Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB...
Transcript of Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB...
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Transmissions numériquesavancées
Jean-Marc Brossier
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INPG TST 3A J.M. Brossier - LIS
Overview
Source coder(compression)
Error Correcting Code (protection)
Modulator (adaptation to medium)
Demodulator Error correction (or detection) Uncompress
Channel: multipath, …
Tx
Rx
Multi-usernoise
Noise
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INPG TST 3A J.M. Brossier - LIS
Source coding (1) ( Analog to digital conversion )
Source coding :• With loss: mp3, mpeg, divx, xvid (mpeg4), …• Lossless: Huffman, Fano-shannon, Lempel-Ziv,
Source message:0 1 1 0 1 1 1 1 1 …
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INPG TST 3A J.M. Brossier - LIS
Source coding (2)Lossless compression : principles
1. The source should be rewritten toachieve uniform probability :Huffman, Fano-shannon, …
2. Remove dependence : e n t r o p -
e n t r o p y : how do you know“y” is missing?
Shannon Entropy (bits) vs. p
X r.v. with two states: 0,1{ } and p X = 1( ) = p = 1! p X = 0( )
H X( ) = ! p log2 p( ) ! 1! p( ) log2 1! p( )
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INPG TST 3A J.M. Brossier - LIS
Channel coding (1)
Transmitter: Receiver:C O D I N G T H E O R Y C P D I N D T O E O R Y
Redundancy at the transmitter side :C O D I N G T H E O R Y CCC OOO DDD III NNN GGG TTT HHH EEE OOO RRR YYY
CCC OPO DDD III NNN GGD TTT OHO EEE OOO RRR YYY C O D I N G T O E O R Y
2 corrected, 1 detected
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INPG TST 3A J.M. Brossier - LIS
Channel coding (2) Tx:
2 code words (length n=3):(000) (111)
k=1 bit
Rx: (001),(010),(100):Detection + correction (000) (110),(101),(011)Detection + correction (111) (000) (111)(Probably) right
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INPG TST 3A J.M. Brossier - LIS
Modulation Modulation: mapping from digital words
to continuous time functions How many orthogonal functions with
support [0,T]? Infinity BT with bandwidth B
Example: B=1/T if a single function isused.
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INPG TST 3A J.M. Brossier - LIS
Band-limited AWGN channels Two facts:
Perfection ( ) means infinite bandwidth Requirement for zero ISI :
The aim: Perfect discrete channel based on perfect finite bandwidth channel
Nyquist criterion:
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INPG TST 3A J.M. Brossier - LIS
Band-limited AWGN channelsIdeal sinc solution
Frequency
Strictly rectangular spectrum hard to implement:• slowly decaying, infinitely long, noncausalwaveform• sensitivity to timing errors• truncation errors (Gibbs phenomenon)
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INPG TST 3A J.M. Brossier - LIS
Band-limited AWGN channelsRoot-Nyquist filtering
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Discrete time baseband model. Additive white gaussian noise (AWGN) Deterministic linear channel Multipath channels Flat fading
Rayleigh: NLOS Rice: LOS Doppler Spectrum
Coherence (Time - Frequency) Problems and solutions
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INPG TST 3A J.M. Brossier - LIS
Real world channels Continuous time
Modulator-demodulator (modem) Noise:
error correction: algebraic, iterative Finite bandwidth:
Nyquist criterion Non ideal frequency response:
Equalization (linear, DFE, Viterbi, OFDM) Random or deterministic
Diversity Multiuser
OFDM, CDMA, MUD
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INPG TST 3A J.M. Brossier - LIS
Baseband model If a system is linear:
Superposition: convolution:
Full system knowledge: What happens to an eigenfunction:
eigenvalue: complex gain.
exp i •[ ] are eigenfunctions
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INPG TST 3A J.M. Brossier - LIS
Baseband model Baseband channel:
Complex impulseresponse
Image of the realchannel (at carrierfrequency)
Translation !
s t( )ei2"#0 t ! S #( )$%#0
= S # & #0( )
sHF
t( ) = Re sI+ is
Q( )
s t( )"#$ %$
ei2"#0 t
'
(
)))
*
+
,,,
= sIcos 2"#
0t( ) & isQ sin 2"#
0t( )!
1
2S # & #
0( ) + S* &# �( )'( *+
Translation !
sHF
| cos 2"#0t( ) $ s
I
sHF
| sin 2"#0t( ) $ s
Q
•e
i2!"0 t ! #$"0
{ }
•e
! i2"#0 t ! $%!#0
{ }
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INPG TST 3A J.M. Brossier - LIS
Baseband model Baseband signals:
Baseband channel:
Tx signal sHF t( ) = Re s t( )ei2!"0 t#$ %&!1
2S " ' "0( ) + S* '" '"0( )#$ %&
Real channel hHF t( ) = Re h t( )ei2!"0 t#$ %&!1
2H " ' "0( ) + H * '" '"0( )#$ %&
Rx signal rHF t( ) = Re r t( )ei2!"0 t#$ %&!1
2R " ' "0( ) + R* '" '"0( )#$ %&
R !( ) = H !( )S !( ) H !( ) baseband frequency response
r t( ) = h t( )* s t( ) h t( ) complex baseband impulse response
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INPG TST 3A J.M. Brossier - LIS
Discrete time baseband modelBaseband impulse response
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INPG TST 3A J.M. Brossier - LIS
Discrete time baseband modelCircular gaussian noise
If n(t) is a white gaussian noise, thediscrete baseband noise
is white, gaussian and circular:
1. Same power on each component
2. Uncorrelated components
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Additive White Gaussian Noise (1)
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Additive White Gaussian Noise (2)
Applications:• Sub carriers of some multicarrier systems• Low speed transmissions, ….
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INPG TST 3A J.M. Brossier - LIS
Channel modelingDeterministic linear channel
Applications:• xDSL (Digital Subscriber Line) systems, cable modems, ….
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Multipath channels (1)
Number of pathsAmplitude
DelayAdditive (gaussian) noise
HF impulse response: LF equivalent impulse response:
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Multipath channels (2)
Parametric baseband representation:
Applications:• UMTS, WiFi (802.11b-g), ….
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INPG TST 3A J.M. Brossier - LIS
Fading scales Distance
outdoor, indoor
Slow fading Log-normal6-10dB, 5 (indoor)-20m (outdoor)
Fast fading Multipath propagation
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Flat Rayleigh fading: NLOS
Path: cluster of micropaths:
NLOS (No Line of Sight, urban) : CLT:
pdf module
pdf phase
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Flat Rayleigh fading
Applications:• Subcarriers of OFDM RF systems
• DAB - Digital Audio Broadcast• DVB-T - Digital Audio Broadcast - , …
• Low speed transmissions, ….
