Estimation and detection from coded signals

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@telin.UGent.be

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

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

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(bk|r) is given by (many terms !) Efficient computation of APPs : sum-product (SP) algorithm + message-passing in factor graph (FG))

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

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)

Factor graph for known q Because of cycles in FG, MAP detection is iterative ak1 ak2 ak3 b1 b2 b3

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)

Receiver structure, q known zk “conventional”MAP detector x r(t) h*(-t) kT (SP algorithm, iterate until convergence) exp(-jqo)

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

g(zK,aK,q) 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

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

Alternative 1 : ML parameter estimation

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

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)

Cramer-Rao bound average over (coded)symbol sequence a (many terms !!) analytical difficulty : Ea[.] “inside” ln(.), so that ln(p(r;q)) is not a simple function of r and q

Modified Cramer-Rao bound Er,a[.] “outside” ln(.)  MCRB easier to evaluate than CRB, when ln(p(r|a;q)) is simple function of a and r (e.g., linear modulation on AWGN channel)

MCRB for phase estimation r = aexp(jq) + w

CRB for phase estimation :uncoded transmission (i.i.d. symbols)  : 1-D summation : 1-D numerical integration

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

CRB, MCRB : numerical results BPSK QPSK (phase estimation) code properties should be exploited during estimation Large SNR : CRB  MCRB Small SNR : CRBuncoded > CRBcoded > MCRB

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

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

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

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

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

EM algorithm : phase estimation (1/2)

EM algorithm : phase estimation (2/2)

EM algorithm : timing estimation (1/2)

EM algorithm : timing estimation (2/2)

Alternative 2 : Simplified sum-product algorithm

g(zK,aK,q) 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

g(zK,aK,q) 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

Simplified sum-product algorithm (n+1)-th iteration of MAP detector aK ak a1 Encoding, interleaving, mapping

Simplified sum-product algorithm :maximization of p(n)(r|q) Similar to ML equation : p(a) is replaced by  EM algorithm to maximize p(n)(r|q) iteratively  SD symbol APP :

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 :

Simplified sum-product algorithm :estimator performance V&V 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

Simplified sum-product algorithm :BER performance BICM : rate 1/2 conv. code, 8-PSK (set part.)1000 symbols, Eb/N0 = 5 dB V&V BER EM carrier phase unknown SP perfect synchr. # MAP detector iterations

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.

Estimation and detection from coded signals