digital communications i: modulation and coding course
Post on 09-Feb-2016
66 Views
Preview:
DESCRIPTION
TRANSCRIPT
Digital Communications I:Modulation and Coding Course
Spring- 2013Jeffrey N. Denenberg
Lecture 9a: Mod/Demod Trade-offs
Lecture 13 2
Last time, we talked about: The properties of Convolutional codes. We introduced interleaving as a
means to combat bursty errors by making the channel seem uncorrelated.
We also studied “Concatenated codes” that simply consist of inner and outer codes. They can provide the required performance at a lower complexity.
Lecture 13 3
Today, we are going to talk about:
Shannon limit Comparison of different modulation
schemes Trade-off between modulation and
coding
Lecture 13 4
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
Lecture 13 5
Error probability plane(example for coherent MPSK and MFSK)
[dB] / 0NEb [dB] / 0NEb
Bit
erro
r pro
babi
lity
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
Lecture 13 6
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)
Lecture 13 7
Nyquist minimum bandwidth requirement
The theoretical minimum bandwidth needed for baseband transmission of Rs symbols per second is Rs/2 hertz.
T21
T21
T
)( fH
f t
)/sinc()( Ttth 1
0 T T2TT20
Lecture 13 8
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
NSWC
power noise Average :[Watt]power signal received Average :]Watt[
Bandwidth :]Hz[
0WNNCES
W
b
Lecture 13 9
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
Lecture 13 10
Shannon limit …C/W [bits/s/Hz]
SNR [bits/s/Hz]
Practical region
Unattainableregion
Lecture 13 11
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.
WC
NE
WC b
02 1log
WNNCES
NSWC
b
0
2 1log
[dB] 6.1693.0log
1
:get we,0or As
20
eNE
WCW
b
0/ NEb
Shannon limit
Lecture 13 12
Shannon limit …
[dB] / 0NEb
W/C [Hz/bits/s]
Practical region
Unattainableregion
-1.6 [dB]
Lecture 13 13
Bandwidth efficiency plane
R<CPractical 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
Lecture 13 14
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)
Lecture 13 15
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
WTWTM
WR
[Hz] /1 ss RTW
][bits/s/Hz log/ 2 MWRb
][bits/s/Hz /log/ 2 MMWRb
Lecture 13 16
Design example of uncoded systems Design goals:
1. The bit error probability at the modulator 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 MRRs [bits/s] R
ssbr RNER
NE
NP
000
)( ,)(0
MPgPNEfMP EBs
E
Input
Output
Lecture 13 17
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
PRNP
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/8modulationMPSK channel limited-Band
MMPP
MNEQMP
RNPM
NEM
NE
WMRRMWR
EB
sE
b
rbs
Cbs
Cb
Lecture 13 18
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
PRNP
W
561
5
0
02
02
0
2
0
00
10103.7)(12
2104.12
exp2
1)16(
44.261)(log)(log
[kHz] 45 [ksym/s] 4.384/960016)/(log16MFSK channel limited-power / small relatively and
[dB] 2.861.61
MPPNEMMP
RNPM
NEM
NE
WMMRMRWMNEWR
RNP
NE
Ek
k
Bs
E
b
rbs
Cbs
bCb
b
rb
Design example of uncoded systems …
Lecture 13 19
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 MRRs
[bits/s] R
ss
ccbr R
NER
NER
NE
NP
0000
)( ,)(0
MPgpNEfMP Ecs
E
Input
Output
Encoder
Decoder
[bits/s] RknRc
)( cB pfP
Lecture 13 20
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
PRNP
W
system limited-power :enough lowNot 10103.7log
)(400032003/9600log/8
modulationMPSK channel limited-Band
96
2
2
MMPP
MRRMWR
EB
bs
Cb
Lecture 13 21
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
Lecture 13 22
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
NE
NEG
25.140003
9600loglog 22
kn
knW
MR
kn
MRR C
bs
Lecture 13 23
Design example of coded systems …
Bandwidth compatible BCH codes
0.41.331061274.36.221131272.27.111201272.36.2251632.28.1157630.28.1126311010 95 BB PPtkn
Coding gain in dB with MPSK
Lecture 13 24
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
96005163
log2
C
bs W
MR
knR
910
1
54
2
4
000
10102.1)1(1
1043102.1
log)(
102.1sin22)( 47.501
jnc
jc
n
tjB
Ec
sE
s
rs
ppjn
jn
P
MMPp
MNEQMP
RNP
NE
Lecture 13 25
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.
