Estimation and detection from coded signals

1
Estimation and detection from coded signals
Joint research - UGent (N. Noels, H. Wymeersch,
H. Steendam, M. Moeneclaey) - UCL (C. Herzet, L.
Vandendorpe)
Presented by Marc Moeneclaey, UGent - TELIN
dept. Marc.Moeneclaey_at_telin.UGent.be
2
Outline

• Coding linear modulation, AWGN channel,
  unknown parameter q (carrier phase, ...)
• MAP detection of information bits q known q
  unknown
• Low-complexity alternatives to MAP detection for
  unknown q iterative ML estimation of q (EM
  algorithm) simplified sum-product algorithm
• Numerical results
• Conclusions

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Coding linear modulation, AWGN channel
codedsymbols
pulse h(t)
infobits
r(t)
encoding,interleaving,mapping
AWGNchannel
linear modulation
b
a
q
b ? a c(b) turbo code, LDPC code, BICM,
.... h(t) square-root Nyquist pulse q unknown
parameter (carrier phase, time delay, freq.
offset, gain)
received signal
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MAP detection, q known
MAP detection (on individual infobits) achieves
minimum BER
r vector representation of r(t)
Assuming q q0 is known, bit APP p(bkr) is
given by
(many terms !)
Efficient computation of APPs sum-product (SP)
algorithm message-passing in factor graph
(FG))
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Factor graph for known q
g(zk,ak,q0)
g(z1,a1,q0)
g(zK,aK,q0)
aK
ak
a1
Encoding, interleaving, mapping
b1
b?
bL
g(zk,ak,q0)
channel observation
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Factor graph for known q
g(zk,ak,q0)
g(z1,a1,q0)
g(zK,aK,q0)
aK
ak
a1
Encoding, interleaving, mapping
b1
b?
bL
g(zk,ak,q0)
channel observation
extrinsic information on ak
pe(ak)
information bit APP
p(b?r)
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Factor graph for known q
Because of cycles in FG, MAP detection is
iterative
ak1
ak3
ak2
b?1
b?2
b?3
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Factor graph, q known
g(zk,ak,q0)
g(z1,a1,q0)
g(zK,aK,q0)
aK
ak
a1
Encoding, interleaving, mapping
b1
b?
bL
g(zk,ak,q0)
channel observation
extrinsic information on ak
pe(n)(ak)
n iteration index of MAP detectoriterate
until convergence
information bit APP
p(n)(b?r)
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Receiver structure, q known
zk
conventionalMAP detector
x
r(t)
h(-t)
kT
(SP algorithm, iterate until convergence)
exp(-jqo)
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MAP detection, q unknown
When q is unknown (uniform distrib.), bit APP is
given by
? conventional MAP detectorhard to compute,
because of integration over q
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Factor graph, q unknown
messages are functions of q
downward messages involve integration over q
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
Encoding, interleaving, mapping
b1
b?
bL
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Alternatives to exact MAP detectionwhen q is
unknown
Alternative 1 ML parameter estimation Provide
estimate of q to conventional MAP detector
Alternative 2 Simplified sum-product
algorithm Approximation of q-dependent messages
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Alternative 1 ML parameter estimation
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Receiver structure when estimating q
Strategy use conventional MAP detector, but
provide estimate (instead of
correct value) of q
zk
conventionalMAP detector
x
r(t)
h(-t)
kT
Estimation of q
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ML parameter estimation
ML estimation of q
For long observation intervals, mean-square error
(MSE) of ML estimate converges to Cramer-Rao
lower bound (CRB) on MSE
Computation of CRB is hard when data symbols are
not a priori known to receiver. Simpler but
less tight bound modified CRB (MCRB) (assumes
data symbols are known to receiver)
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Cramer-Rao bound
average over (coded)symbol sequence a (many
terms !!)
analytical difficulty Ea. inside ln(.), so
that ln(p(rq)) is
not a simple function of r and q
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Modified Cramer-Rao bound
Er,a. outside ln(.) ? MCRB easier to evaluate
than CRB,
when ln(p(raq)) is simple function of a and r
(e.g.,
linear modulation on AWGN channel)
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MCRB for phase estimation
r aexp(jq) w
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CRB for phase estimation uncoded transmission
(i.i.d. symbols)
?
1-D summation
1-D numerical integration
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CRB for phase estimation coded transmission
r aexp(jq) w ?
marginal symbol APPs(from MAP detector
operating on r)
1-D summation per symbol
Monte Carlo simulation
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CRB, MCRB numerical results
BPSK
QPSK
(phase estimation)
? code properties should be exploited
during estimation
Large SNR CRB ? MCRB Small SNR CRBuncoded gt
CRBcoded gt MCRB
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Computation of ML estimate
Direct application of ML estimation is
complicated
many terms !
? compute ML estimate iteratively (EM algorithm)
soft decision (SD) on symbols
symbol APPs computed by MAP detector
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EM algorithm symbol APP computation
outer iteration index i (EM algorithm), inner
iteration index n (MAP detector)
for each i MAP detector is reset and
iterated until convergence (n ? ?)
ak
aK
a1
Encoding, interleaving, mapping
symbol APP
? SD
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Receiver structure for EM algorithm
inner iterations (index n)
r(t)
zk
conventionalMAP detector
x
h(-t)
kT
z
extr. probs
EM compute
k 1, ..,K
outer iterations (index i)
symbol APP
? SD
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Reduced-complexity EM algorithm
For each EM iteration, MAP detector is iterated
till convergence ? computational complexity too
high Solution merging of EM iterations and
MAP detector iterations.For each EM iteration,
only one MAP detector iteration is
performed,without resetting MAP detector ?
reduced complexity (at expense of reduced
convergence speed)
symbol APP
? SD
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EM algorithm exploiting code properties
partial (or no) exploitation of code properties
? degradation of MSE
does not exploit code properties
uses infobit APPs only, assumes parity bits are
uncoded
uses infobit APPs and parity bit APPs
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EM algorithm phase estimation (1/2)
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EM algorithm phase estimation (2/2)
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EM algorithm timing estimation (1/2)
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EM algorithm timing estimation (2/2)
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Alternative 2 Simplified sum-product algorithm
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Simplified sum-product algorithm
Factor graph corresponding to unknown q
messages are functions of q
downward messages involve integration over q
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
Encoding, interleaving, mapping
33
Simplified sum-product algorithm
Messages depending on q are approximated by(n
iteration index of MAP detector)
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
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Simplified sum-product algorithm
Messages depending on q are approximated by(n
iteration index of MAP detector)
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
35
Simplified sum-product algorithm
Messages depending on q are approximated by(n
iteration index of MAP detector)
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
36
Simplified sum-product algorithm
Messages depending on q are approximated by(n
iteration index of MAP detector)
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
37
Simplified sum-product algorithm
Messages depending on q are approximated by(n
iteration index of MAP detector)
q

