Coding Techniques. Manjunatha. P. Professor Dept. of ECE. June 28, J.N.N. College of Engineering, Shimoga.

Transcription

1 Coing Tehniques Mnjunth. P mnjup.jnne@gmil.om Professor Dept. of ECE J.N.N. College of Engineering, Shimog June 8, 3

2 Overview Convolutionl Enoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

3 Overview Convolutionl Enoing Convolutionl Enoer Representtion Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

4 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

5 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

6 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

7 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

8 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

9 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Sequentil Deoing: Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

10 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Sequentil Deoing: Feek Deoing: Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

11 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Sequentil Deoing: Feek Deoing: 8 Turo Coes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

12 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Sequentil Deoing: Feek Deoing: 8 Turo Coes 9 [,, 3, 4, 5] Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

13 Overview Convolutionl Enoing Convolutionl Enoer Representtion 3 Formultion of the Convolutionl Deoing Prolem 4 Properties of Convolutionl Coes: Distne property of onvolutionl oes 5 Systemti n Nonsystemti Convolutionl Coes 6 Performne Bouns for Convolutionl Coes, Coing Gin 7 Other Convolutionl Deoing Algorithms: Sequentil Deoing: Feek Deoing: 8 Turo Coes 9 [,, 3, 4, 5] Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

14 Enoing of Convolutionl Coes Enoing of Convolutionl Coes Enoing of Convolutionl Coes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

15 Enoing of Convolutionl Coes Enoing of Convolutionl Coes Convolutionl oes re ommonly speifie y three prmeters: (n,k,k) where n = numer of outputs k = numer of inputs K = numer of memory registers The quntity k/n lle the oe rte, is mesure of the effiieny of the oe ommonly k n n prmeters rnge from to 8 n K from to. The onstrint length L represents the numer of its in the enoer memory tht ffet the genertion of the n output its. Constrint Length, L = k(k ) Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

16 Enoing of Convolutionl Coes Enoing of Convolutionl Coes A lok igrm of inry rte R=/ nonsystemti feeforwr onvolutionl enoer with memory orer m=3 (n,k,k i.e.,,,3) is sshown in Figure. The enoer onsists of n K= 3-stge shift register together with n= moulo- ers n multiplexer for serilizing the enoer outputs. The mo- er n e implemente s EXCLUSIVE-OR gte. Sine mo- ition is liner opertion, the enoer is liner feeforwr shift register. All onvolutionl enoers n e implemente using liner feeforwr shift register of this type. () V Input U SR SR SR Output () V Figure: onvolutionl enoer(rte=/, K=3) Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

17 Enoing of Convolutionl Coes Enoing of Convolutionl Coes Constrint length K=3, k= input, n= moulo- ers i.e., k/n=/ At eh input it, it is shifte to the leftmost stge n the its in the registers re shifte one position to the right. Connetion vetor for the enoer is s follows: g = g = where in the ith position inites the onnetion in the shift register, n inites no onnetion in the shift register n the moulo- er. u First oe symol Input it m Output rnh wor u seon oe symol Figure: Convolutionl enoer(rte=/, K=3 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 6 / 8

18 Enoing of Convolutionl Coes Enoing of Convolutionl Coes Consier messge vetor m= re inputte one t time in the instnts t,t, n t 3 K-= zeros re inputte t times t 4 n t 5 to flush the register t u = u = t 4 u = u = The output sequene of the enoer is t u = u = t 5 u = u = u = u = t 3 t 6 u = u = Figure: Convolutionlly enoe messge Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 7 / 8

19 Enoing of Convolutionl Coes Impulse Response of the Enoer Impulse Response of the Enoer In impulse response single it t time is pplie to the enoer tht moves through the enoer. Register Brnh wor ontents u u Input it m output Moulo- sum Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 8 / 8

20 Enoing of Convolutionl Coes Polynomil Representtion Enoing of Convolutionl Coes: Polynomil Representtion In ny liner system, time-omin opertions involving onvolution n e reple y more onvenient trnsform-omin opertions involving polynomil multiplition. Sine onvolutionl enoer is liner system, eh sequene in the enoing equtions n e reple y orresponing polynomil, n the onvolution opertion reple y polynomil multiplition. In the polynomil representtion of inry sequene, the sequene itself is represente y the oeffiients of the polynomil Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 9 / 8

21 Enoing of Convolutionl Coes Polynomil Representtion For the given enoer g (X ) represents upper onnetion n g (X ) represents lower onnetion. The out put sequene is foun s follows: g (X ) = + X + X g (X ) = + X U(X ) = m(x )g (X ) interle with m(x )g (X ) m(x )g (X )= ( + X )( + X + X )= + X + X 3 + X 4 m(x )g (X )= ( + X )( + X ) = + X 4 m(x )g (X )= + X + X + X 3 + X 4 m(x )g (X )= + X + X + X 3 + X 4 U(X )= (,)+(,)X+(,)X +(,)X 3 +(,)X 4 U(X )= ( ) ( ) (,) ( ) ( ) Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

