Turbo qualizatio Adré Foseca dos Satos aya Adioel Guimarães Outlie The discrete model of the chael with ISI. The turbo priciple applied to iterative equalizatio ad decodig: Turbo qualizatio. The BCJR algorithm applied to equalizatio ad covolutioal decodig. xample of Turbo qualizatio. Results ad discussio. Turbo qualizatio usig the Iterferece Caceler (IC.
Itroductio Turbo codes were iveted by Berrou, Glavieux ad Thitimajshima i 99 [Ber9]. Turbo qualizatio was proposed first by ouillard, Jezequel, Berrou, Picart, idier, ad Glavieux i 995. A simplified Turbo qualizer was proposed by Glavieux, aot, ad abat i 997. Trasmissio System b covolutioal ecoder c iterleavig c e( t BPSK e r( t ( t MP chael 4
Multipath Chael (MP δ( t τ δ( t δ( t τ 5 Multipath (MP chael c T T g g g N w r g taps of the chael w aditive oise 6
Covetioal system qualizatio ad decodig separately. qualizatio: Filterig (MS, F MS (Viterbi Algorithm r equalizer c deiterleavig c covolutioal decoder b 7 Turbo qualizatio r equalizer c deiterleavig c covolutioal decoder b cooperatio 8 4
MP chael as a Marov chai c T T g g g N s s iput + w r s s iput - s s s s ( time 9 Trasmissio model with MP chael chael/ier ecoder b covolutioal ecoder/outer ecoder c iterleavig c T T g g g N w v r 5
SISO device iput SISO output P og-ielihood Ratio (R l P ( x+ x A hard decisio ca be doe based o the sigal of the R The reliability of the decisio is related to by the magitude of the R. Maximum A Posteriori (MAP qualizer r ( c MAP equalizer a c ( c P c l P c ( + r r ( c P c l P c p l p ( + r ( r ( r c + ( r c e a, usig Bayes' Rule : P c + l P c ( + ( ( c ( c A Priori Iformatio xtrisic Iformatio e ( c + ( c a 6
MAP ecoder Z ( c [ P( c r P( c r... P( c ] P l P p l p N r ( c + ( c ( c + ( c, usig Bayes Rule : P + l P ( c + ( c Z MAP decoder e ( c + ( c a ( c ( b Also, the MAP decoder computes a estimate data as the most liely bit give Z. bˆ of the trasmitted Turbo qualizatio (Tuqu Π c a ( c ( c equalizer r Π ( c decoder b^ 4 7
Turbo qualizatio (Tuqu ( ext c Π r a c equalizer c ext c ext c ( ( Π ( ( c decoder b^ ( c ( c ( c a + ext ( c ( c ( c a + ext r module r module r module ^B 5 Questios??? 6 8
MAP algorithm (BCJR ( b P l P ( b + r b r s s s s iput + iput - usig Bayes' rule : ( b p l p ( b +, r ( b, r P ( a, b P( a b, ( b P( b l b + b p p ( S, S, r ( S, S, r time 7 MAP algorithm s s s s' s s+ s + s r j < r r j > ( s', s, r p( s', s, r, r, r p j< j> 8 9
MAP algorithm Usig Baye's rule : Usig the assumptio that the chael received P p ( s', s, R p( s', s, r, r, r p sequece P P r j> ( a, b P( a b P( b ( s', s, r p( r { s', s, r, r } p( s', s, r, r j> will oly j< ( s', s, R ( s ( s', s ( ' deped o the preset state s: j> j< is memorryless, the future ( s', s, r p( rj> s p( s', s, rj<, r p( rj> s p( { r, s} { s', rj< } p( s', rj< p( r s p( { r, s} s' p( s', r j> j< s j< 9 MAP algorithm s s s s' s s+ s + s r j < r r j > ( s' ( s' s ( s, p( S s', r s ' j< ( s p( r > S s j ( s', s p( { r, S s} S s'
Forward recursive computatio of ( s p( S s r, j< + s s s s' s s ( s p( s, rj<, r p( s', s, rj<, r ( s p( s', s, rj<, r p( { s, r} { s', rj< } p( s', rj< all s' all s' ( s p( { s, r} s' p( s', rj< all s' ' all s Usig Bayes' rule ad the assumptio that the chael is memoryless: ' ( s ( ' s, s ( s ' all s r j < Baye's rule : r ( a, b P( a b P( b P Forward recursive computatio of s s (, ( ( (, r (. (, (. (, +