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Capacity of a flat Rayleigh channel
• Ergodic channel• Non ergodic channel:
• Diversity needed : coding, interleaving, …• Outage probability Random variable
Random capacity close to 0
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Flat Rice fading: LOS
A deterministic component is added: Rice Model
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INPG TST 3A J.M. Brossier - LIS
Channel modeling Jakes Doppler spectrum
Correlation and Doppler spectrum:
Jakes:
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INPG TST 3A J.M. Brossier - LIS
Channel modelingCoherence (time-frequency)
Wide sense stationary (WSS) Uncorrelated scatterers (US)
Spaced-time spaced-frequency correlation function
Coherence bandwidth
Coherence duration
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INPG TST 3A J.M. Brossier - LIS
Some useful links Wireless:
http://grouper.ieee.org/groups/802/11/ IEEE 802.11TM WIRELESS LOCAL AREA NETWORKS - The Working Group for WLAN Standards IEEE 802.16 Working Group on Broadband Wireless Access Standards
http://www.3gpp.org/ The 3rd Generation Partnership Project Agreement. Partners are ARIB, CCSA, ETSI, ATIS, TTA, and TTC.
http://www.etsi.org/ The European Telecommunications Standards Institute (ETSI)
Very-high-bit-rate Digital Subscriber Line:
http://www.vdslalliance.com/ DMT (Discrete MultiTone) based VDSL
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INPG TST 3A J.M. Brossier - LIS
Problems and solutions Parameter estimation:
Timing recovery Carrier synchronization Channel estimation
Channel coding : protection against noise.
Diversity: protection against fading
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INPG TST 3A J.M. Brossier - LIS
Single carrier transmission(selective channel) Optimum receiver Structures:
Linear equalizers ISI cancellers Decision Feedback Equalizer (DFE)
Adaptive equalizers LMS RLS
Maximum Likelihood Sequence Estimation Channel estimation
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INPG TST 3A J.M. Brossier - LIS
Equalization: principlesISI : InterSymbol InterferenceSpectrum equalization
Spectrum equalization
ISI : InterSymbol Interference
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INPG TST 3A J.M. Brossier - LIS
Signal path From the source to the demodulator
output
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INPG TST 3A J.M. Brossier - LIS
Optimum receiver structure
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INPG TST 3A J.M. Brossier - LIS
FIR Structure. Finite Impulse Response
(a.k.a. MA - Moving average)
The output is a linear combination of afinite set of inputs.
Stable
Linear structure
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INPG TST 3A J.M. Brossier - LIS
IIR structure. Infinite Impulse Response
Autoregressive (AR) structure. Inverse of a FIR filter: the output is a
linear combination of a finite set ofoutputs
Stable or NOT
Linear structure
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INPG TST 3A J.M. Brossier - LIS
ARMA structure The output depends on outputs and
inputs Stable or NOT
Linear structure
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INPG TST 3A J.M. Brossier - LIS
Example: channel with 1 zero
Minimum phase Causal, stable inverse: MA or AR
Maximum phase Non causal, stable inverse: MA
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INPG TST 3A J.M. Brossier - LIS
ISI canceller Remove contribution of undesired
symbols
Non linear structure
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INPG TST 3A J.M. Brossier - LIS
Decision Feedback Equalizer(DFE) Division “causal or not”N
on linear structure
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INPG TST 3A J.M. Brossier - LIS
Comparison of 3 structures Equalization ofN
on linear structure
![Page 43: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/43.jpg)
INPG TST 3A J.M. Brossier - LIS
ARMA Wiener filter ARMA implementation of the optimal
Wiener filter
![Page 44: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/44.jpg)
INPG TST 3A J.M. Brossier - LIS
Adaptive equalization:FIR LMS - Least Mean Squares
Objective function:
Minimization:
Gradient: Stochastic approximation:
![Page 45: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/45.jpg)
INPG TST 3A J.M. Brossier - LIS
Adaptive equalization:RLS - Recursive Least Squares
Newton algorithm:
For a quadratic function:
![Page 46: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/46.jpg)
INPG TST 3A J.M. Brossier - LIS
Adaptive equalization:Learning and tracking
Learning
Tracking
![Page 47: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/47.jpg)
INPG TST 3A J.M. Brossier - LIS
IIR LMS IIR Structure:
Objective function:
Adaptive algorithm:
![Page 48: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/48.jpg)
INPG TST 3A J.M. Brossier - LIS
IIR LMS with decision The same idea can be applied to DFE
![Page 49: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/49.jpg)
INPG TST 3A J.M. Brossier - LIS
Maximum LikelihoodSequence Estimation Calculation of the transmitted sequence
that maximizes the likelihood ofobservations.
Efficient implementation: Viterbi Algorithm
![Page 50: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/50.jpg)
Fonctionnement del’algorithme de Viterbi
Exemple d’un code convolutif
![Page 51: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/51.jpg)
INPG TST 3A J.M. Brossier - LIS
Codeur convolutifMachine à état fini
Le codeur effectue uneconvolution sur les donnéestransmises.
Son état interne estdéterminé par L’entrée présente Des (ici 2) variables
internes Les transitions sont pilotées
par l’entrée.
![Page 52: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/52.jpg)
INPG TST 3A J.M. Brossier - LIS
L’automate évolue au coursdu temps
Machine à état fini l’état suivant ne dépend
que de l’entrée et del’état actuel.
Tranche du treillis chaque état peut transiter
vers un autre état àl’instant suivant enfonction de l’entrée.
![Page 53: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/53.jpg)
INPG TST 3A J.M. Brossier - LIS
Maximum LikelihoodSequence Estimation (MLSE) L mots (de 1 bit) transmis engendre L+2 mots (de 2 bits) reçus. On ferme le treillis (2 zéros finaux pour ramener son état à 00)
Le message traverse un canal à bruit additif blanc
On observe y, version bruitée de la sortie du codeur Les transitions de la machine à état fini peuvent être indexées par :
la sortie du codeur les 2 états entre lesquels s’effectue la transition
On recherche la séquence qui maximise la vraisemblance desobservations :
sj, s
j+1
Cj
![Page 54: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/54.jpg)
INPG TST 3A J.M. Brossier - LIS
Notations des transitions surle treillis
![Page 55: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/55.jpg)
INPG TST 3A J.M. Brossier - LIS
Canal binaire symétrique La métrique de branche est donnée par
la distance de Hamming entrel’observation et la sortie du codeur.
La métrique d’un chemin, métriquecumulée, est la somme des métriquesdes branches qui le compose.