22 Enoing of Convolutionl Coes Polynomil Representtion First oe symol u Input it m Output rnh wor u seon oe symol Figure: onvolutionl enoer(rte=/, K=3 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

23 Enoing of Convolutionl Coes Stte Representtion n the Stte igrm Stte Representtion n the Stte igrm Tle: Stte trnsition tle (rte=/, K=3) Input it m First oe symol u Output rnh wor u seon oe symol Input Present Stte Next Stte Output Output it () Sttes represent possile ontents of the rightmost K- register ontent. For this exmple there re only two trnsitions from eh stte orresponing to two possile input its. Soli line enotes for input it zero, n she line enotes for input it one. () () = Input it () = = () Tuple Stte Figure: Stte igrm for rte=/ n K=3 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8 () () = ()

24 Enoing of Convolutionl Coes Stte Representtion n the Stte igrm Brnh wor Input Register Stte t Stte t time t i it ontents time t i time t i+ u u Output sequene: U= Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

25 The Tree igrm Enoing of Convolutionl Coes Stte igrm oes not represent the time history to trk enoer trnsition s funtion of time. Tree igrm provies time history. Enoing is esrie y trversing from left to right. If the input it is zero its rnh wor is foun y moving to the next rightmost rnh in the upwr iretion, n if the input it is one its rnh wor is foun y moving to the next rightmost rnh in the ownwr iretion Assuming initilly the ontents of register re zero, if the input it is zero, its orresponing output is otherwise if the first input it is one its orresponing output is. The limittion of tree igrm is tht the numer of rnhes inreses s funtion of L, where L is the numer of rnh wors. Input Present Next Output Stte Stte Tree igrm t t t 3 t 4 t 5 Figure: Tree representtion Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

26 Enoing of Convolutionl Coes The Trellis igrm The Trellis igrm Soli line enotes for input it zero, n she line enotes for input it one. Noes represent the enoer sttes, I row represents stte = susequent rows orrespon to stte =, = n =. At eh unit of time, the trellis requires K noes to represent the K. Input Present Next Output Stte Stte t t t 3 t 4 t 5 t 6 = = = = Stey Stte Figure: Trellis igrm for rte=/ n K=3 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

27 Enoing of Convolutionl Coes Enoing of Convolutionl Coes: Trnsform Domin Enoing of Convolutionl Coes: Trnsform Domin In ny liner system, time-omin opertions involving onvolution n e reple y more onvenient trnsform-omin opertions involving polynomil multiplition. Sine onvolutionl enoer is liner system, eh sequene in the enoing equtions n e reple y orresponing polynomil, n the onvolution opertion reple y polynomil multiplition. In the polynomil representtion of inry sequene, the sequene itself is represente y the oeffiients of the polynomil Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 6 / 8

28 Enoing of Convolutionl Coes Enoing of Convolutionl Coes: Trnsform Domin For exmple, for (,, m) oe, the enoing equtions eome where u(d) = u + u (D) + u (D )... The enoe sequenes re The genertor polynomils of the oe re After multiplexing, the oe wor eome V () (D) = u(d)g () (D) V () (D) = u(d)g () (D) V () (D) = v () + v () D + v () D +... V () (D) = v () + v () D + v () D +... g () (D) = g () + g () D g () m D m g () (D) = g () + g () D g () m D m V (D) = [v () (D), v () (D)] V (D) = v () (D ) + Dv () (D ) D is ely opertor, n the power of D enoting the numer of time units it is elye with respet to the initil it. The generl formul fter multiplexing (n=numr of output): V (D) = v () (D n ) + Dv () (D n ) D (n ) v (n ) (D n ) Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 7 / 8

29 Enoing of Convolutionl Coes Enoing of Convolutionl Coes: Trnsform Domin Exmple. For (,,3) onvolutionl oe,g () = [ ] n g () = [ ] the genertor polynomils re g () (D) = + D + D 3 g () (D) = + D + D + D 3 For the informtion sequene u(d) = + D + D 3 + D 4, then v () (D) = ( + D + D 3 + D 4 )( + D + D 3 ) v () (D) = ( + D + D 3 + D 4 + D + D 4 + D 5 + D 6 + D 3 + D 5 + D 6 + D 7 ) v () (D) = ( + D + D + D 3 + D 3 + D 4 + D 4 + D 5 + D 5 + D 6 + D 6 + D 7 ) v () (D) = ( + D 7 ) v () (D) = ( + D + D 3 + D 4 )( + D + D + D 3 ) v () (D) = ( + D + D 3 + D 4 + D + D 3 + D 4 + D 5 + D + D 4 + D 5 + D 6 = +D 3 + D 5 + D 6 + D 7 ) v () (D) = + D + D 3 + D 4 + D 5 + D 7 n the oe wor is v(d) = [ + D 7, + D + D 3 + D 4 + D 5 + D 7 ]. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 8 / 8

