1 today, we are going to talk about: shannon limit comparison of different modulation schemes...
TRANSCRIPT
1
Today, we are going to talk about:
Shannon limit Comparison of different modulation
schemes Trade-off between modulation and
coding
2
Goals in designing a DCS
Goals: Maximizing the transmission bit rate Minimizing probability of bit error Minimizing the required power Minimizing required system bandwidth Maximizing system utilization Minimize system complexity
3
Error probability plane(example for coherent MPSK and MFSK)
[dB] / 0NEb [dB] / 0NEb
Bit
erro
r pr
obab
ility
M-PSK M-FSK
k=1,2
k=3
k=4
k=5
k=5
k=4
k=2
k=1
bandwidth-efficient power-efficient
4
Limitations in designing a DCS
Limitations: The Nyquist theoretical minimum
bandwidth requirement The Shannon-Hartley capacity theorem
(and the Shannon limit) Government regulations Technological limitations Other system requirements (e.g
satellite orbits)
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Nyquist minimum bandwidth requirement
The theoretical minimum bandwidth needed for baseband transmission of R
s symbols per
second is Rs/2 hertz.
T2
1
T2
1
T
)( fH
f t
)/sinc()( Ttth 1
0 T T2TT20
6
Shannon limit
Channel capacity: The maximum data rate at which error-free communication over the channel is performed.
Channel capacity of AWGV channel (Shannon-Hartley capacity theorem):
][bits/s1log2
N
SWC
power noise Average :[Watt]
power signal received Average :]Watt[
Bandwidth :]Hz[
0WNN
CES
W
b
7
Shannon limit … The Shannon theorem puts a limit
on the transmission data rate, not on the error probability: Theoretically possible to transmit
information at any rate , with an arbitrary small error probability by using a sufficiently complicated coding scheme
For an information rate , it is not possible to find a code that can achieve an arbitrary small error probability.
CRb bR
CRb
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Shannon limit …C/W [bits/s/Hz]
SNR [bits/s/Hz]
Practical region
Unattainableregion
9
Shannon limit …
There exists a limiting value of below which there can be no error-free communication at any information rate.
By increasing the bandwidth alone, the capacity can not be increased to any desired value.
W
C
N
E
W
C b
02 1log
WNN
CES
N
SWC
b
0
2 1log
[dB] 6.1693.0log
1
:get we,0or As
20
eN
EW
CW
b
0/ NEb
Shannon limit
10
Shannon limit …
[dB] / 0NEb
W/C [Hz/bits/s]
Practical region
Unattainableregion
-1.6 [dB]
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Bandwidth efficiency plane
R<C
Practical region
R>CUnattainable region
R/W [bits/s/Hz]
Bandwidth limited
Power limited
R=C
Shannon limit MPSKMQAMMFSK
510BP
M=2
M=4
M=8M=16
M=64
M=256
M=2M=4M=8
M=16
[dB] / 0NEb
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Power and bandwidth limited systems
Two major communication resources: Transmit power and channel bandwidth
In many communication systems, one of these resources is more precious than the other. Hence, systems can be classified as: Power-limited systems:
save power at the expense of bandwidth (for example by using coding schemes)
Bandwidth-limited systems: save bandwidth at the expense of power (for
example by using spectrally efficient modulation schemes)
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M-ary signaling
Bandwidth efficiency:
Assuming Nyquist (ideal rectangular) filtering at baseband, the required passband bandwidth is:
M-PSK and M-QAM (bandwidth-limited systems)
Bandwidth efficiency increases as M increases.
MFSK (power-limited systems)
Bandwidth efficiency decreases as M increases.
][bits/s/Hz 1log2
bs
b
WTWT
M
W
R
[Hz] /1 ss RTW
][bits/s/Hz log/ 2 MWRb
][bits/s/Hz /log/ 2 MMWRb
Lecture 13 14
Design example of uncoded systems
Design goals:1. The bit error probability at the demodulator output must
meet the system error requirement.2. The transmission bandwidth must not exceed the
available channel bandwidth.
