Estimation and detection from coded signals
description
Transcript of Estimation and detection from coded signals
TELIN
Estimation and detection from coded signals
Presented by Marc Moeneclaey, UGent - TELIN [email protected]
Joint research :- UGent (N. Noels, H. Wymeersch, H. Steendam, M. Moeneclaey) - UCL (C. Herzet, L. Vandendorpe)
TELINOutline
• Coding + linear modulation, AWGN channel, unknown parameter (carrier phase, ...)
• MAP detection of information bits* known* unknown
• Low-complexity alternatives to MAP detection for unknown * iterative ML estimation of (EM algorithm)* simplified sum-product algorithm
• Numerical results• Conclusions
TELIN
Coding + linear modulation, AWGN channel
encoding,interleaving,mapping
infobitscodedsymbols linear
modulationAWGNchannel
r(t)
b a
received signal )t(w)kTt(hae)t(rk
kj
b a = (b) : turbo code, LDPC code, BICM, ....h(t) : square-root Nyquist pulse : unknown parameter (carrier phase, time delay, freq. offset, gain)
pulse h(t)
TELINMAP detection, known
MAP detection (on individual infobits) achieves minimum BER
)|b(pmaxargb̂ k}1,0{bkk
r
Assuming = 0 is known, bit APP p(bk|r) is given by
kb:
0k )),(|(p)|b(pb
barr
r : vector representation of r(t)
Efficient computation of APPs : sum-product (SP) algorithm + message-passing in factor graph (FG))
(many terms !)
TELINFactor graph for known
g(zk,ak,0)g(z1,a1,0) g(zK,aK,0)
a1 ak aK
Encoding, interleaving, mapping
b1 b bL
g(zk,ak,0) channel observation
j
k*k
00kk ezaRe
N2exp),a,z(g
TELINFactor graph for known
g(zk,ak,0)g(z1,a1,0) g(zK,aK,0)
a1 ak aK
Encoding, interleaving, mapping
b1 b bL
g(zk,ak,0)
p(b|r)
channel observation
extrinsic information on ak
information bit APP
j
k*k
00kk ezaRe
N2exp),a,z(g
pe(ak)
TELINFactor graph for known
Because of cycles in FG, MAP detection is iterative
ak1 ak3ak2
b1 b2 b3
TELINFactor graph, known
g(zk,ak,0)g(z1,a1,0) g(zK,aK,0)
a1 ak aK
Encoding, interleaving, mapping
b1 b bL
g(zk,ak,0)
p(n)(b|r)
n : iteration index of MAP detectoriterate until convergence
channel observation
extrinsic information on ak
information bit APP
j
k*k
00kk ezaRe
N2exp),a,z(g
pe(n)(ak)
TELINReceiver structure, known
r(t) h*(-t)kT
exp(-jo)
“conventional”MAP detector
xb̂zk
(SP algorithm, iterate until convergence)
TELINMAP detection, unknown
When is unknown (uniform distrib.), bit APP is given by
kb:k d)),(|(p)|b(p
bbarr
conventional MAP detector
hard to compute, because of integration over
TELINFactor graph, unknown
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
Encoding, interleaving, mapping
b1 b bL
=
messages are functions of
downward messages involve integration over
TELIN
Alternatives to exact MAP detectionwhen is unknown
Alternative 1 : ML parameter estimation Provide estimate of to conventional MAP detector
Alternative 2 : Simplified sum-product algorithmApproximation of -dependent messages
))(ˆ,|b(p
d)|(p),|b(p)|b(p
k
kk
rr
rrr
))(ˆ()|(p rr
TELIN
Alternative 1 : ML parameter estimation
TELINReceiver structure when estimating
Strategy : use conventional MAP detector, but provide estimate (instead of correct value) of
r(t) h*(-t) x
Estimationof
)ˆjexp(
conventionalMAP detector
b̂
kT
zk
TELINML parameter estimation
ML estimation of : )|(pmaxargˆML
r
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)
TELINCramer-Rao bound
);(plnE);(plnECRB 2
221 rr rr
);|(pE)(p);|(p);(p araarr aa
average over (coded)symbol sequence a(many terms !!)