30 Enoing of Convolutionl Coes Enoing of Convolutionl Coes: Trnsform Domin Exmple. n the oe wor is v(d) = [ + D 7, + D + D 3 + D 4 + D 5 + D 7 ]. After multiplexing, the oe wor eome v(d) = v () (D ) + Dv () (D ) v () (D ) = ( + D 4 ) v () (D ) = Dv () (D ) = D( + D + D 6 + D 8 + D + +D 4 ) v () (D ) = Dv () (D ) = D + D 3 + D 7 + D 9 + D + +D 5 v(d) = v () (D ) + Dv () (D ) v(d) = + D + D 3 + D 7 + D 9 + D + D 4 + D 5 The result is the sme s onvolution n mtrix multiplition. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 9 / 8

31 Enoing of Convolutionl Coes Exmple. For (3,,3) onvolutionl enoer Enoing of Convolutionl Coes: Trnsform Domin g () = ( ), g () = ( ), g () = ( ) g () = ( ), g () = ( ), g () = ( ) [ ] + D D + D D For the informtion sequene u () (D) = + D, u () (D) = + D, the enoing equtions give the oewor V (D) = [v () (D), v () (D), v () (D)] [ ] V (D) = [ + D + D D + D, + D] D V (D) = [( + D ).( + D) + ( + D)D, ( + D ).D + ( + D), = ( + D ).( + D) + ( + D)] V (D) = [ + D 3, + D 3, D + D 3 ] After multiplexing, the oe wor eome V (D) = v () (D n ) + Dv () (D n ) D (n ) v (n ) (D n ) v(d) = ( + D 3 ) 3 + D( + D 3 ) 3 + D (D + D 3 ) 3 v(d) = ( + D 9 ) + D( + D 9 ) + D (D 6 + D 9 ) v(d) = + D + D 8 + D 9 + D + D Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

32 Deoing of Convolutionl Coes Deoing of Convolutionl Coes Deoing of Convolutionl Coes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

33 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

34 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Viteri eoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

35 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Viteri eoing Sequentil Deoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

36 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Viteri eoing Sequentil Deoing i) Stk Algorithm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

37 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Viteri eoing Sequentil Deoing i) Stk Algorithm ii) Fno Algorithm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

38 Deoing of Convolutionl Coes Deoing of Convolutionl Coes There re severl ifferent pprohes to eoing of onvolutionl oes. These re groupe in two si tegories. Mximum likely-hoo eoing Viteri eoing Sequentil Deoing i) Stk Algorithm ii) Fno Algorithm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 / 8

39 The Viteri Deoing Algorithm The Viteri Deoing Algorithm The Viteri Deoing Algorithm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

40 The Viteri Deoing Algorithm The Viteri Deoing Algorithm In 967, Viteri introue eoing lgorithm for onvolutionl oes whih hs sine eome known s Viteri lgorithm. Lter, Omur showe tht the Viteri lgorithm ws equivlent to fining the shortest pth through weighte grph. Forney reognize tht it ws in ft mximum likelihoo eoing lgorithm for onvolutionl oes; tht is, the eoer output selete is lwys the oe wor tht gives the lrgest vlue of the log-likelihoo funtion. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

41 The Viteri Deoing Algorithm The Viteri Deoing Algorithm In orer to unerstn Viteris eoing lgorithm, expn the stte igrm of the enoer in time (i.e., to represent eh time unit with seprte stte igrm). Consier (3,, ) oe with G(D) = [ + D, + D, + D + D ] n n informtion sequene of length h=5. The trellis igrm ontins h+m+ time units or levels, n re lele from to h+m. Assuming tht the enoer lwys strts in stte S n returns to stte S, the first m time units orrespon to the enoers eprture from stte S, n the lst m time units orrespon to the enoers return to stte S Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

42 The Viteri Deoing Algorithm The Viteri Deoing Algorithm Not ll sttes n e rehe in the first m or the lst m time units. However, in the enter portion of the trellis, ll sttes re possile, n eh time unit ontins repli of the stte igrm. There re two rnhes leving n entering eh stte. The upper rnh leving eh stte t time unit i represents the input u i =, while the lower rnh represents u i =. Eh rnh is lele with the n orresponing outputs v i, n eh of the h oe wors of length N = n(h + m) is represente y unique pth through the trellis. For exmple, the oe wor orresponing to the informtion sequene u = ( ) is shown highlighte in Figure. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 6 / 8

