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ADVANCES IN CODING FOR THE FADING CHANNEL
EZIO BIGLIERI
Politecnico di Torino (Italy)
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Is Euclidean distance the best criterion?
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MOST OF THE COMMON WISDOM MOST OF THE COMMON WISDOM ON CODE DESIGN ON CODE DESIGN IS BASED ON HIGH-SNR GAUSSIAN CHANNEL:IS BASED ON HIGH-SNR GAUSSIAN CHANNEL:
MAXIMIZE THE MINIMUM EUCLIDEAN DISTANCEMAXIMIZE THE MINIMUM EUCLIDEAN DISTANCE
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FOR DIFFERENT CHANNEL MODELS, FOR DIFFERENT CHANNEL MODELS, DIFFERENT DESIGN CRITERIA MUST BE USEDDIFFERENT DESIGN CRITERIA MUST BE USED
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FOR EXAMPLE, EVEN ON LOW-SNR FOR EXAMPLE, EVEN ON LOW-SNR GAUSSIAN CHANNELS GAUSSIAN CHANNELS MINIMUM-EUCLIDEAN DISTANCE IS NOT MINIMUM-EUCLIDEAN DISTANCE IS NOT THE OPTIMUM CRITERIONTHE OPTIMUM CRITERION
EXAMPLE: Minimum P(e) forEXAMPLE: Minimum P(e) for 4-point, one-dimensional constellation:4-point, one-dimensional constellation:
low SNRlow SNR
high SNRhigh SNR
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WIRELESS CHANNELS DIFFER WIRELESS CHANNELS DIFFER CONSIDERABLY FROM HIGH-SNR CONSIDERABLY FROM HIGH-SNR GAUSSIAN CHANNELS:GAUSSIAN CHANNELS:
SNR IS A RANDOM VARIABLESNR IS A RANDOM VARIABLE
AVERAGE SNR IS LOWAVERAGE SNR IS LOW
CHANNEL STATISTICS ARE NOT GAUSSIANCHANNEL STATISTICS ARE NOT GAUSSIAN
MODEL MAY NOT BE STABLEMODEL MAY NOT BE STABLE
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Modeling the wireless channel
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OPERATIONAL MEANING:OPERATIONAL MEANING:Frequency separation at which two frequency Frequency separation at which two frequency components of TX signal undergo components of TX signal undergo independent attenuationsindependent attenuations
COHERENCE BANDWIDTHCOHERENCE BANDWIDTH
DEFINITION:DEFINITION:
11--------------------------------------------DELAY SPREADDELAY SPREAD
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COHERENCE TIMECOHERENCE TIME
DEFINITION:DEFINITION:
11------------------------------------------------------DOPPLER SPREADDOPPLER SPREAD
OPERATIONAL MEANING:OPERATIONAL MEANING:Time separation at which two time Time separation at which two time components of TX signal undergo components of TX signal undergo independent attenuationsindependent attenuations
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FADING-CHANNEL CLASSIFICATIONFADING-CHANNEL CLASSIFICATION
TTxx
BBcc
TTcc
BBxx
flat flat in in timetime
flat flat in time and in time and frequencyfrequency
flat in flat in frequencyfrequency
selective selective in timein timeand frequencyand frequency
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MOST COMMON MODEL FOR FADINGMOST COMMON MODEL FOR FADING
• channel is frequency-flatchannel is frequency-flat• channel is time-flat (fading is “slow”)channel is time-flat (fading is “slow”)
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• FREQUENCY-FLAT CHANNEL:FREQUENCY-FLAT CHANNEL:
Fading affects the received signal as a Fading affects the received signal as a multiplicative processmultiplicative process
Received signal:Received signal:
r t R t j t x t n t( ) ( )exp ( ) ( ) ( ) Gaussian process: Gaussian process: RR Rayleigh or Rice Rayleigh or Rice
MOST COMMON MODEL FOR FADINGMOST COMMON MODEL FOR FADING
transmitted signal
noise
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MOST COMMON MODEL FOR FADINGMOST COMMON MODEL FOR FADING
• SLOW FADING :SLOW FADING :
Fading is approximately constantFading is approximately constantduring a symbol durationduring a symbol duration
Received signal:Received signal:
r t R j x t n t t T( ) exp ( ) ( ), 0
This is constant over a symbol interval
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COHERENT DEMODULATIONCOHERENT DEMODULATION
Received signal:Received signal:
TttntxRtr 0),()()(
Phase term is estimated and compensated for
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CHANNEL-STATE INFORMATIONCHANNEL-STATE INFORMATION
The value of the fading attenuation is the “channel-state information”
