms
These are
strongly
correlated to thesignal vector
coefficients sm
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Correlation Demodulator
Received Signal model Additive White Gaussian Noise (AWGN)
Distortion
Pattern dependant noise
Attenuation Inter symbol Interference
Crosstalk
Feedback
)()()( tntstr m!
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Additive White Gaussian Noise
(AWGN)
02
1)()(
)()()(
Nff
tntstr
ssrr
m
*!*
!
02
1)( Nfnn !*
i.e., the noise is flat in Frequency domain
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Correlation Demodulator
Consider each
demodulator
output
kmk
T
k
T
km
T
kk
ns
dttftn
dttfts
dttftrr
!
!
!
0
0
0
)()(
)()(
)()(
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Correlation Demodulator
Noise components
kmkmN
dtftfN
ddtftfntnEnnE
T
mk
T T
mkmk
{!
!
!
!
0
2
1
)()(2
1
)()()]()([)(
0
00
0 0
X
XXX
{nk} are uncorrelatedGaussian random
variables
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Correlation Demodulator
Correlator outputs
!
!
-
!
!!
!
N
Mm
N
sr
Np
snsr
N
k
mkk
Nm
mkkmkk
,,2,1
)(exp
)(
1)|(
)()(
1 0
2
2/
0
-
Tsr
Have mean = signal
For each of the M codes
Number of basis functions (=2 for QAM)
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Matched filterDemodulator
Use filters whose
impulse response is
the orthonormal
basis of signal
Can show this is
exactly equivalentto
the correlationdemodulator
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Matched filterDemodulator
We find that this
Demodulator
Maximizes the SNR
Essentially show that
any other function
thanf1() decreases
SNR as is not as wellcorrelated to
components ofr(t)
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? A
? A221
2
1 0
2
0
1 0
2
2/0
2min
)(min
)(ln
2
1max
)(exp
)(
1max)|(max
mm
N
k
mkk
N
k
mkk
N
k
mkk
Nm
sr
N
srNN
N
sr
Np
ssrr
sr
!
!
-
!
-
-
!
!
!
!
T
T
The optimal Detector
Maximum Likelihood (ML):
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? A ? A
Mm
m
m
mmmm
-,2,1
2max
2max2min
222
!
-
!
!
Isr
ssrssrr
The optimal Detector
Maximum Likelihood (ML):
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Optimal Detector
Can show that
!
!!!!
T
m
N
n
T
nm
T
n
N
n
mnnm
dttstr
dttftsdttftrsr
0
1 001
)()(
)()()()(sr
so
-
!
-
2
)()(max2
max0
m
T
mm
m dttstr IIsr
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Optimal Detector
Thus get new type of correlation demodulatorusing symbols notthe basis functions:
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Alternate Optimal rectangular QAM
Detector
Mlevel QAM = 2 x level PAM signals
PAM = Pulse Amplitude Modulation
M
tftgtf
As
tfstftgAts
c
g
gmm
m
cmm
T
T
I
I2cos)(
2)(
2
1
)(2cos)()(
!
!
!!
Mm
dMms gm
,,2,1
)12(2
1
-!
! I
g
e dd I2)(min !
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2
2
2
2
)12(2max
2
2max2
max
2
2
2
gg
m
g
m
m
g
mm
m
ddsr
Mmd
A
II
I
II
!
-
!
-
!
-
sr
srsr
The optimal PAM Detector
For PAM
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The optimal PAM Detector
ms
r
isrp 1p isr
22
2 )(min
eg
m
ddsr ! I
2
)(
min
ed
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Optimal rectangular QAM Demodulator
d= spacing of rectangular griddis gi )12(
21 ! I
T
dt0
)(-
T
dt0
)(-
v
v
1s
/
Selectsifor which
22 gd I-
tftgtf cg
TI
2cos)(2
)(1 !
tftgtf cg
TI
2sin)(2
)(2 !
Ms
im ss !1
1s/
Selectsifor which
2
2 gd I
-im ss !2
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Probability of Error for rectangular
M-ary QAM
Related to error probability of PAM
-
"
! 2
21 gmM
d
srPM
M
P
I
Accounts for endsms
r
M
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Probability of Error for rec. QAM
Assume Gaussian noise
-
|
-
!
!
-
g
0
2
0
2
2/
/
0
Q2
2erfc
2
2
22
02
N
d
N
d
dxeN
d
sr
g
g
d
Nxg
mg
I
I
II
T
2
2
gd Imsr
0
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Probability of Error for rectangular
M-ary QAM
Error probability of PAM
-
!0
2
2Q21
Nd
MMP g
M
I
M
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SNR for M-ary QAM
Related to PAM
For PAM find average energy in equallyprobable signals
M
g
M
m
g
M
m
mav
dM
MmM
d
M
I
I
II
2
1
2
2
1
)1(
6
1
)12(2
1
!
!
!
!
!
M
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SNR for M-ary QAM
Related to PAM
T
dM
TP
g
avav
I
I
2
)1(6
1!
!
M
Find average Power
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SNR for M-ary QAM
Related to PAM
? A
? AMN
dM
MN
NT
TSNR
NSNR
g
av
avbb
av
20
2
20
0
0
log)1(
6
1
log
I
I
I
I
!
!
!!
!
M
Find SNR
Then SNR per bit
(ratio of powers)
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SNR for M-ary QAM
Related to PAM
? A
? A
-
!
!
)1(log6Q21
)1(log6
2
202
MSNRM
MMP
MM
SNRNd
b
M
bgI
M
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SNR for M-ary QAM
Related to PAM
Now need to get
M-ary QAM fromPAM
M
M=16
M=8
M=4
M
=2
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SNR for M-ary QAM
Related to PAM
? A
2
)1(
log3Q2
111
)1(1
QAM
PAM
2
2
2
b
b
b
MM
SNRSNR
M
SNRM
M
M
PP
!
-
-
!
!
M
(1- probability of no QAM error)
(Assume power in each PAM)
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SNR for M-ary QAM
Related to PAMMProbability of Symbol Error for QAM
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
-6
-4-20 2 4 6 8 1
01214
16
18
20
22
24
SNR per bit (dB)
P
robabiltyofsymbolErrorPM
256
64
16
4
? A
2
2
)1(log3Q2111
-
-
!
MSNRM
MMP bM M=
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