43 The Viteri Deoing Algorithm The Viteri Deoing Algorithm In the generl se of n (n, k, m) oe n n informtion sequene of length kh, there re k rnhes leving n entering eh stte, n kh istint pths through the trellis orresponing to the kh oe wors. Now ssume tht n informtion sequene u = (u,..., u h ) of length kh is enoe into oe wor v = (v, v,..., v h+m ) of length N = n(h + m), n tht sequene r = (r, r,..., r h+m ) is reeive over isrete memoryless hnnel (DMC). Alterntively, these sequenes n e written s u = (u,..., u kh ), v = (v, v,..., v N ), r = (r, r,..., r N ), where the susripts now simply represent the orering of the symols in eh sequene. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 7 / 8

44 The Viteri Deoing Algorithm The Viteri Deoing Algorithm As generl rule of etetion, the eoer must proue n estimte of the oe wor v se on the reeive sequene r. A mximum likelihoo eoer (MLD) for DMC hooses s the oe wor v whih mximizes the log-likelihoo funtion log P(r v). Sine for DMC P(r v) = It follows tht log P(r v) = h+m i= h+m i= P(r i v i ) = log P(r i v i ) = N i= N i= where P(ri vi) is hnnel trnsition proility. P(r i v i ) (.) log P(r i v i ) (.3) This is minimum error proility eoing rule when ll oe wors re eqully likely Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 8 / 8

45 The Viteri Deoing Algorithm The Viteri Deoing Algorithm The log-likelihoo funtion log P(r v) is lle the metri ssoite with the pth v, n is enote M(r v). The terms log P(r i v i ) in the sum of Eqution (.3) re lle rnh metris, n re enote M(r i v i ), wheres the terms log P(r i v i ) re lle it metris, n re enote M(r i v i ). The pth metri M(r v) n e written s M(r v) = h+m i= M(r i v i ) = N i= M(r i v i ) The eision me y the log-likelihoo funtion is lle the soft-eision. If the hnnel is e with AWGN, soft-eision eoing les to fining the pth with minimum Eulien istne. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 9 / 8

46 The Viteri Deoing Algorithm The Viteri Deoing Algorithm A prtil pth metri for the first j rnhes of pth n now e expresse s The following lgorithm, when pplie to the reeive sequene r from DMC, fins the pth through the trellis with the lrgest metri (i.e., the mximum likelihoo pth). The lgorithm proesses r in n itertive mnner. At eh step, it ompres the metris of ll pths entering eh stte, n stores the pth with the lrgest metri, lle the survivor, together with its metri. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

47 The Viteri Deoing Algorithm The Viteri Deoing Algorithm The Viteri Algorithm Step. Beginning t time unit j = m, ompute the prtil metri for the single pth entering eh stte. Store the pth (the survivor) n its metri for eh stte. Step. Inrese j y. Compute the prtil metri for ll the pths entering stte y ing the rnh metri entering tht stte to the metri of the onneting survivor t the preeing time unit. For eh stte, store the pth with the lrgest metri (the survivor), together with its metri, n eliminte ll other pths. 3 Step 3. If j < h + m, repet step. Otherwise, stop. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

48 The Viteri Deoing Algorithm The Viteri Deoing Algorithm G(X ) = [ + X + X, + D ] Input it m u u First oe symol seon oe symol Output rnh wor Figure: onvolutionl enoer of R=/ n K=3 Input Present Next Output Stte Stte Output it () Input it () = () Tuple Stte () = = () () () = () Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 3 / 8

49 The Viteri Deoing Algorithm The Viteri Deoing Algorithm The trellis igrm is rerwn y leling eh rnh with the Hmming istne etween reeive oe symol n the rnh wor. Figure shows messge sequene m the orresponing oewor sequene U n noise orrupte reeive sequene Z. The rnh wors of the Trellis re known priori to oth enoer n eoer. At time t the reeive oe is, stte trnsition with n output of n is ompre with reeive oe n its Hmming istne is of n is mrke on tht rnh. Similrly t time t the reeive oe is, stte trnsition with n output of n is ompre with reeive oe n its Hmming istne is of n is mrke on tht rnh. Input it Output it () () = () = = () () () = () Input t m: () t t t 3 t 4 t 5 t 6 = Trnsmitte it Z: Reeive it Z: = = = Stey Stte = = = = t t t 3 t 4 t 5 t 6 Brh metri Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 33 / 8