g(zk,ak,q)
g(z1,a1,q)
aK
ak
a1
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Simplified sum-product algorithm
(n1)-th iteration of MAP detector
aK
ak
a1
Encoding, interleaving, mapping
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Simplified sum-product algorithm
(n1)-th iteration of MAP detector
aK
ak
a1
Encoding, interleaving, mapping
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Simplified sum-product algorithm maximization
of p(n)(rq)
Similar to ML equation p(a) is replaced by
? EM algorithm to maximize p(n)(rq) iteratively
? SD
symbol APP
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Simplified sum-product algorithm receiver
stucture
for each n iterate EM algorithm until
convergence (i ? ?)
outer iterations (index n)
zk
conventionalMAP detector
x
r(t)
h(-t)
kT
z
extr. probs.
Compute
(i ? ?)
inner iterations (index i)
? SD
symbol APP
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Simplified sum-product algorithm estimator
performance
VV
BICM rate 1/2 conv. code, 8-PSK (set
part.)1000 symbols, Eb/N0 5 dB
carrier phase unknown
SP
EM
MCRB
MAP detector iterations
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Simplified sum-product algorithm BER performance
BICM rate 1/2 conv. code, 8-PSK (set
part.)1000 symbols, Eb/N0 5 dB
VV
BER
EM
carrier phase unknown
SP
perfect synchr.
MAP detector iterations
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Conclusions
Two alternatives for MAP detection when q is
unknown

• Iterative ML estimation of q by means of EM
  algoritm

symbol APP during i-th EM iteration
one MAP detector iterationfor each EM iteration

• Simplified SP algorithm (combined with EM
  algorithm)

symbol APP during i-th EM iteration of n-th MAP
decoder iteration
EM algorithm iterated till convergencefor each
MAP detector iteration
Both algorithms yield similar MSE and BER after
convergence, but simplified SP algorithm requires
considerably less MAP decoder iterations to
converge.