Bacward recursive computatio of ( s p( r S ' ' j> s ' s' ( s p( rj> s' p( { rj>, s} s' all s p( { rj>, r, s} s' all s p( rj> { s, s', r} p { r, s} all s p( rj> {} s p( { r, s} s' all s s ( s' ( s ( s' s, all s r ( s' s s+ s + P P s Baye's rule : r j > ( a, b P( a b P( b ({ a, b} c P( a ( b, c P( b c Bacward recursive computatio of s ( + (, s + + ( + (, + ( r + (. (, (. (, + + + + + 4
MAP algorithm ( s, s'. ( s', ( s' ( s', s ( s s. all s' all s ( ( s s (, ( (, (, ( + + (, s + + ( + ( r r + ( (. (, + (. (, ( (. (, + (. (, + + + + 5 MAP algorithm s iput + ( b P l P ( b+ r b r s s s iput - time ( b l b + b p p ( S, S, r ( S, S, r l b + b ' ' ( s ( s, s ( s ' ' ( s ( s, s ( s 6
Questios??? 7 MAP qualizer ( c P c l P c ( + r r r ( c MAP equalizer a c ' ( s s p { r, S s} ( S s', Usig Baye's rule : Baye's rule : P P ( a, b P( a b. P( b ({ a, b} c P( a ( b, c. P( b c ' ( s, s p r { s, s' }. P( s s' is govered by the output symbol is govered by the iput symbol 8 4
MAP qualizer c ' ( s, s p r { s, s' } p,47,47,85. P( s s' s s s s. ( s ', s p( r v. P( c r v exp( r v σ σ w T T v r g,47 ; g,85; g,47.69.85.85.85 iput + iput -..85.69 s s s s..69.85 v.85.85..85.69 9 MAP qualizer ( s', s P( c exp( ( r v σ iformatio ext c σ obtaied from the extrisic of the decoder : ext ( c P ( c + ext c l ext( c P( c + exp ( ext( c + P( c + exp( ( c P c ext ( c P c exp + exp ( c. ext( c ( ( c ext P c l P c l P c ( + ( P( c ( + + exp ( ( c ext c {,} 5
MAP qualizer exp + exp ( c. ( c ( ( c ext ( s', s exp ( r v σ c {,} ext σ (c versus P (c+ (c versus P(c- 8 8 6 6 4 4 (c - (c - -4-4 -6-6 -8-8 -....4.5.6.7.8.9 P(c+ -....4.5.6.7.8.9 P (c- MAP qualizer implemetatio [Koetter] s s s s..69.85.85.85..69 iput + iput -.85 s s s s..69.85.85.85..85.69 A ( +, A( { P} { B} i, j, i, j, ( s, s, i j compoetwise product of A ad P 6
MAP qualizer implemetatio Iput : Matrices P, A ( +, A( -, B ( +, B (-, f e b. Iitializatio: the first colum of vector f ad the last of vector b are iitialized as for every lies. Recursively f b P P T f b + compute of f ad b :,,, N, N-,, Output : for,...,n T f B ( + c r l f T B b + ( b + ( c r l c+ c ' ' ( s ( s, s ( s ' ' ( s ( s, s ( s MAP ecoder Z ( c P l MAP decoder ( c+ Z ( c Z ( c ( b e + P ( c ( c a P ( c r exp + exp ( c. ext ( c ( c ext c {,} Z [ P( c r P( c r... P( c r ] T N 4 7
MAP ecoder c b T T s s / / / / / c ' ( s, s p { r, S s} Usig Baye's rule : ( S s' ' ( s, s p r { s, s' } P( s s' s s / / / is govered by the output symbol ( s s P( b P( c c r P( c c r i, j, i, j, i, j is govered by the iput symbol there is o a priori iformatio of the iformatio bits 5 MAP ecoder implemetatio [Koetter] A b (, A ( b s s / / / / / A (, A ( c c s s / / / A (, A ( c c 6 8
9 7 MAP ecoder implemetatio T T T b B f b B f r b P b f P f e b f B B A A P + + + + + + c,...,n,, N- N l for : Output,,,,.,,,,, Matrices : Iput ad, for c c b Iitializatio: the first colum of vector f ad the last of vector b are iitialized as for every lies. : ad compute of y Recursivel b f + ' ' ' ',, l c c s s s s s s s s c r 8 MAP implemetatio For a practical implemetatio, the vectors forward ad bacward eed to be ormalized to avoid uderflow. The MAP algorithm ca be implemeted i the log domai (og-map-algorithm for computatioal simplicity.