50 The Viteri Deoing Algorithm The Viteri Deoing Algorithm Input t m: t t t 3 t 4 t 5 t 6 = Trnsmitte it Z: = = = Stey Stte Reeive it Z: t t t 3 t 4 t 5 t 6 = = = = Brh metri At eh time there re K sttes, n t eh stte two pths re entering (two pths leving). Computing the metris for the two pths entering eh stte n eliminting one of them n this is one for eh of K sttes t time t i then the eoer moves to time t i+ n repets the proess. = t t Stte metris T = = Stte metris t t t 3 T =3 = T = = T =3 = T = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 34 / 8

51 The Viteri Deoing Algorithm The Viteri Deoing Algorithm Input t m: t t t 3 t 4 t 5 t 6 = Trnsmitte it Z: = = = Stey Stte Reeive it Z: t t t 3 t 4 t 5 t 6 = = = = Brh metri At eh time there re K sttes, n t eh stte two pths re entering (two pths leving). Computing the metris for the two pths entering eh stte n eliminting one of them n this is one for eh of K sttes t time t i then the eoer moves to time t i+ n repets the proess. = = Stte metris t t = T = t Stte metris t =T = T = T = Stte metris t t t 3 = T =3 Stte metris t t t 3 = = T T=3 =3 = = = T = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 34 / 8 = T = T =3 T =

52 The Viteri Deoing Algorithm The Viteri Deoing Algorithm t t t 3 t 4 t 5 t 6 = Input t m: Trnsmitte it Z: = = = Stey Stte Reeive it Z: t t t 3 t 4 t 5 t 6 = = = = Brh metri At time t 3 two pths iverging from eh stte. As result two pths entering eh stte time time t 4. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 35 / 8

53 The Viteri Deoing Algorithm = T = = t t t 3 t 4 t 5 t 6 = = = = = T = Stey Stte = = = t t The Viteri t 3 Deoing Algorithm T =3 Input t m: Trnsmitte it Z: Reeive it Z: t t t 3 t 4 t 5 t 6 = = = = T =3 T = T = Brh metri At time t 3 two pths iverging from eh stte. As result two pths entering eh stte time time t 4. Lrger umultive pth metri entering eh stte n e eliminte. In se if there re two pths hving sme umultive pth metri, then one pth is selete ritrrily. = t t t 4 t 3 = t t t 3 t 4 Stte metris T =3 = = = = T =3 T = = = T = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 35 / 8

54 t t The Viteri Deoing Algorithm The Viteri t 3 Deoing Algorithm = T = = T t t t 3 t 4 t 5 t =3 6 = Input t m: Stte metris Stte metris t t t Trnsmitte it Z: t t 3 = = TT = = = = = Reeive it Z: T =3 T =3 t t t 3 t 4 t 5 t 6 = = T = = = T =3 = Brh = T = metri = = = T = = T = = Stey Stte = T = At time t 3 two pths iverging from eh stte. As result two pths entering eh stte time time t 4. Lrger umultive pth metri entering eh stte n e eliminte. In se if there re two pths hving sme umultive pth metri, then one pth is selete ritrrily. t t t 3 t 4 t t t 3 t Stte metris 4 = t = t t 3 t 4 t t t t Stte metris 3 4 T =3 = = T =3 = = T =3 = = T =3 = = = = T = T = = = = = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 35 / 8 T = T =

55 t t t The 3 t Viteri Deoing 4 t Algorithm The t Viteri t Deoing 3 t Algorithm 4 = = T =3 t t t 3 t 4 t 5 t 6 = = =Input t m: T =3 Trnsmitte it Z: = Reeive it Z: t t t 3 t 4 t 5 t 6 = = = T = = = Brh metri = = T = = = = Stey Stte = t t t 3 t 4 t 5 = t t t 3 t 4 t 5 Stte metri T = = = = = T = T =3 = = T = Mnjunth. t P (JNNCE) t t t Coing Tehniques t Junet8, 3 t 36 / 8 t

56 t t t The 3 t Viteri Deoing 4 t Algorithm The t Viteri t Deoing 3 t Algorithm 4 = = T = = T =3 t t t 3 t 4 t 5 t 6 = = =Input t m: T =3 t t t 3 t 4 t t t t Stte metris Trnsmitte it Z: 3 4 = = T = =3 Reeive it Z: t t t 3 t 4 t 5 t 6 = = = T = = = T =3 = = Brh metri = = = T = T = = = = = = T = = Stey Stte = = = = t t t 3 t 4 t 5 t t t 5 = = = = t 3 t 4 t t t 3 t 4 t 5 Stte metri = t t t 3 t 4 t 5 Stte metris T = = T = = T = = T = = T =3 = T =3 = T = = T = Stte metris t 3 t 4 t t t 5 = t t t t 5 t 6 6 = T = Mnjunth. t P (JNNCE) t t t Coing Tehniques t Junet8, 3 t 36 / 8 t t 3 t 4