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INPG TST 3A J.M. Brossier - LIS
Algorithme de Viterbi
Etat initial00
Information001
SéquenceTransmise
00 00 11 01 11
SéquenceReçue
10 01 11 01 11
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INPG TST 3A J.M. Brossier - LIS
Channel estimation Viterbi for equalization required the
knowledge of the channel impulseresponse: Pilot sequence (midamble) Blind estimation Semi-blind techniques
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INPG TST 3A J.M. Brossier - LIS
Synchronization Carrier synchronization Timing recovery Strongly related to channel estimation
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INPG TST 3A J.M. Brossier - LIS
Origin and consequences of aphase misadjustment
Problems:• Doppler• Electronics
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INPG TST 3A J.M. Brossier - LIS
Carrier synchronizationBPSK (Binary PSK)
Squared loop:
Costas loop:
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INPG TST 3A J.M. Brossier - LIS
Carrier synchronizationProblem and notations
Basic model:
Notations:
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INPG TST 3A J.M. Brossier - LIS
Digital Costas Loop (1)BPSK
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INPG TST 3A J.M. Brossier - LIS
Digital Costas Loop (2)BPSK
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INPG TST 3A J.M. Brossier - LIS
Decision Feedback Loop (1)DFL - BPSK
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INPG TST 3A J.M. Brossier - LIS
Decision Feedback Loop (2)DFL - BPSK
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INPG TST 3A J.M. Brossier - LIS
Decision Feedback Loop (3)DFL - Generalization
Cost function:
Adaptive algorithm:
![Page 67: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/67.jpg)
INPG TST 3A J.M. Brossier - LIS
Fourth power loopQAM Phase estimation
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INPG TST 3A J.M. Brossier - LIS
Timing synchronizationBasic algorithm
Adaptive algorithm approach
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INPG TST 3A J.M. Brossier - LIS
Timing synchronizationBasic algorithm implementation
Estimation of the derivative: Shannon interpolation Finite difference approximation
Fully digital implementation requires: Digital interpolation (multirate filters, …) 2 samples/symbol
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INPG TST 3A J.M. Brossier - LIS
Related to channel estimation Phase and timing are special cases of
channel estimation.
Separation, why? Several time scales Simplicity History
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INPG TST 3A J.M. Brossier - LIS
Advanced modulationschemes OFDM
Principles Discrete time formulation DMT for deterministic channels:
Bit loading Water filling
OFDM over random channels: COFDM
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INPG TST 3A J.M. Brossier - LIS
OFDM: Principles (1)Why and how
Why? Equalizers face many problems: Non minimum phase channels. Spectral nulls. Complexity (Viterbi) or suboptimality.
How? Multicarrier approach:1. Several (for speed) low rate (to avoid ISI) transmissions.2. Non interfering sub transmissions.3. The Transmitter helps.4. Low complexity (FFT)
![Page 73: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/73.jpg)
INPG TST 3A J.M. Brossier - LIS
OFDM: Principles (2)Analog point of view (f)
1. Several low rate transmissions.Analog point of view, frequency-domain vision.
Divide the bandwidth Parallelize low band transmissions
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INPG TST 3A J.M. Brossier - LIS
OFDM: Principles (3) Analog point of view (t)
Short symbols:ISI
Long symbols:ISI freetransmission
1. Several low rate transmissions.Analog point of view, time-domain vision.
![Page 75: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/75.jpg)
INPG TST 3A J.M. Brossier - LIS
OFDM: Principles (4)Non interfering transmissions
2. Non interfering sub transmissions. Sub streams orthogonality Distinct supports (too strong) Orthogonal carriers:
Infinite length: frequencies are always orthogonal.
Finite length: minimum carrier spacing: 1/T
x t( ) | y t( ) = X !( ) |Y !( )
ei2!"1t | e
i2!"2 t
T
=ei2! "1!"2( )T
!1
i2! "1!"
2( )= 0" "
1!"
2=k
T,k #!
ei2!"1t | ei2!"2 t = # "1!"2( )" Orthogonality for "1 # "2
![Page 76: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/76.jpg)
INPG TST 3A J.M. Brossier - LIS
OFDM:Discrete time formulation
!k
=k
T,k = 0!N!1" Shannon: !
S=N
T"T
S=T
N
tn
= nTS
= nT
N
#
$
%%%%%
&
%%%%%
exp i2"!ktn
( ) = exp i2"k
T
'
()))*
+,,, n
T
N
'
()))
*
+,,,
'
()))
*
+,,,,
= expi2"kn
N
'
()))
*
+,,,
an
k exp i2"!kt( )
k=0
N!1
-tn =n
T
N
= an
k expi2"kn
N
'
()))
*
+,,,
k=0
N!1
-
Analog formulation:
For symbol n:
Discrete time formulation:
an
kexp i2!"
kt( )
k=0
N!1
"
ankp t!nT( )ei2!"k t
k=0
N!1
"n=0
+#
"
Efficient implementation:
TF!1
an
0 ,an
1 ,!,an
N!1{ } = TF an
{ }
![Page 77: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/77.jpg)
INPG TST 3A J.M. Brossier - LIS
DMT:DMT=OFDM + bit loading
1 bits
4 bits
2 bits
Carrier 0
Carrier 1
Carrier K
DMT signal
Tones
SNR b(i)
an
k exp i2!"kt( ) with "
k=k
T,k = 0!N!1
k=0
N!1
"
an
0ei2!"0 t
an
N!1ei2!"
N!1t
Discrete Multi Tones
an
1ei2!"1t
![Page 78: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/78.jpg)
INPG TST 3A J.M. Brossier - LIS
DMT:Bit loading (1)
Channel:
Energy of an even QAM:
Symbol error rate:
EQ = E x2( ) =
M !1
6dQ
2 where
dQ minimum distance of the QAM
M number of states of the QAM
"
#$$
%$$
PQ ! 4Qdout
2
2N0
"
#
$$$$$
%
&
'''''= 4Q 3(( ) with (=
dout2
6N0
=H
2
dQ2
6N0
y= Hx+ n with
H complex gain of a flat channel
x !QAM
n gaussian noise with E n2
= !2
= N0
"
#
$$$$
%
$$$$
PE for the real component: 2Qdout
2N0
!
"####
$
%
&&&&& with Q x( ) =
1
2!e't2 /2
dtx
+(
)dout minimum distance at the noise free channel output: dout = H dQ
Pbit = KbQdout2
2N0
!
"
#####
$
%
&&&&&
![Page 79: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/79.jpg)
INPG TST 3A J.M. Brossier - LIS
DMT:Bit loading (2)
Gap for a given Pe :
Bit loading:
!= !0
+ !m"!
cdB( )
Pe
/ dim <PQ
2!">"
0 with "
0=
1
3Q#1
PQ
4
$
%&&&
'
())))
*
+
,,
-
.
//
2
!out
=EQH
2
N0
EQ =M !1
6dQ2 " M =1+
6EQ
dQ2
=1+EQ H
2
N0
#
$
%%%%%
&
'
(((((
6N0
dout2
#
$%%%%
&
'
((((
b= log2 M( ) = log2 1+
!out
!
"
#$$$
%
&'''
1 bit = 3 dBMargin Coding gain
![Page 80: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/80.jpg)
INPG TST 3A J.M. Brossier - LIS
DMT:Water filling
Two channels example: Total power: PT
Cj = log2 1+ ! j( ) = log2 1+" j
2
# j2
!