Questios??? 9 Tuqu example b covolutioal ecoder c iterleavig c MP chael e(t 4X4 c c T T b T T,47,47,85 g g 5 7 c w v r 4
Tuqu example b covolutioal ecoder c iterleavig c BPSK e(t b c [ ] [ ] c [ ] 4 Tuqu example a c equalizer r c ext c ext c ( ext c ( ( Π Π ext ( c ( ( c decoder b^ p,, r module r module r module ^B r equalizer c deiterleavig c covolutioal decoder b p 4
Tuqu example (p r equalizer c deiterleavig c covolutioal decoder b r ( c MAP equalizer a c 4 Tuqu example (p s.69.69.69.69 s s.85.85.85...85.85.85...85.85.85...85.85.85.. s c.85.69 r, r, 5 r,.85 ( s', s P( c exp( ( r v σ.69 σ.85.69.85.69 r 4,8 [ ] 44
Tuqu example (p ( (, (, (, (, ( ( (,,,,,,,,,,,,,,,,,,,,,,,, 45 Tuqu example (p ( (, (, (, (, ( ( (,,,,,,,,,,,,,,,,,,,,,,,, ( (. (, + (. (, ( (. (, + (. (, ( (. (, + (. (, ( (. (, + (..,955+,955.,97+,97 46
Tuqu example (p ( (, (, (, (, ( ( (,,,,,,,,,,,,,,,,,,,,,,,, ( (. (, + (. (,,955.,5+ ( (. (, + (. (,,97.,595 +, ( (. (, + (. (,,97.,54+ ( (. (, + (.,97.,84+, 7,4,6 47 Tuqu example (p ( (, (, (, (, ( ( (,,,,,,,,,,,,,,,,,,,,,,,, 4 4 4 4 ( (. (, + (. (, ( (. (, + (. (,,78 ( (. (, + (. (,,7 ( (. (, + (., 5,9 48 4
Tuqu example (p ( (, (, (, (, ( ( (,,,,,,,,,,,,,,,,,,,,,,,, 5 5 5 5 ( 4(. 4(, + 4(. 4(, ( 4(. 4(, + 4(. 4(,,8 ( 4(. 4(, + 4(. 4(,,9 ( (. (, + (., 5 4 4 4 4, 49 Tuqu example (p (, (, (, (, 5 (,,,,,,,,,,,,,,,,,,,,,,,, 5 ( 5 ( 5 ( 4 4 4 4 9 ( 5(. 4(, + 5(. 4(,,. 9 ( 5(. 4(, + 5(. 4(,,5. 9 ( 5(. 4(, + 5(. 4(,,4. 9 ( (. (, + (.,5. 5 4 5 4 5 5
Tuqu example (p (, (, (, (, 5 (,,,,,,,,,,,,,,,,,,,,,,,, 5 ( 5 ( 5 ( 9 ( 4(. (, + 4(. (,,. 9 ( 4(. (, + 4(. (,,. 9 ( 4(. (, + 4(. (,,. 9 ( (. (, + (.,. 4 4 5 Tuqu example (p (, (, (, (, 5 (,,,,,,,,,,,,,,,,,,,,,,,, 5 ( 5 ( 5 ( ( (. (, + (. (,,4. ( (. (, + (. (,,. ( (. (, + (. (,,. ( (. (, + (.,. 5 6
Tuqu example (p (, (, (, (, 5 (,,,,,,,,,,,,,,,,,,,,,,,, 5 ( 5 ( 5 ( ( (. (, + (. (,,. ( (. (, + (. (,,. ( (. (, + (. (,,. ( (. (, + (.,. 5 Tuqu example (p (, (, (, (, 5 (,,,,,,,,,,,,,,,,,,,,,,,, 5 ( 5 ( 5 ( ( c r l c+ c p p ( S, S, r ( S, S, r l c+ c ' ' ( s ( s, s ( s ' ' ( s ( s, s ( s 54 7
Tuqu example (p ( ( ( (,, (,,,,, ( ( ( ( ( c r l c+ c p p ( S, S, r ( S, S, r 55 Tuqu example (p ( ( ( (,,, ( ( ( c + (,, r (. (,. ( + (. (,. ( p S S + (. (,. ( + (. (,. ( 56 8
Tuqu example (p ( (, ( ( (,, ( ( (, ( c (,, r (. (,. ( + (. (,. ( p S S + (. (,. ( + (. (,. ( 57 Tuqu example (p ( ( ( (,, (,,,,, ( ( ( ( (,, p S S r l c+,754 (,, p S S r c ( c r 58 9
Tuqu example (p r ( c MAP equalizer a c c [ ] [,754,874,774,79 ] c leads to wrog decisios 59 Tuqu example (p r equalizer c deiterleavig c covolutioal decoder b Z MAP decoder ( c ( b 6
Tuqu example (p Z P [ P( c r P( c r... P( c ] N r exp c. ext ( c r ( c r c {,} + exp ( c r ext Z MAP decoder ( c ( b [,754 4,454,999,5 ] c ( [,48,988,87,976 ] Z c ( [,85,7,7,84 ] Z c 6 Tuqu example (p s s / / / / / s s Z. Z. Z.Z Z Z. Z Z Z Z. s s / / / s s Z. Z Z Z. Z.Z Z [ P( c r P( c r... P( c ] N r ( s s P( b. P( c c r. P( c c r i, j, i, j, i, j there is o a priori iformatio of the iformatio bits 6