M-arymodulator
M-arydemodulator
][symbols/s log2 M
RRs [bits/s] R
ssbr RN
ER
N
E
N
P
000
)( ,)(0
MPgPN
EfMP EB
sE
Input
Output
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Design example of uncoded systems …
Choose a modulation scheme that meets the following system requirements:
5
0
10 [bits/s] 9600 [dB.Hz] 53
[Hz] 4000 with channel AWGNAn
Bbr
C
PRN
P
W
56
2
50
02
02
0
2
10103.7log
)(
102.2)/sin(/22)8(
67.621
)(log)(log
[Hz] 4000 [sym/s] 32003/9600log/8
modulationMPSK channel limited-Band
M
MPP
MNEQMP
RN
PM
N
EM
N
E
WMRRM
WR
EB
sE
b
rbs
Cbs
Cb
16
Choose a modulation scheme that meets the following system requirements:
5
0
10 [bits/s] 9600 [dB.Hz] 48
[kHz] 45 with channel AWGNAn
Bbr
C
PRN
P
W
561
5
0
02
02
0
2
0
00
10103.7)(12
2104.1
2exp
2
1)16(
44.261
)(log)(log
[kHz] 45 [ksym/s] 4.384/960016)/(log16
MFSK channel limited-power / small relatively and
[dB] 2.861.61
MPPN
EMMP
RN
PM
N
EM
N
E
WMMRMRWM
NEWR
RN
P
N
E
Ek
k
Bs
E
b
rbs
Cbs
bCb
b
rb
Design example of uncoded systems …
17
Design example of coded systems Design goals:
1. The bit error probability at the decoder output must meet the system error requirement.
2. The rate of the code must not expand the required transmission bandwidth beyond the available channel bandwidth.
3. The code should be as simple as possible. Generally, the shorter the code, the simpler will be its implementation.
M-arymodulator
M-arydemodulator
][symbols/s log2 M
RRs
[bits/s] R
ss
ccbr R
N
ER
N
ER
N
E
N
P
0000
)( ,)(0
MPgpN
EfMP Ec
sE
Input
Output
Encoder
Decoder
[bits/s] Rk
nRc
)( cB pfP
18
Design example of coded systems …
Choose a modulation/coding scheme that meets the following system requirements:
The requirements are similar to the bandwidth-limited uncoded system, except that the target bit error probability is much lower.
9
0
10 [bits/s] 9600 [dB.Hz] 53
[Hz] 4000 with channel AWGNAn
Bbr
C
PRN
P
W
system limited-power :enough lowNot 10103.7log
)(
400032003/9600log/8
modulationMPSK channel limited-Band
96
2
2
M
MPP
MRRM
WR
EB
bs
Cb
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Design example of coded systems
Using 8-PSK, satisfies the bandwidth constraint, but not the bit error probability constraint. Much higher power is required for uncoded 8-PSK.
The solution is to use channel coding (block codes or convolutional codes) to save the power at the expense of bandwidth while meeting the target bit error probability.
dB 16100
9
uncoded
bB N
EP
20
Design example of coded systems
For simplicity, we use BCH codes. The required coding gain is:
The maximum allowed bandwidth expansion due to coding is:
The current bandwidth of uncoded 8-PSK can be expanded by still 25% to remain below the channel bandwidth.
Among the BCH codes, we choose the one which provides the required coding gain and bandwidth expansion with minimum amount of redundancy.
dB 8.22.1316)dB()dB()dB(00
coded
c
uncoded
b
N
E
N
EG
25.140003
9600
loglog 22
k
n
k
nW
M
R
k
n
M
RR C
bs
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Design example of coded systems …
Bandwidth compatible BCH codes
0.41.33106127
4.36.22113127
2.27.11120127
2.36.225163
2.28.115763
0.28.112631
1010 95 BB PPtkn
Coding gain in dB with MPSK
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Design example of coded systems …
Examine that the combination of 8-PSK and (63,51) BCH codes meets the requirements:
[Hz] 4000[sym/s] 39533
9600
51
63
log2
C
bs W
M
R
k
nR
910
1
54
2
4
000
10102.1)1(1
1043
102.1
log
)(
102.1sin2
2)( 47.501
jnc
jc
n
tjB
Ec
sE
s
rs
ppj
nj
nP
M
MPp
MN
EQMP
RN
P
N
E
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Effects of error-correcting codes on error performance
Error-correcting codes at fixed SNR influence the error performance in two ways:1. Improving effect:
The larger the redundancy, the greater the error-correction capability
2. Degrading effect: Energy reduction per channel symbol or coded bits
for real-time applications due to faster signaling. The degrading effect vanishes for non-real time
applications when delay is tolerable, since the channel symbol energy is not reduced.
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Bandwidth efficient modulation schemes
Offset QPSK (OQPSK) and Minimum shift keying Bandwidth efficient and constant envelope
modulations, suitable for non-linear amplifier M-QAM
Bandwidth efficient modulation Trellis coded modulation (TCM)
Bandwidth efficient modulation which improves the performance without bandwidth expansion