analytical difficulty : Ea[.] “inside” ln(.), so that ln(p(r;)) is not a simple function of r and
TELINModified Cramer-Rao bound
);|(plnE);|(plnEMCRB 2
2
,
2
,1 arar arar
Er,a[.] “outside” ln(.) MCRB easier to evaluate than CRB, when ln(p(r|a;)) is simple function of a and r (e.g., linear modulation on AWGN channel)
TELINMCRB for phase estimation
r = aexp(j) + w Iww2
N][E 0T
k
k*k
0
)]jexp(raRe[N2);|(pln ar
kk
*k
02
2
)]jexp(raRe[N
2);|(pln ar
0
s
k
2ka
02
2
,1
NKE2|a|E
N2);|(plnEMCRB
k
arar
TELIN
CRB for phase estimation :uncoded transmission
k
k )a(p)(p a
k
kkak
k )];a|r(p[E);r(p);(pk
r
2
kkar1 )];a|r(p[ElnEKCRB
kk
][Ekr : 1-D numerical integration
(i.i.d. symbols)
][Eka : 1-D summation
TELIN
CRB for phase estimation :coded transmission
araarr );|(p);|(pln);(pln
kk
*k
0
)]jexp(raIm[N2);|(pln ar
kk
*k
0
)]jexp(r);(a~Im[N2);(pln rr
ka
kkkk );|a(pa);|(pa);(a~ rrara
marginal symbol APPs(from MAP detector operating on r)
21 );(plnECRB rr
][E r : Monte Carlo simulation
][E |akr : 1-D summation per symbol
r = aexp(j) + w
TELINCRB, MCRB : numerical results
Large SNR : CRB MCRB Small SNR : CRBuncoded > CRBcoded > MCRB
s
0
LE2NMCRB
1.0
1.5
2.0
2.5
3.0
-6 -4 -2 0 2 4 6 8 10 12
Es/N0 (dB)
CR
B/M
CR
B
Uncoded, M=2, L=1001Uncoded, M=4, L=1001Turbo, M=2, I=334, L=1002, r=1/3Turbo, M=4, I=L=1001, r=1/2LDPC, M=2, L=1000, r=1/2LDPC, M=4, L=500, r=1/2
M-PSKPhase estimation
(phase estimation)
code properties should be exploited during estimation
QPSKBPSK
TELINComputation of ML estimate
Direct application of ML estimation is complicated :
b
barr )),(|(p)|(p many terms !
compute ML estimate iteratively (EM algorithm) :
kk
*)i(k
)i()1i( za~argˆ,|),|(pEmaxargˆ rar
soft decision (SD) on symbols]ˆ,|a[Ea~ )i(k
)i(k r
symbol APPs computed by MAP detector
)i(
iMLˆlimˆ
TELINEM algorithm : symbol APP computation
a1 ak aK
Encoding, interleaving, mapping
)ˆ,a,z(g )i(kk )ˆ,a,z(g )i(
KK )ˆ,a,z(g )i(11
for each i : MAP detector is reset and iterated until convergence (n )
)a(p k)n,i(
e
)a(p)ˆ,a,z(g)ˆ,|a(p k),i(
e)i(
kk)i(
krsymbol APP : )i(
ka~ SD
)ˆ,a,z(g )i(kk
outer iteration index i (EM algorithm), inner iteration index n (MAP detector)
TELINReceiver structure for EM algorithm
r(t)h*(-t) x
),(ˆ b
)i(ˆje
)1i(ˆ EM : compute
conventionalMAP detectorkT
zk
z
outer iterations (index i))a(plim k
)n,i(en
extr. probs
)a(p)ˆ,a,z(g)ˆ,|a(p k),i(
e)i(
kk)i(
krsymbol APP : )i(
ka~ SD
inner iterations (index n)
k = 1, ..,K
TELINReduced-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)
)a(p)ˆ,a,z(g)ˆ,|a(p k)1,i(
e)i(
kk)i(
k rsymbol APP : )i(ka~ SD
TELINEM 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 APPs1
10
0 0.5 1 1.5 2 2.5 3Es/No (dB)
MSE
/MC
RB
, CR
B/M
CR
B
V&V
sDD(i) 20it
sDD(ii) 15it
CRB
CRBuncoded
Turbo code, rate 1/2, 4-PSK
Phase estimation
TELINEM algorithm : phase estimation (1/2)
1.E-03
1.E-02
1.E-01
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Eb/N0 (dB)
MSE MCRB
V&VEM (10 it)
Turbo code (I=240, rate 1/3), BICM, QPSKPhase estimation
TELINEM algorithm : phase estimation (2/2)
1.E-03
1.E-02
1.E-01
1.E+00
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Eb/N0 (dB)
FER
perfect synchronizationV&VEM (10 it)
Turbo code (I=240, rate 1/3), BICM, QPSKPhase estimation
TELINEM algorithm : timing estimation (1/2)
1.E-04
1.E-03
1.E-02
1.E-01
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Eb/N0 (dB)
MSE
MCRBO&MEM with 1 O&M init
Turbo code (I=240, rate 1/3), BICM, QPSKNo pilot symbols
timing estimation(assume perfect frame synchronization)
TELINEM algorithm : timing estimation (2/2)
1.E-03
1.E-02
1.E-01
1.E+00
-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5
Eb/N0 (dB)
FER
perfect synchronizationO&MEM (10 it)
