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
frequency
Rectangular grid
1
Nc
1
Pilot distance in
time
frequency direction: Nl=10
Nk
Pilot distance between
OFDM symbols: Nk=10
Ns
Nl
data symbol
pilot symbol
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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:
H n,l ,(i ) 
Rn,l
S n ,l

Rn,l Sn*,l
S n ,l
2
 H n ,l 
Z n ,l
S n ,l
Replacing unknown Sn,l by the expectations
(soft symbol and soft variance):
H n ,l ,(i ) 
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Rn ,l Sn*,l ,(i )
ES
n ,l ,( i )
5
Channel Estimation (CE)
Filtering localized estimates yields final estimates of the complete CSI:
Hˆ
  
H
, T  P  D, n  1, , N , l  1, , N ,
n ,l ,( i )
n ',l 'Tn ,l
n ',l ', n ,l ,( i )
n ',l ',( i )
n,l
c
s
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
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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
LA   Ac  nA
nA 
N ( A , A2 )
A 
 A2
2
Probability density function of LA

p A ( | C  c) 
e
( 
 A2
2
c )2
2
2 A
2 A
A-priori mutual information

1
2 p A ( | C  c)
I A (C ; LA )     p A ( | C  c)  log 2
d
2 c 1 
p A ( | C  1)  p A ( | C  1)
I A (C; LA ) monotonically increasing, reversible function of σA
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EXIT Charts
Steps for EXIT chart computation
1.
Variance of a-priori L-values from a-priori information
A 
2.
I A1 (C ; LA )
A-priori L-value
I A (C ; LA )  1 



pE ( | C  c)  log 2 (1  e  )d 
LA   Ac  nA
3.
Input a-priori L-value and simulated “channel”-value to component
4.
Measure extrinsic information at output of component with histogram
estimator

1
2 pE ( | C  c)
I E (C ; LE )     pE ( | C  c)  log 2
d
2 c 1 
pE ( | C  1)  pE ( | C  1)
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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
LA   Ac  nA
nA 
N ( A , A2 )
A 
 A2
2
Probability density function of LA

p A ( | C  c) 
e
( 
 A2
2
c )2
2
2 A
2 A
A-priori BER
0
1
PA (C ; LA )     c  p A ( | C  c)d 
2 c 1 
PA (C; LA ) monotonically increasing, reversible of σA
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BERT Charts
Steps for BERT chart computation
1.
Variance of a-priori L-values from a-priori BER
 A  PA1(C; LA )
2.
1
 
PA (C; LA )  erfc  A 
2
 8
A-priori L-value
LA   Ac  nA
3.
Input a-priori L-value and simulated “channel”-value to component
4.
Measure extrinsic BER at output of component by hard decision
1 N 1  cn  sgn( LE ,n )
PE (C; LE )  
N n1
2
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Comparison of EXIT and BERT Charts
EXIT chart computation
BERT chart computation
1.
Variance of a-priori L-values
 A  I A1(C; LA )
 A  PA1(C; LA )
2.
A-priori L-value
LA   Ac  nA
Input a-priori L-value and simulated “channel”-value to component
4. Measure extrinsic BER / information at output of component
3.
1 N 1  cn  sgn( LE ,n )
PE (C; LE )  
N n1
2

1
I E (C ; LE )     pE ( | C  c)
2 c1 
 log 2
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2 pE ( | C  c)
d
pE ( | C  1)  pE ( | C  1)
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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
fD,max
0.025Δf ≈ 100 Hz
Interleaver length
199980
τmax
20 μs
Interleaver type
random
τrms
0.001τmax
Pilot spacing frequency
10
Pilot spacing time
10
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Exponential Channel model with
Jakes’ Doppler fading
…
time
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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 Channel
Histogram 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
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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!
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