Lecture 13 26
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
Lecture 13 27
Course summary In a big picture, we studied:
Fundamentals issues in designing a digital communication system (DSC) Basic techniques: formatting, coding,
modulation Design goals:
Probability of error and delay constraints Trade-off between parameters:
Bandwidth and power limited systems Trading power with bandwidth and vise versa
Lecture 13 28
Block diagram of a DCS
FormatSourceencode
Channelencode
Pulsemodulate
Bandpassmodulate
FormatSourcedecode
Channeldecode
Demod. SampleDetect
Channel
Digital modulation
Digital demodulation
Lecture 13 29
Course summary – cont’d In details, we studies:
1. Basic definitions and concepts Signals classification and linear systems Random processes and their statistics
WSS, cyclostationary and ergodic processes Autocorrelation and power spectral density
Power and energy spectral density Noise in communication systems (AWGN) Bandwidth of signal
2. Formatting Continuous sources
Nyquist sampling theorem and aliasing Uniform and non-uniform quantization
Lecture 13 30
Course summary – cont’d1. Channel coding
Linear block codes (cyclic codes and Hamming codes) Encoding and decoding structure
Generator and parity-check matrices (or polynomials), syndrome, standard array
Codes properties: Linear property of the code, Hamming
distance, minimum distance, error-correction capability, coding gain, bandwidth expansion due to redundant bits, systematic codes
Lecture 13 31
Course summary – cont’d Convolutional codes
Encoder and decoder structure Encoder as a finite state machine, state
diagram, trellis, transfer function Minimum free distance, catastrophic codes,
systematic codes Maximum likelihood decoding:
Viterbi decoding algorithm with soft and hard decisions
Coding gain, Hamming distance, Euclidean distance, affects of free distance, code rate and encoder memory on the performance (probability of error and bandwidth)
Lecture 13 32
Course summary – cont’d1. Modulation
Baseband modulation Signal space, Euclidean distance Orthogonal basic function Matched filter to reduce ISI Equalization to reduce channel induced ISI Pulse shaping to reduce ISI due to filtering at
the transmitter and receiver Minimum Nyquist bandwidth, ideal Nyquist
pulse shapes, raise cosine pulse shape
Lecture 13 33
Course summary – cont’d Baseband detection
Structure of optimum receiver Optimum receiver structure
Optimum detection (MAP) Maximum likelihood detection for equally likely
symbols Average bit error probability Union bound on error probability Upper bound on error probability based on
minimum distance
Lecture 13 34
Course summary – cont’d Passband modulation
Modulation schemes One dimensional waveforms (ASK, M-PAM) Two dimensional waveforms (M-PSK, M-
QAM) Multidimensional waveforms (M-FSK)
Coherent and non-coherent detection Average symbol and bit error probabilities Average symbol energy, symbol rate,
bandwidth Comparison of modulation schemes in terms of
error performance and bandwidth occupation (power and bandwidth)
Lecture 13 35
Course summary – cont’d1. Trade-off between modulation and coding
Channel models Discrete inputs, discrete outputs
Memoryless channels : BSC Channels with memory
Discrete input, continuous output AWGN channels
Shannon limits for information transmission rate Comparison between different modulation and
coding schemes Probability of error, required bandwidth, delay
Trade-offs between power and bandwidth Uncoded and coded systems
Lecture 13 36
Information about the exam: Exam date:
8th of March 2008 (Saturday) Allowed material:
Any calculator (no computers) Mathematics handbook Swedish-English dictionary
A list of formulae that will be available with the exam.
top related