"
####
$
%
&&&&&
C = C0 +C1
!0
2=PT
2+"1
2 !"0
2
2
"
#$$$$
%
&''''
!1
2=PT
2!"1
2 !"0
2
2
"
#$$$$
%
&''''
(
)
******
+
******
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INPG TST 3A J.M. Brossier - LIS
DMT: implementation Modem implementation
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INPG TST 3A J.M. Brossier - LIS
COFDM Random channels:
Repetitions: > Coherence time > Coherence bandwidth
Interleaving Channel coding
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INPG TST 3A J.M. Brossier - LIS
Advanced modulationschemes CDMA (Direct sequence)
Conventional receiver Multi user Multiuser detection
![Page 84: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/84.jpg)
INPG TST 3A J.M. Brossier - LIS
DS-SS: PrincipleDirect Sequence Spread Spectrum
A code is a pulse shaping with: A wide band. Good correlation and cross correlation
properties.
00
Frequency
carrier
Code
Frequency
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INPG TST 3A J.M. Brossier - LIS
DS-SSModulation
Time Domain Frequency Domain
![Page 86: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/86.jpg)
INPG TST 3A J.M. Brossier - LIS
DS-SSTransmitter and Receiver
![Page 87: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/87.jpg)
INPG TST 3A J.M. Brossier - LIS
DS-SSDemodulation
Robustness, low band interferers
![Page 88: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/88.jpg)
INPG TST 3A J.M. Brossier - LIS
Multi User Detection
y t( ) = Akbk j[ ]sk t! jTS !!k( )+ n t( )j=!M
M
"k=1
K
"
Symbols, user k
Code, user k
Asynchronism
k ! 0,TS
[ ]
Noise
K users
![Page 89: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/89.jpg)
INPG TST 3A J.M. Brossier - LIS
y t( ) = Akbk j[ ]sk t! jTS !!k( )+ n t( )j=!M
M
"k=1
K
"
Multi User Detection
Synchron: !i, j!"
#$ = si t( )
0
TS
% s j t( )dt!
"&&
#
$'' i, j
yk = Akbk + Ajbj! jk + nk
j!k" with yk = y t( )sk t( )dt, k = 1!K
0
TS
#
yT = y1,!, yK( ), bT = b
1,!,bK( ), A= diag A
1,!,Ak( )
nk = n t( )sk t( )dt0
TS
!
y= RAb+ n, Noise covariance "n
= !2R
Symbols, user k
Code, user k
Asynchronism, k ! 0,TS
[ ]
NoiseK users
![Page 90: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/90.jpg)
INPG TST 3A J.M. Brossier - LIS
Mono User performance
y t( ) = Abs t( )+ n t( ),t ! 0,TS[ [
P =Q
A
!
!"###$%&&& for device: b!= sign y t( )s t( )dt
0
TS
'!"##
$%&&&
P1
=1
2Q
A1!A
2!
"
"
#$$$
%
&'''+
1
2Q
A1+ A
2!
"
"
#$$$
%
&'''(Q
A1!A
2!
"
"
#$$$$
%
&
''''
1 user
2 users
K users Near-Far
![Page 91: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/91.jpg)
INPG TST 3A J.M. Brossier - LIS
Multi User Detection Optimum receiver
Maximization:
yk = Akbk + Ajbj! jk + nkj!k
"
![Page 92: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/92.jpg)
INPG TST 3A J.M. Brossier - LIS
Bloc transmission (1)General model
Input
Output
![Page 93: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/93.jpg)
INPG TST 3A J.M. Brossier - LIS
Bloc transmission (2)IBI Free transmission
Guard interval to prevent IBI and …
x i( ) = H
0
Tu i( )+ H1
Tu i!1( )+ noise
u i( )! Tu i( )
Two usual choices: Cyclic Prefix: frequency domain equalizer but sensitivity to zeros near the FFT grid.
Zero Padding: symbol identifiability but high receiver complexity.
GUARD (Size L) USEFUL PART (Size N)
Size P=N+L
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INPG TST 3A J.M. Brossier - LIS
Bloc transmission (3)Cyclic Prefix, Zero Padding
Time
Block 1Guard Guard Block 2
100101111 0100111010100101111 Next
Previous 0100111010100101111 000000000
Channel impulse response
Cyclic Prefix
Zero Padding
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INPG TST 3A J.M. Brossier - LIS
Bloc transmission (4)Cyclic Prefix
Add Cyclic Prefix (CP) to bloc x= x0!xN!1[ ] :
x= x0!xN!1[ ]" u= TCPx= xN!L+1!xN!1 | x0!xN!L+1!xN!1[ ]
with TCP =ICP
IN
#
$%%
&
'((, N + L( ))N matrix
P = N + L and ICP last lines of IL
RCP = 0N)L IN[ ]
"H = RCPH0TCP with "Hkl = h k! l( ) N[ ]
F "HF!1 = DH = diag H e
j0( )!H ej2!
N!1
N*
+,,,,
-
.
////
#
$
%%%
&
'
(((
with Fkn =1
Nexp !i2!
kn
N
*
+,,,
-
.///
*
+,,,
-
.////
00k ,n0N!1
and H exp i2! f( )( ) = hn exp !i2! fn( )n=0
L
1
![Page 96: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/96.jpg)
INPG TST 3A J.M. Brossier - LIS
Usual DS-CDMA
[ ]•
!!!!
"
#
$$$$
%
&
•
•
•
•
=
!!!!
"
#
$$$$
%
&
•
•
•
•
TIME DOMAIN:
Tdata
+1 -1
-1
-1
+1
+1
SpreadingSequence
Spreadedsignal
Tc
Take one bit and spread it using a code (spreading sequence)
Matrix form
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INPG TST 3A J.M. Brossier - LIS
Multi-codes DS-CDMA
!"
#$%
&
•
•
!!!!
"
#
$$$$
%
&
•
•
•
•
•
•
•
•
=
!!!!
"
#
$$$$
%
&
•
•
•
•
TIME DOMAIN+1 -1
First spreading sequence
Second spreadingsequence
Multi-codes signal
Each column is a code
Take N bits and spread them using N codes (spreading sequences)
Matrix form
![Page 98: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/98.jpg)
INPG TST 3A J.M. Brossier - LIS
OFDM
!!!
"
#
$$$
%
&
•
•
!!!
"
#
$$$
%
&
••
••
=
!!!
"
#
$$$
%
&
•
•
!
"
!#!
"
!
OFDM SYMBOL
SQUARE FFT MATRIX BITS TO BETRANSMITTED
Each column is a sub-carrier,it is also a « code »A kind of multi-codes …
Matrix form
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INPG TST 3A J.M. Brossier - LIS
Block Spreading
!!!
"
#
$$$
%
&
•
•
!!!!
"
#
$$$$
%
&
••
••
=
!!!!
"
#
$$$$
%
&
•
•
!
"
!!
!!
"
!
!
