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

TELIN Outline 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 infobits coded symbols linear modulation AWGN channel r(t) ba received signal 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) 

TELIN MAP detection,  known MAP detection (on individual infobits) achieves minimum BER Assuming  =  0 is known, bit APP p(b k |r) is given by r : vector representation of r(t) Efficient computation of APPs : sum-product (SP) algorithm + message-passing in factor graph (FG)) (many terms !)

TELIN Factor graph for known  g(z k,a k,  0 ) g(z 1,a 1,  0 ) g(z K,a K,  0 ) a1a1 akak aKaK Encoding, interleaving, mapping b1b1 b bLbL g(z k,a k,  0 ) channel observation

TELIN Factor graph for known  g(z k,a k,  0 ) g(z 1,a 1,  0 ) g(z K,a K,  0 ) a1a1 akak aKaK Encoding, interleaving, mapping b1b1 b bLbL g(z k,a k,  0 ) p(b |r) channel observation extrinsic information on a k information bit APP p e (a k )

TELIN Factor graph for known  Because of cycles in FG, MAP detection is iterative a k1 a k3 a k2 b 1 b 2 b 3

TELIN Factor graph,  known g(z k,a k,  0 ) g(z 1,a 1,  0 ) g(z K,a K,  0 ) a1a1 akak aKaK Encoding, interleaving, mapping b1b1 b bLbL g(z k,a k,  0 ) p (n) (b |r) n : iteration index of MAP detector iterate until convergence channel observation extrinsic information on a k information bit APP p e (n) (a k)

TELIN Receiver structure,  known r(t)h*(-t) kT exp(-j  o ) “conventional” MAP detector x zkzk (SP algorithm, iterate until convergence)

TELIN MAP detection,  unknown When  is unknown (uniform distrib.), bit APP is given by  conventional MAP detector hard to compute, because of integration over 

TELIN Factor graph,  unknown g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK Encoding, interleaving, mapping b1b1 b bLbL =  messages are functions of  downward messages involve integration over 

TELIN Alternatives to exact MAP detection when  is unknown Alternative 1 : ML parameter estimation Provide estimate of  to conventional MAP detector Alternative 2 : Simplified sum-product algorithm Approximation of  -dependent messages

TELIN Alternative 1 : ML parameter estimation

TELIN Receiver structure when estimating  Strategy : use conventional MAP detector, but provide estimate (instead of correct value) of  r(t)h*(-t) x Estimation of  conventional MAP detector kT zkzk

TELIN ML parameter estimation ML estimation of  : 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)

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

TELIN Modified Cramer-Rao bound E r,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)

TELIN MCRB for phase estimation r = aexp(j  ) + w

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

TELIN CRB for phase estimation : coded transmission marginal symbol APPs (from MAP detector operating on r) : Monte Carlo simulation : 1-D summation per symbol r = aexp(j  ) + w 

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

TELIN 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

TELIN EM algorithm : symbol APP computation a1a1 akak aKaK Encoding, interleaving, mapping for each i : MAP detector is reset and iterated until convergence (n   ) symbol APP :  SD outer iteration index i (EM algorithm), inner iteration index n (MAP detector)

TELIN Receiver structure for EM algorithm r(t) h*(-t) x EM : compute conventional MAP detector kT zkzk z outer iterations (index i) extr. probs symbol APP :  SD inner iterations (index n) k = 1,..,K

TELIN 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

TELIN 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

TELIN EM algorithm : phase estimation (1/2)

TELIN EM algorithm : phase estimation (2/2)

TELIN EM algorithm : timing estimation (1/2)

TELIN EM algorithm : timing estimation (2/2)

TELIN Alternative 2 : Simplified sum-product algorithm

TELIN Simplified sum-product algorithm g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK Encoding, interleaving, mapping =  messages are functions of  downward messages involve integration over  Factor graph corresponding to unknown 

TELIN Simplified sum-product algorithm g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK =  Messages depending on  are approximated by (n : iteration index of MAP detector)

TELIN Simplified sum-product algorithm g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK =  Messages depending on  are approximated by (n : iteration index of MAP detector)

TELIN Simplified sum-product algorithm g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK =  Messages depending on  are approximated by (n : iteration index of MAP detector)

TELIN Simplified sum-product algorithm Messages depending on  are approximated by (n : iteration index of MAP detector) g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK = 

TELIN Simplified sum-product algorithm Messages depending on  are approximated by (n : iteration index of MAP detector) g(z k,a k,  ) g(z 1,a 1,  ) g(z K,a K,  ) a1a1 akak aKaK = 

TELIN Simplified sum-product algorithm a1a1 akak aKaK Encoding, interleaving, mapping (n+1)-th iteration of MAP detector

TELIN Simplified sum-product algorithm a1a1 akak aKaK Encoding, interleaving, mapping (n+1)-th iteration of MAP detector

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

TELIN Simplified sum-product algorithm : receiver stucture for each n : iterate EM algorithm until convergence (i   ) r(t)h*(-t) x Compute conventional MAP detector extr. probs. kT zkzk z inner iterations (index i) symbol APP :  SD (i   ) outer iterations (index n)

TELIN Simplified sum-product algorithm : estimator performance # MAP detector iterations V&V EM SP 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 algoritm symbol APP during i-th EM iteration one MAP detector iteration for 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 convergence for each MAP detector iteration Conclusions