Tuqu example (p b b [ ] Hard decisios of the iformatio bits : ˆ b [ ] [ ] c Hard decisios of the coded bits : cˆ Z MAP decoder ( c [ ] 6 Tuqu example (p r a c equalizer c ext c ext c ( ( ext c ( Π Π ( ( c decoder b^ r ( c MAP equalizer a c ( s', s exp + exp ( c. ext( c ( ( c ext. exp σ ( ( r v σ 64
Tuqu example (p p c [ ] [,754,874,774,79 ] c p leads to wrog decisios [ ] c ( c [,967 4,777,979,88 ] errors corrected 65 Tuqu example (p Z P [ P( c r P( c r... P( c ] N r exp c. ext ( c r ( c r c {,} + exp ( c r ext Z MAP decoder ( c ( b [,985 4,859,99,95 ] c Z Z ( [,87,9864,8,98 ] c ( [,77,6,789,97 ] c 66
Tuqu example (p b ˆ [ ] Hard decisios of the iformatio bits : b [ ] [ ] c Hard decisios of the coded bits : cˆ Z MAP decoder ( c ( b [ ] 67 Tuqu example (p p Hard decisios of the iformatio bits : ˆ b p Hard decisios of the iformatio bits : ˆ b [ ] Hard decisios of the coded bits : cˆ [ ] Hard decisios of the coded bits : cˆ [ ] [ ] 68 4
Questios??? 69 volutio of the R s 7 5
Results - - iteratio iteratio iteratio iteratio 4 iteratio 5 coded AWGN chael without IS I qualizatio ad decodig without priors B R - -4-5 Chael without ISI -6 4 5 6 7 7 b/no Tuqu usig MAP The MAP algorithm applied i turbo equalizatio is adequate to low spectral efficiecy modulatios ad chaels exhibitig a low delay spread. I 997, Glavieux proposed a Iterferece Caceller i the Turbo qualizer for chaels with strog delay spread ad high order modulatios. 7 6
Turbo qualizatio with IC λ R F F ext c ( Π c ext c ( ext Π ext ( c ( decoder ( c b^ Matched filter ISI estimator r F c _ λ F 7 Turbo qualizatio with IC r F c _ λ F The iput of the filter F is : λ exp + exp { c} P( c +.+ P( c (. ( ( c ext( c + tah ( ( c + exp( ( c 74 7
Tuqu example λ R F F ext c ( Π c ext c ( ext Π ext ( c ( decoder ( c b^ p,, r module r module r module ^B r equalizer c deiterleavig c covolutioal decoder b p 75 Adaptive Turbo qualizatio r F c _ λ F For each stage, the equalizer is updated accordig to the mea - square - error (MS criterio [aot]: MS W + + W { λ } c µ. R + µ. R ( s cˆ ( s cˆ 76 8
Results [aot] 77 Tuq, other approaches Turbo equalizatio usig bloc codes Turbo equalizatio usig turbo codes Turbo equalizatio usig joit chael estimatio ad MAP equalizatio (BCJR. Turbo equalizatio applied i multi-user detectio of CMA systems. 78 9
Mai refereces C. ouillard, M. Jezequel, C. Berrou, A Picart, P. idier, ad A Glavieux. Iterative Correctio of Itersymbol Iterferece: Turbo qualizatio. uropea Tras. O Telecomm., vol. 6, pp. 57 5, Sep-Oct 995.. Hazo, T. H. iew, ad B.. Yeap. Turbo Codig, Turbo equalizatio ad Space-Time Codig for Trasmissio Over Fadig Chaels. Joh Wiley ad Sos, Ic G. Bauch, H. Khorram ad J. Hageauer. Iterative qualizatio ad ecodig i Mobile Commuicatio Systems. d PMCC 97 ad rd ITG-Fachtagug Telecomm, Mobile Kommuiatio, Bo, Germay, Oct. 997. 79 Mai refereces A. Glavieux, C. aot, ad J. abat. Turbo qualizatio Over a Frequecy Selective Chael. Proc. Of the Iter. Symposium o Turbo codes, Brest, Frace, pp. 96, September 997. C. aot, A. Glavieux ad J. abat. Turbo qualizatio: Adaptative qualizatio ad Chael ecodig Joitly Optimized. I Joural o Selected Areas i Commuicatios, vol.9, º9, september, T. Michael., Koetter. Ralf., A.. Siger. Turbo qualizatio. I Sigal Processig Magazie, Feb 8 4
Questios??? 8 4