57 The Viteri Deoing Algorithm The Viteri Deoing Algorithm t t t 3 t 4 t 5 = Input t m: Trnsmitte it Z: Reeive = it Z: t t t 3 t 4 t 5 t 6 = = = Brh metri == = t t t 3 t 4 t 5 Stte metris =t t t 3 t 4 t 5 t T 6 = = = = = = = T = T =3 T = Stey Stte = = = = t t t 3 t 4 t 5 t 6 t t t 3 t 4 t 5 t 6 = = = = Stte me T = T = T = T = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 37 / 8

58 The Viteri Deoing Algorithm The Viteri Deoing Algorithm = = T = t t t 3 t 4 t 5 t t t 3 t 4 t 5 Stte metris = =t t t 3 t 4 t 5 t T 6 Input t m: = = Trnsmitte t it Z: t t 3 t 4 t 5 t t t 3 t 4 t 5 Stte metris = = T Reeive = it Z: = = = T t t t 3 t 4 t 5 t 6 = = = = T = = = T =3 = Brh metri = = T =3 == = = T = = = T = = Stey Stte t t t 3 t 4 t 5 t t = t 6 t 3 t 4 t 5 t = 6 = = = = = = = = = Stte me t t t 3 t 4 Stte metris t 5 t t 6 t = t 3 t 4 t 5 t 6 T = T = = T = T = T = = T = T = = T = Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 37 / 8

59 The Viteri Deoing Algorithm Prokis IV Eition-8-6 G(D) = [, +D, +D+D ] The Viteri Deoing Algorithm Tle: Stte trnsition tle for the (3,,)enoer k= SR SR SR Figure: R=/3 onvolutionl enoer with memory m= 3 n=3 Input Present Stte Next Stte Output () () Tuple Stte s s s s 3 () () () () () () Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 38 / 8

60 The Viteri Deoing Algorithm The Viteri Deoing Algorithm () () () () () () () () S 3 S 3 S 3 S 3 S S S S S S S S S S S S S S Time units S S S S Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 39 / 8

61 The Viteri Deoing Algorithm The Viteri Deoing Algorithm () () S 3 S 3 () () () S 3 () S 3 () () () S S S S S () () () () () () S () S S S S () S S S S S S S S () () Input: S 3 () X 4 S 3 () 6 S 3 () X 4 S 3 () () () () () () S () S S S S S X X X () () () () 3 X S S S S S () () 4 3 X 4 X S S X S X S S S X S () () () r= X Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

62 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

63 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm The eoer strts t the origin noe with the threshol T= n the metri vlue M=. It looks forwr to the est of the k sueeing noes, i.e., with lrgest metri. If M f is the metri of the forwr noe eing exmine n if M f T then the eoer moves to this noe. It heks for the en of tree hs een rehe, otherwise threshol tightening is performe if the noe is exmine for the first time i.e., T is inrese y the lrgest multiple of threshol inrement so tht new threshol oes not exee the urrent metri. If the noe hs een exmine previously, no threshol tightening is performe. Then the eoer gin looks forwr the the est sueeing noe. If M f < T, the eoer looks kwr to the preeing noe. If M is the metri of the kwr noe eing exmine, n if M T, then the T is lowere y n the look forwr to the est noe step is repete. If M T, the eoer moves k to the preeing noe. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 4 / 8

64 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

65 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

66 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 3 At timet 3 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

67 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 3 At timet 3 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 4 At timet 4 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

68 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 3 At timet 3 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 4 At timet 4 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. 5 Count 3 is turnroun riterion eoer ks out tries lternte pth y erementing ount y i. e., ount= t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

69 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 3 At timet 3 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 4 At timet 4 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. 5 Count 3 is turnroun riterion eoer ks out tries lternte pth y erementing ount y i. e., ount= 6 At timet 4 reeives symol n moves rnh eoer moves ownwr n eoes s it gin it is isgreement with ount = 3. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

70 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm At timet reeives symol eoer moves ownwr n eoes s it. At timet reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 3 At timet 3 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount =. 4 At timet 4 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. 5 Count 3 is turnroun riterion eoer ks out tries lternte pth y erementing ount y i. e., ount= 6 At timet 4 reeives symol n moves rnh eoer moves ownwr n eoes s it gin it is isgreement with ount = 3. 7 Count 3 is turnroun riterion ll the lterntive pths hve trverse eoer ks out tries lternte pth y erementing ount i. e., ount= moves to the noe t t 3 level. t t t 3 t 4 t 5 t 6 Z= Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 43 / 8

71 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

72 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

73 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. At timet 5 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