This may be:
• Unknown to transmitter and receiver• Known to receiver only (through pilot tones, pilot symbols, …)• Known to transmitter and receiver
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0.00001
0.0001
0.001
0.01
0.1
1
0 10 20 30
GAUSSIAN CHANNEL
RAYLEIGH FADING
signal-to-noise ratio (dB)
bit e
rror
pr o
babi
li ty,
bi n
a ry
a nti p
oda l
si g
nal s
performance of uncoded modulation over the fading channelperformance of uncoded modulation over the fading channelwith coherent demodulationwith coherent demodulation
EFFECT OF FADING ON ERROR PROBABILITIESEFFECT OF FADING ON ERROR PROBABILITIES
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Optimum codes for the frequency-flat, slow fading channel• Euclid vs. Hamming• How useful is an “optimum code”?
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MOST COMMON MODEL FOR CODINGMOST COMMON MODEL FOR CODING
Our analysis here is concerned with the Our analysis here is concerned with the frequency-flat, slow,frequency-flat, slow,
FULLY-INTERLEAVED CHANNELFULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates aas the de-interleaving mechanism creates afading channel in which the random variablesfading channel in which the random variablesRR in adjacent intervals are in adjacent intervals are independentindependent
Our analysis here is concerned with the Our analysis here is concerned with the frequency-flat, slow,frequency-flat, slow,
FULLY-INTERLEAVED CHANNELFULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates aas the de-interleaving mechanism creates afading channel in which the random variablesfading channel in which the random variablesRR in adjacent intervals are in adjacent intervals are independentindependent
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DESIGNING OPTIMUM CODESDESIGNING OPTIMUM CODES
Chernoff bound on the pairwise error probability over the Rayleigh fading channel with high SNR:
Most relevant parameter: Hamming distanceMost relevant parameter: Hamming distance
Px xk k
k
dH
( )| |
( , )
x xx x
1
14
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2
Signal-to-noise ratioHamming distance
Product distance
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Design criterion:Design criterion:
Maximize Hamming distance among signalsMaximize Hamming distance among signals
Design criterion:Design criterion:
Maximize Hamming distance among signalsMaximize Hamming distance among signals
A consequence:
In trellis-coded modulation, avoid “parallel transitions”as they have Hamming distance = 1.
DESIGNING OPTIMUM CODESDESIGNING OPTIMUM CODES
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If we maximize Hamming distance among If we maximize Hamming distance among signals strange effects occur. For signals strange effects occur. For example:example:
If we maximize Hamming distance among If we maximize Hamming distance among signals strange effects occur. For signals strange effects occur. For example:example:
DESIGNING OPTIMUM CODESDESIGNING OPTIMUM CODES
4PSK4PSK
if fading acts if fading acts independentlyindependentlyon I and Q parts:on I and Q parts:
Effect of a deep fade Effect of a deep fade on Q part (one bit is on Q part (one bit is lost)lost)
Rotated 4PSKRotated 4PSK(same Euclidean distance)(same Euclidean distance)
if fading acts if fading acts independentlyindependentlyon I and Q parts:on I and Q parts:
Effect of a deep fade Effect of a deep fade on Q parton Q part(no bit is lost)(no bit is lost)
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DESIGNING OPTIMUM CODESDESIGNING OPTIMUM CODES
Problems with optimum fading codes:
• The channel model may be unknown, or incompletely known• The channel model may be unstable
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ROBUST CODESROBUST CODES
In these conditions, one should look for robust, rather than optimum,
coding schemes
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• BICM as a robust coding scheme
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A ROBUST SCHEME: BICMA ROBUST SCHEME: BICM
interleaving is done at bit level demodulation and decoding are separated
encoder bitinterleaver
modulator
channel demod.