SYMBOL NON SQUARE MATRIX BITS TO BETRANSMITTED
A generalization of both OFDM and CDMA
Matrix form
![Page 100: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/100.jpg)
INPG TST 3A J.M. Brossier - LIS
Some Factorizations …MC-CDMA
!= F
H
CA GENERALIZED MC-
CDMA
LEADS TO THE USUAL MC-CDMA C = c
1[ ]
FFT matrix
Column of Σ are non-binary valued codes.
![Page 101: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/101.jpg)
INPG TST 3A J.M. Brossier - LIS
Some Factorizations …Generalized MultiCarrier
!= F
H
"# GENERALIZEDMC
FULL COLUMN RANKSPREADING MATRIX
MAPPING ON A SUBSET OF CARRIERS
IFFT
Factorize Σ in order to fight againstthe three major issues:
• Noise: error correcting code
• ISI: e.g. CP-DMT
• MUI: e.g. Vandermonde
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INPG TST 3A J.M. Brossier - LIS
What about the receiver ? Zero Forcing: pseudo-inverse
Matched Filter: Hermitian
!= F
H
"#
!#
!
H
Σ NEED TO BE FULLCOLUMN RANK
![Page 103: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/103.jpg)
INPG TST 3A J.M. Brossier - LIS
OFDM – CDMA combinationsIn a wireless context
MUI sensitivity: MUD needed
Selective Fading Zeros near the FFTgrid.
Weakness
Fading resilience MUI resilience Equalization
Strength
CDMAOFDM
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INPG TST 3A J.M. Brossier - LIS
COMPARISON serial vs block Serial:
Minimum phasechannel?
Zeros near the unitcircle?
IIR response for asample rateequalizer
Block:
Symbol detectability(ZP)
Equalization (CP)
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INPG TST 3A J.M. Brossier - LIS
Trends Multidimensional processing
SISO - Single Input Single Output Channel capacity (deterministic, random) Random channels and diversity
SIMO - Single Input Multiple Outputs Beam forming Diversity
MIMO - Multiple Inputs Multiple Outputs
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INPG TST 3A J.M. Brossier - LIS
SISO channels (1)Single Input – Single Output
Flat frequency response: frequency non selective channels
Several kinds of time selective channels: is a time independent complex number.
is random and changes from one channel use to another one.
is random but remains constant over a large block: the channel remains constant during a codeword duration.
!
k= !
!
k
!
k
![Page 107: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/107.jpg)
INPG TST 3A J.M. Brossier - LIS
SISO channels (2)Limit performance
Capacity
A well defined number for deterministic AWGN channels.
A random variable for non-ergodic random channel: The capacity is limited by the lowest SNR values. If the lowest value is zero, e.g.
Rayleigh channel, the capacity is zero too! We must accept “channel outage”. What capacity can be achieved for almost all
transmissions (typically 99%)? This is the 1%-outage capacity.
Ck= log
21+
!k
2
" 2
#
$%
&
'( bits
![Page 108: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/108.jpg)
INPG TST 3A J.M. Brossier - LIS
SISO channels (3)Bit Error Rate
For an AWGN channel, the error probability decreases exponentially asSNR increases.
For a Rayleigh channel, it is only inversely proportional to the SNR: theerror probability is dominated by worst cases. Some “diversity” is needed.
Type I: no coding at the Tx. Antenna array with Maximum Ratio Combining (MRC) exploits the available sensors to
maximize the SNR. Path diversity and rake receivers.
Type II: the transmitter repeats the symbols in some way (frequency, time, space, …),the receiver combines several sources of information.
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INPG TST 3A J.M. Brossier - LIS
SIMO channels (1)Single Input – Multiple Outputs
Strongly correlated sensors outputs. Beamforming, spatial filtering. Fourier based methods. High resolution angle estimation. Adaptive antennas.
Independent sensors outputs. Diversity. SNR improvement. The Rx observes independent replica of the transmitted signal. Fading mitigation.
Identifiability improvement. The vector channel is invertible if there are no common zeros between sub-channels.
Tx
Rx
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INPG TST 3A J.M. Brossier - LIS
SIMO channels (2)Beamforming
r t( ) = a
s t( )s t ! "( )!
s t ! n !1( )"( )
#
$
%%%%%
&
'
(((((
Plane wave
r! =
1
ei"
!
ei n#1( )"
$
%
&&&&
'
(
))))
s
Narrowband approximation: a usual Fourier transform.
![Page 111: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/111.jpg)
INPG TST 3A J.M. Brossier - LIS
MIMO channels
Channel is represented by an n x m matrix A:each Rx antenna receives a linear combinationof transmitted signals.
Each path is associated to a frequency nonselective – flat - channel (perhaps subcarriers)
Txm antennas
Rxn antennas
y=Hx+n
![Page 112: Transmissions numériques avancéesjean-marc.brossier/cours-com-num.pdf · INPG TST 3A J .MB rosie-LIS Maximum Likelihood Sequence Estimation (MLSE) L mots (de 1 bit) transmis engendre](https://reader030.fdocuments.us/reader030/viewer/2022040606/5eb5411dff6a93581a23ef36/html5/thumbnails/112.jpg)
INPG TST 3A J.M. Brossier - LIS
MIMO channelsProblems
Spatial interference - non diagonalterms - are to be suppressed: Multi User Detection like techniques Equalization Linear and non-linear techniques.
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INPG TST 3A J.M. Brossier - LIS
MIMO channelsUsual assumptions
are independent random variables: Coherence time much greater than symbol duration: Hij are r.v. (no
random process) Antennas are spaced sufficiently apart: Hij are independent.
Fading conditions Rayleigh fading valid for rich scattering environment – no line-of-sight
(LOS). are independent Gaussian circular random variables. Rice fading (LOS between Tx and Rx)H
ij
Hij
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INPG TST 3A J.M. Brossier - LIS
Parallel additive gaussianchannels Energy constraint:
Waterfilling solution:
E = En
n=1
N
!
C = log2 1+En
!n
2
!
"####
$
%
&&&&n=1
N
'
!n
2+E
n= µ for !
n
2< µ
En
= 0 for !n
2> µ
()**
+**
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INPG TST 3A J.M. Brossier - LIS
Singular Values Decomposition of aMIMO channel
Channel model:
SVD:
y=Hx+n
=UDV+x+n
y!=U+y=Dx!+U
+n
with x!=V+x and n!=U
+n
y!n = dn x!n + n!n!C = log2 1+
dn2En
2!2
"
#$$$$
%
&''''
n=1
N
(
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INPG TST 3A J.M. Brossier - LIS
Optimum transmission system
Tx: Matrix processing using V Rx: Matrix processing using U
+
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INPG TST 3A J.M. Brossier - LIS
Unknown CSI Uniform distribution of the energy:
C = log2 1+dn
2E
2m!2
!
"####
$
%&&&&
n=1
N
'
= log2 det In +E
2m!2H
+H
!