74 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. At timet 5 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. At this ount eoer k up n reset the ounter= timet 5 reeives symol there re two rnhes n n tries lternte pth s with isgreement of n gin ount = 3. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

75 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. At timet 5 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. At this ount eoer k up n reset the ounter= timet 5 reeives symol there re two rnhes n n tries lternte pth s with isgreement of n gin ount = 3. The eoer ks out of this pth n sets ount=, ll the lternte pths hve trverse t t 5 level, so eoer returns to noe t t 4 n resets ount= t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

76 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. At timet 5 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. At this ount eoer k up n reset the ounter= timet 5 reeives symol there re two rnhes n n tries lternte pth s with isgreement of n gin ount = 3. The eoer ks out of this pth n sets ount=, ll the lternte pths hve trverse t t 5 level, so eoer returns to noe t t 4 n resets ount= 3 At t 4 with t s n pth with mkes ount=3 eoer k to the time t noe t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

77 Sequentil Deoing: The Fno Deoing Algorithm Sequentil Deoing: The Fno Deoing Algorithm 8 At timet 3 reeives symol there re two rnhes n n untrie one is eoer moves ownwr n eoes s it with isgreement ount =. 9 At timet 4 reeives symol n moves rnh eoer moves upwr n eoes s it with greement with ount =. At timet 5 reeives symol there re two rnhes n eoer moves upwr ritrrily n eoes s it with isgreement ount = 3. At this ount eoer k up n reset the ounter= timet 5 reeives symol there re two rnhes n n tries lternte pth s with isgreement of n gin ount = 3. The eoer ks out of this pth n sets ount=, ll the lternte pths hve trverse t t 5 level, so eoer returns to noe t t 4 n resets ount= 3 At t 4 with t s n pth with mkes ount=3 eoer k to the time t noe 4 At timet reeives symol n now follows the rnh wor with isgreement of with ount =. t t t 3 t 4 t 5 t 6 Z= Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 44 / 8

78 Feek Deoing Feek Deoing Feek Deoing Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 45 / 8

79 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

80 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

81 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

82 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

83 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

84 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. 6 The next step is to exten the lower prt of the tree. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

85 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. 6 The next step is to exten the lower prt of the tree. 7 In the next step slie over two oe symols. t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

86 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. 6 The next step is to exten the lower prt of the tree. 7 In the next step slie over two oe symols. 8 Upper-hlf metris:,4,3,3 t t t 3 t 4 t 5 t 6 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

87 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. 6 The next step is to exten the lower prt of the tree. 7 In the next step slie over two oe symols. 8 Upper-hlf metris:,4,3,3 9 Lower-hlf metris: 3,,4,4 t t t 3 t 4 t 5 t 6 Z= Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

88 Feek Deoing Feek Deoing Beginning with the first rnh the eoer omputes L Hmming pth metris n eies the it is zero if the minimum istne is in the upper prt of the tree n eies the it is one if the minimum istne is in the lower prt of the tree. Assuming the reeive sequene s Z= n exmining the eight pths from time t through t 3 n omputing the metris of these eight pths. 3 Upper-hlf metris: 3,3,6,4 4 Lower-hlf metris:,,,3 5 The minimum metri is ontine in the lower prt of the tree therefore the first oe it is one. 6 The next step is to exten the lower prt of the tree. 7 In the next step slie over two oe symols. 8 Upper-hlf metris:,4,3,3 9 Lower-hlf metris: 3,,4,4 The sme proeure is ontinue until the entire messge is eoe. t t t 3 t 4 t 5 t 6 Z= Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 46 / 8

89 Struturl Properties of Convolutionl Coes Struturl Properties of Convolutionl Coes Struturl Properties of Convolutionl Coes Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 47 / 8

90 Struturl Properties of Convolutionl Coes Struturl Properties of Convolutionl Coes Grphilly, there re three wys to represent onvolution enoer, in whih we n gin etter unerstning of its opertion. These re: Stte Digrm Tree Digrm 3 Trellis Digrm Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 48 / 8

91 Struturl Properties of Convolutionl Coes Struturl Properties of Convolutionl Coes The stte igrm is grph of the possile sttes of the enoer n the possile trnsitions from one stte to nother. The igrm shows the possile stte trnsitions. Eh irle in the stte igrm represents stte. At ny one time, the enoer resies in one of these sttes. The lines to n from it shows the stte trnsition tht re possile s its rrive. Assuming tht the enoer is initilly in stte S (ll-zero stte), the oe wor orresponing to ny given informtion sequene n e otine y following the pth through the stte igrm n noting the orresponing outputs on the rnh lels. Following the lst nonzero informtion lok, the enoer is return to stte S y sequene of m ll-zero loks ppene to the informtion sequene. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 49 / 8