bitdeinterlea
ver
decoder
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Bit interleaving may increase Hamming distance among code words at the price of a slight decrease of Euclideandistance (robust solution if channel model is not stable)
Bit interleaving may increase Hamming distance among code words at the price of a slight decrease of Euclideandistance (robust solution if channel model is not stable)
A ROBUST SCHEME: BICMA ROBUST SCHEME: BICM
Separating demodulation and decoding is a considerabledeparture from the “Ungerboeck’s paradigm” , which statesthat demodulation and decoding should be integratedin a single entity for optimality
But this may not be true if the channel is not Gaussian!
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BICM idea is that Hamming distance (and hence performance over the fading channel)can be increased by making it equal to the smallest number of bits (rather than channel symbols)along any error event:
10
00 00 00
11 11
correct path
concurrent path
TCM: Hamming distance is 3BICM: Hamming distance is 5TCM: Hamming distance is 3BICM: Hamming distance is 5
A ROBUST SCHEME: BICMA ROBUST SCHEME: BICM
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BICM DECODER USES MODIFIED “BIT METRICS”
With TCM, the metric associated with symbol s is
p(r | s)
With BICM, the metric associated with bit b is
pb
( | )( )
r ssSi
where is the set of symbols whose label is b in position iSi ( )b
01
00
10
11 S1(0)EXAMPLE:EXAMPLE:
A ROBUST SCHEME: BICMA ROBUST SCHEME: BICM
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A ROBUST SCHEME: BICMA ROBUST SCHEME: BICM
The performance of BICM with ideal interleaving depends on the following parameters:
• Minimum binary Hamming distance of the code selected• Minimum Euclidean distance of the constellation selected
• A powerful modulation scheme• A powerful code (turbo codes, …)
so we can combine:
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ENCODERMEMORY
BICM TCMdE
2 dE2
Hd Hd
2 1.2 3 2 1
3 1.6 4 2.4 2
4 1.6 4 2.8 2
5 2.4 6 3.2 2
6 2.4 6 3.6 3
7 3.2 8 3.6 3
8 3.2 8 4 3
EXAMPLEEXAMPLE: 16QAM, 3bits/2 dimensions
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Turn the fading channel into Turn the fading channel into a Gaussian channel, and use standard codesa Gaussian channel, and use standard codesTurn the fading channel into Turn the fading channel into a Gaussian channel, and use standard codesa Gaussian channel, and use standard codes
• Antenna diversity• Channel inversion as a power-allocation technique
ANTENNA DIVERSITY & CHANNEL INVERSIONANTENNA DIVERSITY & CHANNEL INVERSION
Possible solution to the”robustness problem”:
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Antenna diversity
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ANTENNA DIVERSITY (order M)ANTENNA DIVERSITY (order M)ANTENNA DIVERSITY (order M)ANTENNA DIVERSITY (order M)
• The fading channel becomes Gaussian as
• Codes optimized for the Gaussian channel perform well on the Rayleigh channel if M is large enough
• Branch correlation coefficients up to 0.5 achieve uncorrelated performance within 1 dB
• The error floor with CCI decreases exponentially with the product of times the Hamming distance of the code used
M
M
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EXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTSEXPERIMENTAL RESULTS
Performance was evaluated for the following coding schemes:
J4: 4-state, rate-2/3 coded 8-PSK optimized for Rayleigh-fading channels U4 & U8: Ungerboeck’s rate-2/3 coded 8-PSK with 4 and 8 states optimized for the Gaussian channel Q64: “Pragmatic” concatenation of the “best” binary rate-1/2 64-state convolutional code (171, 133) mapped onto Gray-encoded 4-PSK
J4: 4-state, rate-2/3 coded 8-PSK optimized for Rayleigh-fading channels U4 & U8: Ungerboeck’s rate-2/3 coded 8-PSK with 4 and 8 states optimized for the Gaussian channel Q64: “Pragmatic” concatenation of the “best” binary rate-1/2 64-state convolutional code (171, 133) mapped onto Gray-encoded 4-PSK
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10-8
10-6
10-4
10-2
100
5 10 15 20 25 30 35
J4, M=16U4, M=16
J4, M=4 U4, M=4
J4, M=1
U4, M=1
BER
Eb/N0 (dB)
EXPERIMENTAL RESULTSEXPERIMENTAL RESULTS
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• The block-fading channel
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Most of the analyses are concerned with the Most of the analyses are concerned with the
FULLY-INTERLEAVED CHANNELFULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates a as the de-interleaving mechanism creates a virtually memoryless coding channel.virtually memoryless coding channel.