"###
$
%&&&
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INPG TST 3A J.M. Brossier - LIS
Space-Time ProcessingAlamouti scheme,Tx diversity
Alamouti processing (1998):
y0, y1[ ]= h
0x0
+ h1x1,!h
0x1
* + h1x0
*"#
$%+ n
0,n1[ ]
h0
*!h
1
h1
*h0
"
#
&&&
$
%
'''
y0
!y1
*
"
#
&&
$
%
''= h
0
2+ h
1
2( )x0 + h0
*n0
+ h1n1
*( ), h02
+ h1
2( )x1 + h0
*n1
* + h1
*n0( )"
#&$
%'
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INPG TST 3A J.M. Brossier - LIS
Space-Time ProcessingComplex case: Alamouti scheme
Full data rate, delay optimal code for N=2 only.
Alamouti scheme
Antenna 2
Antenna 1
Time 2Time 1
x
0
x
1
!x
1
"
x
0
!
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INPG TST 3A J.M. Brossier - LIS
Capacity of a MIMO channelKnown CSI
When m sensors are available at the transmitter and n at the receiver, thechannel capacity writes:
Cmn
= log det In+!
T
mAA
*
"
#
$$$$
%
&
''''
(
)*
+
,- with !
T=!2
PT
. 2
Cmn
=i=1
K
! log 1+"2
# 2$
X ,i
2
$H ,i
%
&'
(
)*
$H ,i
singular values of A and $X ,i
= PT
i=1
m
!
The channel can be reduced to K (rank of A) parallelsub-channels:
r = !Ax + n
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INPG TST 3A J.M. Brossier - LIS
Capacity of a MIMO channelUnknown CSI
Cmn
=i=1
K
! log 1+a
2
" 2#
X ,i#
H ,i
2$
%&
'
()
#X ,i
=P
T
m
Known CSI (waterfilling)
Unknown CSI
Cmn
=i=1
K
! log 1+"2
# 2$
X ,i
2
$H ,i
%
&'
(
)*
$X ,i
= + ,# 2
a2$
H ,i
2
-
./
0
12
+
+ is chosen to satisfy the constraint $X ,i
= PT
i=1
m
!
3
4
55
6
55
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INPG TST 3A J.M. Brossier - LIS
Capacity of a MIMO channel For high SNR, capacity increases
almost linearly with K. Examples:
1 Rx, little to be gain for more than 4 Tx. 2 Rx, little to be gain for more than 6 Tx.
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INPG TST 3A J.M. Brossier - LIS
Space only processing (1) Transmission of a x-scaled version of symbol s
through each antenna. Each symbol is transmittedonly once.
Received vector: r = Axs + n
s: transmitted symbol. x: scaling vector. A: m x n channel matrix. n: additive white Gaussian noise vector.
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INPG TST 3A J.M. Brossier - LIS
Space only processing (2) With a linear combiner z, decisions are based on:
D = z
*Axs + n( )
Channel State Information (CSI) needed atthe transmitter side.
Optimum solution: x: principal right singular vector of A. z: principal left singular vector of A.
Optimum SNR is given by: SNR
max=!
max( A
"A)
#2
Es
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (1) N channel use to transmit symbol s (time diversity) At time k (from 1 to N), symbol s is scaled using a scaling vector
Decision variable:
SNR at the linear combiner output:
kx
D =1
Nk=1
N
! zk
"Ax
ks + n
k
#$%
&'(
SNR =N
! 2
2
tr ARxz
"#$
%&'
tr Rzz
"#$
%&'
Es
r
k= Ax
ks + n
k
Rab=
1
Na
kb
k
*
k=1
N
!
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (2)
Optimum solution: Optimum SNR: The channel must be known at the receiver
end. Two usual contraint:
Trace constraint: CSI available at the Tx, samesolution than for space only processing.
Max Eigenvalue constraint: no CSI at the Tx.
z
k! Ax
k
SNR =N
! 2tr AR
xxA
"#$%
&'( E
s
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (3) Tx is allowed to transmit in all possible
directions. Maximum power in each direction lower than
a given threshold: constraint on themaximum eigenvalue of
Optimum solution: R
xx=
1
N k=1
N
! xkx
k
"
SNR
max= N
tr(A!A)
"2
Es
R
xx= I
m
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (4) One possible choice to get
N must be at least m: date rate decreases by afactor N: cycling is not efficient.
Solution: transmission of N symbols by channeluse and being able to separate symbols at theRx side.
xk= me
k with e
k is the m x 1 vector having 1 in
position k and 0 elsewhere.
R
xx=
1
N k=1
N
! xkx
k
"= I
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (5)Real case
Coding matrix Xi : Column k of matrix Xi is the scaling vector forthe transmission of symbol i at time k. Noting N the noise matrix,the received matrix is:
R = A
i=1
N
!Xis
i+ N
Di= ! A
"AX
iX
i
"#
$%
&
'( s
i+
j) i
*! tr A"AX
jX
i
"#
$%
&
'(
+,-
.-
/0-
1-s
j
Full data rate and delay optimal solutions - using Hurwitz-Radonfamily - exist for N=2,4,8.
(complex numbers, quaternion, octonion) Full data rate solution exist for other values of N but they are no
longer delay optimal.
X
iX
i
!= I and X
iX
j
!= "X
jX
i
! for i # j
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INPG TST 3A J.M. Brossier - LIS
Space-Time Processing (5)Complex case: rate 3/4
Complex coder can outperform the realcoder. Transmission of m=3 complexsymbols with 4 channel use.
Higher rates with ? N !m
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INPG TST 3A J.M. Brossier - LIS
Space-Time Coding (5)
R = AX + NTime
Spa
ce
Received matrix Channel Transmitted matrixNoise matrix
Optimum ML estimation of the transmitted signal:
X
*= argmin
X
R ! AX2
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INPG TST 3A J.M. Brossier - LIS
Space-Time Coding (5)Code design criterion (Tarok)
P x ! x '( ) " #i
i=1
$
%&
'()
*+
,n
ES
4N0
&
'()
*+
,$nPair-wise error probability:
Coding gain Diversity gain
! " m : Rank of matrix D of differences x! x̂
Tarok like design: D full rank for all differences. The achieved diversity is n times the min rank over
the set of codewords pair. Maximize the product of nonzero eigenvalues.
Problems: This bound is valid for very high SNR only. the waterfall region becomes more and more extended as we increase the
number of antennas.
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INPG TST 3A J.M. Brossier - LIS
Space-Time Coding (5)Asymptotic analysis (Biglieri)
m finite, n large: asymptotically, the PEP (pair-wiseerror probabilities) depends only on the Euclidiandistance between code words and not on thedeterminant or the rank. The fading channel tends toa Gaussian channel with no spatial interference.
