Advances in coding for the fading channel

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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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• 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


transmitted signal

noise

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• SLOW FADING :SLOW FADING :

Fading is approximately constantFading is approximately constantduring a symbol durationduring a symbol duration


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


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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• 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,


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.


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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:


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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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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• 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)


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


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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:


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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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• 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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• The block-fading channel

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Most of the analyses are concerned with the Most of the analyses are concerned with the


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


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)


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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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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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• 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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• 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.

70

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

Advances in coding for the fading channel

Technology

fading acts independently

er cu rves

maximize hamming distance

fading channel

ci ty

ro vi

hamming distance

code word