HOWEVER,HOWEVER,
in practical applications such as digital cellularin practical applications such as digital cellularspeech communication, the delay introduced by speech communication, the delay introduced by long interleaving is intolerablelong interleaving is intolerable
Most of the analyses are concerned with the Most of the analyses are concerned with the
FULLY-INTERLEAVED CHANNELFULLY-INTERLEAVED CHANNEL
as the de-interleaving mechanism creates a as the de-interleaving mechanism creates a virtually memoryless coding channel.virtually memoryless coding channel.
HOWEVER,HOWEVER,
in practical applications such as digital cellularin practical applications such as digital cellularspeech communication, the delay introduced by speech communication, the delay introduced by long interleaving is intolerablelong interleaving is intolerable
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FACTSFACTS
In many wireless systems:In many wireless systems:
Typical Typical Doppler spreads range from 1 Hz to 100 HzDoppler spreads range from 1 Hz to 100 Hz (hence coherence time ranges from 0.01 to 1 s)(hence coherence time ranges from 0.01 to 1 s)
Data rates range from 20 to 200 kbaudData rates range from 20 to 200 kbaud
Consequently, at leastConsequently, at least L=20,000 x 0.01 = 200 symbolsL=20,000 x 0.01 = 200 symbols are affected approximately by the same fading gainare affected approximately by the same fading gain
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FACTSFACTS
Consider transmission of a code word of length Consider transmission of a code word of length n.n.
For each symbol to be affected by an independentFor each symbol to be affected by an independentfading gain, interleaving should be usedfading gain, interleaving should be used
The actual time spanned by the interleaved code The actual time spanned by the interleaved code word becomes at least nLword becomes at least nL
The delay becomes very largeThe delay becomes very large
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FACTSFACTS
In some applications, large delays are unacceptableIn some applications, large delays are unacceptable(real time speech: 100 ms at most)(real time speech: 100 ms at most)
Thus, an n-symbol code word Thus, an n-symbol code word is affected by less than n independent fading gainsis affected by less than n independent fading gains
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BLOCK-FADING CHANNEL MODELBLOCK-FADING CHANNEL MODEL
This model assume that the This model assume that the fading-gain processfading-gain processis piecewise constantis piecewise constant on blocks of N symbols. on blocks of N symbols.
It is modeled as a sequence of independentIt is modeled as a sequence of independentrandom variables, each of which is the fading gainrandom variables, each of which is the fading gainin a block.in a block.
A code word of length n is spread over A code word of length n is spread over MM blocks blocksof of NN symbols each, so that symbols each, so that n=NMn=NM
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..
..N N N N
n=NM
1 2
3
M
• Each block of length N is affected by the same fading.
• The blocks are sent through M independent channels.
• Interleaver spreads the code symbols over the M blocks.