2 Tx antennas and 4 Rx antennas is often enough.
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INPG TST 3A J.M. Brossier - LIS
Space-Time Coding (5)Asymptotic analysis
Low complexity receivers: Zero Forcing (pseudo-inverse) and MMSE
For m finite, n large: quasi ML performance. For m,n large with a fixed ratio m/n:
ML receiver performance remains the same. Bad performance of linear receivers when n is close
to m: trade-off between ML complexity and thecomplexity due to a large number of antennas.
DFE like structures: D-BLAST V-BLAST
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Graphes factorielset
communications numériques
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Modèles graphiques• Historique :
• Représentation graphique d’un code :• Gallager (1963), Tanner (1981)• Wiberg (1995) Ajout de variables cachées, interprétation Bayésienne.
• Graphes factoriels (graphes de Tanner généralisés) Application auxfonctions
• Aji & McEliece 1997, 2000 : beaucoup d’algorithmescalculent des marginales d’un produit de fonctions
• Graphes factoriels• Applicable lorsque la solution exacte est d’une complexité
rédhibitoire.• Liens avec d’autres approches : MRF (Markov Random Fields),
réseaux Bayésiens (BP est un cas particulier de SP)
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Marginalisationd’un produit de fonctions
• Fonction de n variables :
• Algorithme de calcul de marginales efficace :• Exploiter les factorisations
• Réutiliser les sommes partielles
g x1,!, xn( ) :S = A1!!!An" R
n marginales gi xi( ) = g x1,!, xn( )~ xi
#
g x1,!, xn( ) = f j X j( )
j!J
"
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Modèle graphiqueGraphe factoriel
•Graphe biparti
•Arbre enraciné en 1
g x1, x
2, x
3, x
4( ) = fa x1( ) fb x
1, x
2( ) fc x2 , x3( ) fd x2, x
4( )
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Graphes et modélisation
• Approche comportementale• Configurations valides des variables
• Approche probabiliste• Représentation d’une probabilité conjointe
• Approche mixte• Exemple : codage canal et
communications numériques.
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Approche comportementaleExemple d’un code bloc linéaire
• Fonction indicatrice d’un ensemble B
• Calcul par une série de tests successifs 1Bx1,!, x
n( ) = x
1,!, x
n( )! B"#
$%
H =
110010
011001
101100
!
"
####
$
%
&&&&
n
! "###
n' k
1code x1,!, xn( ) = x1! x
2! x
5= 0[ ] x2 ! x3! x6 = 0[ ] x1! x3! x4 = 0[ ]
P1^ P
2^!^ P
n[ ]= P
1[ ] P
2[ ]! P
n[ ]
GRAPHES ET MODÉLISATION : APPROCHE COMPORTEMENTALE
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Approche comportementaleExemple des modèles de Markov cachés
• Modèle d’état• Représentation des transitions autorisées
GRAPHES ET MODÉLISATION : APPROCHE COMPORTEMENTALE
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Approche comportementaleDéfinition du comportement par un treillis
• Section i :• Arêtes (i) vers (i-1) : variable
visible• Comportement local :
(si-1,xi,si)• Les contrôles sont les
indicatrices descomportements locaux.
• Treillis : comportement dansl’espace s,x
GRAPHES ET MODÉLISATION : APPROCHE COMPORTEMENTALE
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Approche probabiliste• Représentation de distributions de
probabilités :• Expression des indépendances et des
indépendances conditionnelles.• Exemples :
• Calcul des probabilités a posteriori en codage.• Chaînes de Markov cachées.
GRAPHES ET MODÉLISATION : APPROCHE PROBABILISTE
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Approche probabilisteQuelques factorisations
courantes• Forme directe
• Bayes
• Markov
f x
1, x
2,!, xn( )
f x1, x
2,!, xn( ) = f x j | x1,!, x j!1( )
j=1
n
"
f x1, x
2,!, xn( ) = f x j | x j!1( )
j=1
n
"
GRAPHES ET MODÉLISATION : APPROCHE PROBABILISTE
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Marginalisation récursive de
g x, x1,!, xN !1( )
~ x
" = F1x,X
1( )X1
"#
$%&
'(! FK x,XK( )
XK
"#
$%&
'(
= Fi x,Xi( )~ x
"i=1
K
)
F1x,X
1( )~ x
! = f1x, x
1,!, xL( )G1 x1,X11( )!GL xL ,X1L( )
~ x
!
= f1x, x
1,!, xL( ) G
1x1,X
11( )X11
!"
#$%
&'! GL xL ,X1L( )
X1L
!"
#$%
&'x1 ,!,xL
!
= f1x, x
1,!, xL( ) Gi xi ,X1i( )
~ xi
!i=1
L
("
#$%
&'~ x
!
x1,!, x
L,X11,!,X1L partition de X1
X1,!,XK
est une partition de x1,!, xN !1
g x, x1,!, xN !1( ) = Fi x,Xi( )
i=1
K
"
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Arbre factoriel : arbre decalcul
Le message d’un nœud v vers une arête e est le produit de la fonction locale en v (Id pour nœud detype variable) par le résumé en e des messages reçus par les autres arêtes.
Cas particuliersLes nœuds de type variable de degré deux ne font rien.
Le résumé est superflu pour les fonctions de une seule variable.
Transformations élémentairesArbre factoriel Arbre de calcul
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Arbre factoriel : arbre decalcul
• Pour un arbre, le graphe encode :• la factorisation• L’algorithme de calcul des marginales :
• Récursif du haut vers le bas.• Calcul du bas vers le haut.
• Vision imagée du fonctionnement :• Les nœuds sont des processeurs qui
communiquent entre eux par des canaux (arêtes)• Les messages décrivent des marginales.
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MarginalisationAlgorithme « somme-produit » pour un noeud
• Le nœud i est pris comme racine• Chaque variable feuille envoie la fonction identité à son père.• Chaque fonction feuille envoie la description de f à son père.
• Les nœuds attendent les messages de tous leurs enfantspour calculer le message à destination du père :
• Un nœud de type variable envoie le produit des messagesprovenant de ses enfants.
• Un nœud de type fonction envoie vers son père x le résuméen x du produit par f des messages provenant de sesenfants.
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Marginalisations multiplesAlgorithme « somme-produit »
•L’application de l’algorithmeà l’ensemble des variablesest (très) redondante :
•Il suffit de calculer deuxmessages par arête.
•Une seule règle : le messaged’un nœud v vers une arête eest le produit de la fonctionlocale en v (Id pour nœud detype variable) par le résumé ene des messages reçus par lesautres arêtes.
mx! f x( ) = mh!x x( )w"n x( )\ f{ }#
mf!x x( ) = f X( ) mw! f w( )w"n f( )\ x{ }#
$
%
&&&&
'
(
)))))~ x
*
X = n f( )
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Algorithme « somme-produit ».
