Ger manAerospace Center
Transfer Chart Analysis of Iterative OFDM Receivers with Data Aided Channel Estimation
Stephan Sand, Christian Mensing, and Armin Dammann
German Aerospace Center (DLR)
3rd COST 289 Workshop, Aveiro, Portugal, 12th July
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Outline
System model
Frame structure
Channel estimation (CE)
Extrinisic information transfer (EXIT) Charts
Bit-error rate transfer (BERT) Charts
Comparison of BERT and EXIT charts
Simulation results
Conclusions & outlook
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System Model: OFDM System with Iterative Receiver
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Frame Structure
Burst transmission
Rectangular grid
Pilot distance in
frequency direction: Nl=10
Pilot distance between
OFDM symbols: Nk=10
1
1
Nc
Ns
Nk
Nl
data symbol
pilot symbol
frequency
time
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Channel Estimation (CE)
Initial iteration (i=0) only pilot symbols: Pilot aided channel estimation (PACE)
Afterwards (i>0) additionally data estimates:Pilot and data aided iterative channel estimation (ICE)
Localized estimates for the channel transfer function at pilot or data symbol positions, i.e., the least-squares (LS) estimate:
Replacing unknown Sn,l by the expectations (soft symbol and soft variance):
*, , , ,
, ,( ) ,2, ,,
n l n l n l n ln l i n l
n l n ln l
R R S ZH H
S SS
, ,( )
*, , ,( )
, ,( )
n l i
n l n l in l i
S
R SH
E
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Channel Estimation (CE)
Filtering localized estimates yields final estimates of the complete CSI:
where ωn’,l’,n,l,(i) is the shift-variant 2-D impulse response of the filter. Tn,l is the set of initial estimates that are actually used for filtering.
Filter design:
Knowledge of the Doppler and time delay power spectral densities
(PSDs)
optimal 2-D FIR Wiener filter
Separable Doppler and time delay PSDs
two cascaded 1-D FIR Wiener filters perform similar than 2-D
FIR Wiener filter
,
, ,( ) ', ', , ,( ) ', ',( ) ,', '
ˆ , , 1, , , 1, , ,n l
n l i n l n l i n l i n l c sn l
H H n N l N
T
T P D
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EXIT Charts
Benefits
Mutual information flow between inner and outer receiver
Independent computation for inner and outer receiver
Arbitrary combination of inner and outer receiver
Prediction of “turbo cliff“ position and BER possible
Assumptions
Log-likelihood ratio values (L-values): Gaussian distributed random variables
Interleaver depth large: uncorrelated L-values
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EXIT Charts
A-priori L-values: independent Gaussian random variable
Probability density function of LA
A-priori mutual information
monotonically increasing, reversible function of σA
A A AL c n 2
2A
A 2( , )A A An N
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1 2 ( | )( ; ) ( | ) log
2 ( | 1) ( | 1)A
A A Ac A A
p C cI C L p C c d
p C p C
22
2
( )2
2
( | )2
A
A
c
AA
ep C c
( ; )A AI C L
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EXIT Charts
Steps for EXIT chart computation
1. Variance of a-priori L-values from a-priori information
2. A-priori L-value
3. Input a-priori L-value and simulated “channel”-value to component
4. Measure extrinsic information at output of component with histogram estimator
12( ; ) ( ; ) 1 ( | ) log (1 )A A A A A EI C L I C L p C c e d
A A AL c n
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1 2 ( | )( ; ) ( | ) log
2 ( | 1) ( | 1)E
E E Ec E E
p C cI C L p C c d
p C p C
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BERT Charts
Benefits
BER flow between inner and outer receiver
Independent computation for inner and outer receiver
Arbitrary combination of inner and outer receiver
Prediction of “turbo cliff“ position and BER possible
Assumptions
Log-likelihood ratio values (L-values): Gaussian distributed random variables
Interleaver depth large: uncorrelated L-values
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BERT Chart
A-priori L-values: independent Gaussian random variable
Probability density function of LA
A-priori BER
monotonically increasing, reversible of σA
A A AL c n 2
2A
A 2( , )A A An N
0
1
1( ; ) ( | )
2A A Ac
P C L c p C c d
22
2
( )2
2
( | )2
A
A
c
AA
ep C c
( ; )A AP C L
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BERT Charts
Steps for BERT chart computation
1. Variance of a-priori L-values from a-priori BER
2. A-priori L-value
3. Input a-priori L-value and simulated “channel”-value to component
4. Measure extrinsic BER at output of component by hard decision
1 1( ; ) ( ; ) erfc
2 8A
A A A A AP C L P C L
A A AL c n
,
1
1 sgn( )1( ; )
2
Nn E n
E En
c LP C L
N
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Comparison of EXIT and BERT Charts
BERT chart computation
1. Variance of a-priori L-values
2. A-priori L-value
3. Input a-priori L-value and simulated “channel”-value to component
4. Measure extrinsic BER / information at output of component
A A AL c n
1( ; )A A AP C L 1( ; )A A AI C L
1
2
1( ; ) ( | )
2
2 ( | ) log
( | 1) ( | 1)
E E Ec
E
E E
I C L p C c
p C cd
p C p C
EXIT chart computation
,
1
1 sgn( )1( ; )
2
Nn E n
E En
c LP C L
N
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Simulation Results: Scenario
Bandwidth 4.004 MHz
Subcarriers 1001
FFT length 1024
Sampling duration Tspl 3.1 ns
Guard interval TGI 205 Tspl
Subcarrier spacing Δf 4 kHz
OFDM symbols / Frame 101
Modulation QPSK, linear mapping
Coding Conv. code, R=1/2, (23,37)
Information bits 99986
Interleaver length 199980
Interleaver type random
Pilot spacing frequency 10
Pilot spacing time 10
fD,max 0.025Δf ≈ 100 Hz
τmax 20 μs
τrms 0.001τmax
time…
Exponential Channel model with Jakes’ Doppler fading
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Simulation Results: AWGN Channel BERT
Acronyms:
PCE: perfect channel estimation
DMOD: demodulator
DCOD: decoder
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Simulation Results: AWGN Channel EXIT
Acronyms:
PCE: perfect channel estimation
DMOD: demodulator
DCOD: decoder
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Simulation Results: Exponential ChannelHistogram of L-values at demodulator output
No Gaussian
distribution of
L-values
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Simulation Results: Exponential Channel BERT
Acronyms:
PCE: perfect channel estimation
ICE: iterative channel estimation
DMOD: demodulator
DCOD: decoder
BERT: DCOD too
pessimistic due to
Gaussian
assumption!
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Simulation Results: Exponential Channel EXIT
Acronyms:
PCE: perfect channel estimation
ICE: iterative channel estimation
DMOD: demodulator
DCOD: decoder
ICE system
trajectory dies out:
independence
assumption violated
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Simulation Results: Exponential Channel BER Plot
Acronyms:
PACE: pilot aided channel estimation
PCE: perfect channel estimation
ICE: iterative channel estimation
DMOD: demodulator
DCOD: decoder
@ 7dB:
ICE reaches PCE
after 5 iterations
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Conclusions & Outlook
Iterative receiver including pilot and data aided channel estimation
BERT and EXIT charts:
simpler computation of BERT charts
direct prediction of BERs in BERT charts
Simulation results indicate:
BERT charts too pessimistic due to Gaussian assumption of decoder
EXIT charts more robust against Gaussian assumption
ICE reaches PCE after a few iterations
Outlook:
A-posteriori feedback in ICE to improve convergence
Thank you!