(McEliece and Stark, 1984 -- Knopp, 1997)
BLOCK-FADING CHANNEL MODELBLOCK-FADING CHANNEL MODEL
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BLOCK-FADING CHANNEL MODELBLOCK-FADING CHANNEL MODEL
Special cases:Special cases:
M=1 (or N=n)M=1 (or N=n) the entire code word the entire code word is affected by the is affected by the same fading gainsame fading gain (no interleaving)(no interleaving)
M=n (or N=1)M=n (or N=1) each symbol is affected each symbol is affected by an independentby an independent fading gainfading gain (ideal interleaving)(ideal interleaving)
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BLOCK-FADING CHANNEL MODELBLOCK-FADING CHANNEL MODEL
The delay constraints determinesThe delay constraints determinesthe maximum Mthe maximum M
The choice makes the channelThe choice makes the channelergodic, and allows Shannon’s channelergodic, and allows Shannon’s channelcapacity to be defined (more on this later)capacity to be defined (more on this later)
M
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System where this model is appropriateSystem where this model is appropriate:
t
f
MM=4 (half-rate GSM)=4 (half-rate GSM)
1
2
3
4
1
2
3
4
1
2
GSM with frequency hoppingGSM with frequency hopping
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IS-54 with time-hoppingIS-54 with time-hopping
MM=2=2
11 22 11
System where this model is appropriate:System where this model is appropriate:
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COMPUTING ERROR PROBABILITIESCOMPUTING ERROR PROBABILITIES
““Channel use” is now the transmissionChannel use” is now the transmissionof a block of N coded symbolsof a block of N coded symbols
From Chernoff bound we have, over From Chernoff bound we have, over Rayleigh block-fading channels:Rayleigh block-fading channels:
Mm m Nd
P0
2 4/1
1)ˆ( XX
Squared Euclidean distancebetween coded blocks
Set of indices in which coded symbols differ
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COMPUTING ERROR PROBABILITIESCOMPUTING ERROR PROBABILITIES
)ˆ,(2
2 44
1
1)ˆ(
XX
XXHd
Mmmd
P
Signal-to-noise ratio Hamming block-distance
Product distance
For high SNR:For high SNR:
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Relevant parameter for Relevant parameter for designdesign
Minimum Hamming block-distance D between
code words on block basis:
Error probability decreases with Error probability decreases with exponent Dexponent Dminmin
(also called: (also called: code diversitycode diversity))
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00
4 binary symbols4 binary symbols 4 binary symbols4 binary symbols
Block #1 Block #2Block #1 Block #2
01
10
00
11
01
11
11
10
11
00 00 00
DDminmin=2=2
EXAMPLE (EXAMPLE (NN=4)=4)
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Bound on Bound on DDminmin
With S-ary modulation, Singleton bound holds for a rate-R code:
S
RMD
2min log
11
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Example: Coding in GSMExample: Coding in GSM
+
+
Rate-1/2 convolutional code (0.5 bits/dimension)Rate-1/2 convolutional code (0.5 bits/dimension)used in GSM with M=8. It has dused in GSM with M=8. It has dfreefree=7=7
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ddfreefree path is: path is: {0...011010011110...0}
Symbols in each one of the 8 blocks:Symbols in each one of the 8 blocks:1: 0...0110...02: 0...0110...03: 0...0000...04: 0...0100...05: 0...0000...06: 0...0000...07: 0...0100...08: 0...0100...0
Dmin=5
Example: Coding in GSMExample: Coding in GSM
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This code is optimum!This code is optimum!
With full-rate GSM, With full-rate GSM, RR=0.5 bits/dim, =0.5 bits/dim, MM=8, =8, SS=2. Hence:=2. Hence:
5min D
achieved by the code. (With S=4 the upper bound achieved by the code. (With S=4 the upper bound would increase to 7).would increase to 7).
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Power control
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PROBLEM:PROBLEM:How to encode if CSI is known at How to encode if CSI is known at the transmitter (and at the receiver)the transmitter (and at the receiver)
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Assume Assume RR is known to transmitter is known to transmitter and receiverand receiver
)()( tsR
tx
((channel inversion) then the fading channel channel inversion) then the fading channel is turned into a is turned into a Gaussian channelGaussian channel
)()()( tntxRtr We have:We have:
If:If:
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Channel inversion is common Channel inversion is common in spread-spectrum systemsin spread-spectrum systemswith near-far imbalancewith near-far imbalance
PROBLEM: For Rayleigh fading channels the averagePROBLEM: For Rayleigh fading channels the average transmitted power would be infinite.transmitted power would be infinite. SOLUTION: Use average-power constraint.SOLUTION: Use average-power constraint.