Turbo code (I=240, rate 1/3), BICM, QPSKTiming estimation
TELIN
Alternative 2 : Simplified sum-product algorithm
TELINSimplified sum-product algorithm
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
Encoding, interleaving, mapping
=
messages are functions of
downward messages involve integration over
Factor graph corresponding to unknown
TELINSimplified sum-product algorithm
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
k
k)n(
e)n(
ek
kk )a(p)(p),a,z(g);|(p aar
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
=
)a(p k)n(
e
Messages depending on are approximated by(n : iteration index of MAP detector)
)ˆ( )n(
TELINSimplified sum-product algorithm
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
k
k)n(
e)n(
ek
kk )a(p)(p),a,z(g);|(p aar
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
=
)a(p k)n(
e
Messages depending on are approximated by(n : iteration index of MAP detector)
)ˆ( )n(
TELINSimplified sum-product algorithm
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
k
k)n(
e)n(
ek
kk )a(p)(p),a,z(g);|(p aar
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
=
)a(p k)n(
e
)|(p )n( r
Messages depending on are approximated by(n : iteration index of MAP detector)
)ˆ( )n(
TELINSimplified sum-product algorithm
Messages depending on are approximated by(n : iteration index of MAP detector)
)ˆ( )n(
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
k
k)n(
e)n(
ek
kk )a(p)(p),a,z(g);|(p aar
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
=
)a(p k)n(
e
)ˆ( )n(
)|(p )n( r
TELINSimplified sum-product algorithm
Messages depending on are approximated by(n : iteration index of MAP detector)
)ˆ( )n(
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
k
k)n(
e)n(
ek
kk )a(p)(p),a,z(g);|(p aar
g(zk,ak,)g(z1,a1,) g(zK,aK,)
a1 ak aK
=
)a(p k)n(
e
)ˆ( )n(
)|(p )n( r
)ˆ,a,z(g )n(kk
TELINSimplified sum-product algorithm
a1 ak aK
Encoding, interleaving, mapping
)ˆ,a,z(g )n(kk
(n+1)-th iteration of MAP detector
TELINSimplified sum-product algorithm
a1 ak aK
Encoding, interleaving, mapping
)ˆ,a,z(g )n(kk )a(p k
)1n(e
(n+1)-th iteration of MAP detector
TELIN
Simplified sum-product algorithm :maximization of p(n)(r|)
a
aarr )(p);|(pmaxarg)|(pmaxargˆ )n(e
)n()n(
Similar to ML equation : p(a) is replaced by )(p )n(e a
EM algorithm to maximize p(n)(r|) iteratively )i,n(
i
)n( ˆlimˆ
kk
*)i,n(k
k
)i,n(kk
)1i,n( za~arg]ˆ,|,a,z(g[Emaxargˆ r
symbol APP : )a(p)ˆ,a,z(g)ˆ;|a(p k)n(
e)i,n(
kk)i,n(
k)n( r )i,n(
ka~ SD
TELIN
Simplified sum-product algorithm :receiver stucture
for each n : iterate EM algorithm until convergence (i )
r(t) h*(-t) x)(ˆ b
),1n(ˆje
)}a(p{ k)n(
e
)i,n(̂Compute
conventionalMAP detector
extr. probs.
kT
zk
z
inner iterations (index i)
symbol APP : )a(p)ˆ,a,z(g)ˆ;|a(p k)n(
e)i,n(
kk)i,n(
k)n( r )i,n(
ka~ SD
(i )
outer iterations (index n)
TELIN
Simplified sum-product algorithm :estimator performance
# MAP detector iterations
V&V
EMSP
MCRB
BICM : rate 1/2 conv. code, 8-PSK (set part.)1000 symbols, Eb/N0 = 5 dB
carrier phase unknown
TELIN
Simplified sum-product algorithm :BER performance
perfect synchr.
SP
EM
V&V
# MAP detector iterations
BER
BICM : rate 1/2 conv. code, 8-PSK (set part.)1000 symbols, Eb/N0 = 5 dB
carrier phase unknown
TELIN
Two alternatives for MAP detection when is unknown
•Iterative ML estimation of by means of EM algoritmsymbol APP during i-th EM iteration
)a(p)ˆ,a,z(g k)n(
e)i,n(
kk
)a(p)ˆ,a,z(g k)1,i(
e)i(
kk 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
Both algorithms yield similar MSE and BER after convergence, but simplified SP algorithm requires considerably less MAP decoder iterations to converge.
EM algorithm iterated till convergencefor each MAP detector iteration
Conclusions