Exemple.• On commence aux feuilles
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Quelques cas particulierscélèbres
Algorithme SP en signal, IA,communication
• Sur des chaînes :• Algorithme « forward/backward » (BCJR, MAP)• Algorithme de Viterbi bidirectionnel• Lisseur de Kalman
• Sur des arbres :• Algorithme « Belief Propagation » de Pearl.• Décodeurs itératifs (turbo-codes, LDPC). Instance
de l’algorithme BP sur un graphe à cycles longs.• Certains algorithmes de FFT
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Algorithme MAP[BCJR, forward/backward …]
•Loi conjointe sachantl’observation :
•Probabilités proportionnellesaux fonctions marginales :
•Application possible del’algorithme « somme-produit »
gy u, s, x( ) = Ti si!1, xi ,ui , si( )
i=1
n
" f yi | xi( )i=1
n
"
p ui | y( )! gy u, s, x( )~ui
"
Entrées u
Etats s liés parcontrôles locaux
Sorties x
Observations
Modèle mixte comportemental/probabiliste
Ti si!1, xi ,ui , si( )
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Algorithme MAP en tant quecas particulier de « somme-
produit »Algorithme générique
mf!x x( ) = f X( ) mw! f w( )w"n f( )\ x{ }#
$
%
&&&&
'
(
)))))~ x
*
X = n f( )
mTi!u
i
ui
( ) "!! ! ui
( )
msi!T
i+1si
( ) "!! " si
( ), forward
msi!T
i
si
( ) "!! # si
( ), backward
mxi!T
i
xi
( ) "!! $ xi
( )
! si
( ) = Tisi!1, x
i,u
i, s
i( )! s
i!1( )~ s
i
" " xi
( )
# si!1( ) = T
isi!1, x
i,u
i, s
i( )# s
i( )
~ si!1
" " xi
( )
$ ui
( ) = Tisi!1, x
i,u
i, s
i( )! s
i!1( )~u
i
" " xi
( )
Algorithme MAP
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AlgorithmeMAP, BCJR,
forward/backward …• Etape 1 :
initialisation• Initialisation des
feuilles.• Les nœuds de type
variable d’ordre 2ne font quetransmettre.
• Deux récurrencesparallèles :
• directe etrétrograde.
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AlgorithmeMAP, BCJR,
forward/backward …• Etape 2 :
• Propagation desmessages directs etrétrogrades
! si
( ) = msi!T
i+1si
( ) = ! e( )e"E
isi( )# " e( )
# si$1( ) = m
si!T
i
si
( ) = # e( )e"E
isi$1( )# " e( )
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Turbo-synchronisation dephase
• Observation d’un mot du code :
yk = xke
i!k + bk , k = 0!N "1
Mot code
Observation
Phase Bruit
p a,! | y( )" p a( ) p !( ) p y | a,!( )
= x #C[ ] p !( ) p y | x,!( )
But : sachant les observations, retrouvez les symboles :
A priori Vraisemblance
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Turbo-synchronisation dephase
•Observation
•A priori•Phase (Markov caché)
•Code
Bornes Cramer Rao Bayésiennesen ligne et hors ligne
yk = xkei!k + nk , k = 0!N "1
xk version codée des symboles ak
!k = !k"1
+ wk , wk! N 0,#w
2( ) i.e. p !( )
1Cx1,!, x
n( ) = x
1,!, x
n( )!C"#
$%
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Turbo-synchronisation de phaseModèle graphique du problème
• Complexité rédhibitoire dela solution optimale
• Le graphe présente descycles.
• La résolution parpropagation de croyanceitérative (BP) fournit unebonne approximation.
p a,! | y( )" x #C[ ] p !0( ) p !k |!k$1( )
k=0
N $1
% p yk | xk ,!k( )k=0
N $1
%
Factorisation de la trajectoirede phase.
Factorisation de la vraisemblance.
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Turbo-estimation de phase Mise en œuvre pratique
• Voies vers une implantation del’algorithme
• Discrétiser la phase• Paramétrer les densités de probabilités.• Méthodes particulaires.
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Bloc ou séquentiel ?• Traitements « par bloc » :
• Complexes et performants• Naturels dans les normes récentes• Hypothèses rigides (paramètres constants, …)
• Traitements séquentiels type algorithmes adaptatifs:
• Simples et peu performants• Naturels en « flot continu »• Hypothèses peu contraignantes : robustesse
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INPG TST 3A J.M. Brossier - LIS
Trends Iterative algorithms
Turbo-codes Turbo everything
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INPG TST 3A J.M. Brossier - LIS
Coding Emitted/Received word:
Decoder
X = X1,!,Xn
( )!Y = Y1,!,Y
n( ) received word
!= !1,!,!n
( ) reliability
"#$$
%$$
Error vector: Zm =Y & Xm
Weight of the error vector: W Zm( ) = Z
i
m
i=1
n
'
Incomplete decoder: codeword Xm = X1
m ,!,Xn
m( ) close (Hamming) to Y = Y1,!,Yn
( ) / W Zm( )!
dmin "1
2
#
$##
%
&%%
- one word if W Zm( )!
dmin "1
2
#
$##
%
&%% else no word
- Right decision if the number of errors is less than dmin "1
2
#
$##
%
&%%
Complete decoder: minmW Y ' X
m( ) can provide a codeword even if the number of errors is greater than dmin "1
2
#
$##
%
&%%
Complete soft decoder: minmW!Y ' X
m( ) with analog weight W!Zm( ) = !
iZi
m
i=1
n
(
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INPG TST 3A J.M. Brossier - LIS
Chasing Hard decoding: XA
Use the hard decoder to find asmall number of words and selectthe minimum analog distance Modify Y using test sequences: Y’ Select minimum Euclidian
distance
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INPG TST 3A J.M. Brossier - LIS
Reliability For a bit j
! dj( ) = logP aj = +1 R( )P aj ="1 R( )
#
$
%%%%%
&
'
(((((
P aj = ±1 R( ) = P E = CiR( )
Ci)Sj
±1
* with Sj±1 set of word / cj
i = ±1
! dj( ) = log
P R E = Ci( )
Ci)Sj
+1
*
P R E = Ci( )
Ci)Sj
"1
*
#
$
%%%%%%%%%
&
'
((((((((((
with P R E = Ci( ) =
1
2!"
#
$%%%
&
'((((
n
exp "R"Ci
2
2"2
#
$
%%%%%%
&
'
(((((((
! dj( )+1
2"2R"C"1( j )
2
" R"C+1( j )2( )
C±1( j ) codewords in Sj
±1 at minimum Euclidian distance / R
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INPG TST 3A J.M. Brossier - LIS
Product code
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INPG TST 3A J.M. Brossier - LIS
Iterative decoding
Decision:
R= r1,!,rn
( ) = E+G
E : transmitted codeword
G : gaussian noise
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INPG TST 3A J.M. Brossier - LIS
Gain de codage
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INPG TST 3A J.M. Brossier - LIS
Turbo Concatenation of SISO (soft input soft
output) algorithms can be extended toall blocks (equalization, multi-userdetection, synchronization, …)