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CODING FOR THE FADING CHANNELCODING FOR THE FADING CHANNEL
• Using multiple antennas
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MULTIPLE-ANTENNA MODEL
(Single-user) channel with t transmit and r receive antennas:
t r
H
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CHANNEL CAPACITY
RATIONALE: Use space to increase diversity (Frequency and time cost too much)
Each receiver sees the signals radiated from the t transmit antennas
Parameter used to assess system quality:CHANNEL CAPACITY
(This is a limit to error-free bit rate, providedby information theory)
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CHANNEL CAPACITY
Assume that transmission occurs in frames:these are short enough that the channel is essentially unchanged during a frame, although it might change considerably from oneframe to the next (“quasi-stationary” viewpoint)
We assume the channel to be unknown to the transmitter, but known to the receiver
However, the transmitter has a partial knowledgeof the channel quality, so that it can choose the transmission rate
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CHANNEL CAPACITY
Now, the channel varies with time from frameto frame, so for some (small) percentage offrames delivering the desired bit rate at thedesired BER may be impossible.
When this happens, we say that a channel outagehas occurred. In practice capacity is a randomvariable.
We are interested in the capacity that can be achieved in nearly all transmissions (e.g., 99%).
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CHANNEL CAPACITY
1%-outage capacity(upper curves)for Rayleigh channelvs. SNR and number of antennasNote: at 0-dB SNR,25 b/s/Hz are available with t=r=32!
t=r(SNR is P/N at each receive antenna)
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CHANNEL CAPACITY
1%-outage capacityper dimension(upper curves)for Rayleigh channelvs. SNR and number of antennas
t=r
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ACHIEVABLE RATES
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SPACE-TIME CODING
Consider t =2 and r =1.
Denote s0 the signal from antenna 0 and s1 the signal from antenna 1During the next symbol period -s1* is transmitted by antenna 0 s0* is transmitted by antenna 1
(Alamouti, 1998)
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SPACE-TIME CODING
The signals received in two adjacent time slots are
r r t h s h s n
r r t T h s h s n
0 0 0 1 1 0
1 0 1 1 0 1
( )
( )
The combiner yields
~
~s h r h r
s h r h r
0 0 0 1 1
1 1 0 0 1
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SPACE-TIME CODING
So that:~
~
s h h s
s h h s
0 0
2
1
2
0
1 0
2
1
2
1
noise
noise
A maximum-likelihood detector makes a decision on s0 and s1. This scheme has the sameperformance as a scheme with t =1, r =2 andmaximal-ratio combining.
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SPACE-TIME CODING
t =2r =1
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SPACE-TIME CODING
SNR (dB)
MRRC=maximum-ratio receive combining
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SPACE-TIME CODING
The performance of this system with t =2and r =1 is 3-dB worse than with t =1 and r =2plus MRRC.
This penalty is incurred because the curves are derived under the assumption that each TXantenna radiates half the energy as the singletransmit antenna with MRRC.
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SPACE-TIME CODING
Consider two transmit antennas
Example:Space-time code achieving diversity 2 withone receive antenna (“2-space-time code”),and diversity 4 with two receive antennas
(Tarokh, Seshadri, Calderbank, et al.)
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SPACE-TIME CODING
Label xy means that signal x is transmitted on antenna 1, whilesignal y is (simultaneously) transmitted on antenna 2
00 01 02 03
10 11 12 13
20 21 22 23
30 31 32 33
2-space-time code4PSK4 states2 bit/s/Hz
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SPACE-TIME CODING
• If yjn denotes the signal received at antenna j
at time n, the branch metric for a transition labeled q1 q2 … qt is
y h qjn
i j ii
t
j
r
,
11
2
(note that channel-state information is neededto generate this metric)
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SPACE-TIME CODING
For wireless systems with a small numberof antennas, the space-time codes ofTarokh, Seshadri, and Calderbank provideboth coding gain and diversity
Using a 64-state decoder these comewithin 2—3 dB of outage capacity
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TURBO-CODED MODULATION
(Stefanov and Duman, 1999)
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TURBO-CODED MODULATION
BER for severalturbo codesand a 16-statespace-time code