92 Struturl Properties of Convolutionl Coes Trellis Digrm It looks like tree struture with emerging rnhes hene it is lle trellis. A trellis is more instrutive thn tree in tht it rings out expliitly the ft tht the ssoite onvolutionl enoer is finite stte mhine. The onvention use in Figure to istinguish etween input symol n is s follows. A oe rnh proue y n input is rwn s soli line, wheres oe rnh proue y n input is rwn s she line. Eh input sequene orrespons to speifi pth through the trellis. The trellis ontins (L+K) levels, where L is the length of the inoming messge sequene, n K is the onstrint length of the oe. The levels of the trellis re lele s j=,, L+K-. The first (K-) levels orrespons to the enoer s eprture from the initil stte, n the lst (K-) levels orrespons to the enoer s return to the stte. Not ll these stte n e rehe in these two portions of the trellis. After the initil trnsient, the trellis ontins four noes t eh stge, orresponing to the four sttes. After the seon stge, eh noe in the trellis hs two inoming pths n two outgoing pths. Of the two outgoing pths, orrespons to the input it n the other orrespons input it. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

93 Struturl Properties of Convolutionl Coes Signl Flow Grph: Signl Flow Grph: Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

94 Struturl Properties of Convolutionl Coes Signl Flow Grph: Signl Flow Grph: Any system opertion n e represente y set of liner lgeri equtions. The lgeri equtions represent the reltionship etween system vriles. A signl flow grph is grphil representtion of reltionship etween vriles of set of liner equtions. x + x 3 = x x + fx 4 = x 3 ex + x 3 = x 4 gx 3 + hx 4 = x 5 f 3 4 e g h 5 Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 5 / 8

95 Struturl Properties of Convolutionl Coes Signl Flow Grph: Bsi Terminologies: Noe: Noe represents the system vrile. Brnh: The line joining x n x forms the rnh. Brnh j-k origintes t noe j n termintes upon noe k, the iretion from j to k eing inite y n rrowhe on the rnh. Eh rnh j-k hs ssoite with it quntity lle the rnh gin n eh noe j hs n ssoite quntity lle the noe signl. Soure (Input) Noe: A soure is noe hving only outgoing rnhes. Sink (output) Noe: A sink is noe hving only inoming rnhes. Pth: A pth is ny ontinuous suession of rnhes trverse in the inite rnh iretions. Forwr pth: A forwr pth is pth from soure to sink long whih no noe is enountere more thn one. Loop: A loop is pth whih origintes n termintes t the sme noe ( lose pth without rossing the sme point more thn one). Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 53 / 8

96 Struturl Properties of Convolutionl Coes Signl Flow Grph: Non-Touhing Loops: Loops re si to e non-touhing if they o not psses through ommon noe. Or two loops re non-touhing or non-interting if they hve no noes in ommon. Feek Loop: A feek loop is pth tht forms lose yle long whih eh noe is enountere one per yle. Pth Gin: A pth gin is the prout of the rnh gins long tht pth. Loop gin: The loop gin of feek loop is the prout of the gins of the rnhes forming tht loop. The gin of flow grph is the signl ppering t the sink per unit signl pplie t the soure. To fin the grph gin, first lote ll possile sets of non-touhing loops n write the lgeri sum of their gin prouts s the enomintor of. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 54 / 8

97 Struturl Properties of Convolutionl Coes Signl Flow Grph: Mson s Gin Fomul: The generl expression for grph gin my e written s G k k k G = where G k = gin of the kth forwr pth = P m + P m P m m m m P mr = Gin prout of the mth possile omintion of non touhing loops = loop gins + omintion of two non touhing loops omintion of 3 non touhing loops... k = The vlue of for tht prt of the grph not touhing the kth forwr pth Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 55 / 8

98 Struturl Properties of Convolutionl Coes Signl Flow Grph: The Trnsfer Funtion of Convolutionl Coe: For every onvolutionl oe, the trnsfer funtion gives informtion out the vrious pths through the trellis tht strt from the ll-zero stte n return to this stte for the first time. Aoring to the oing onvention esrie efore, ny oe wor of onvolutionl enoer orrespons to pth through the trellis tht strts from the ll-zero stte n returns to the ll-zero stte. The performne of onvolutionl oe epens on the istne properties of the oe. The free istne of onvolutionl oe is efine s the minimum Hmming istne etween ny two oe wors in the oe. A onvolutionl oe with free istne free n orret t errors if n only if free is greter thn t. The free istne n e otine from the stte igrm of the onvolutionl enoer. Any nonzero oe sequene orrespons to omplete pth eginning n ening t the ll-zero stte. The stte igrm is moifie to grph lle s signl flow grph with single input n single output. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 56 / 8

99 Struturl Properties of Convolutionl Coes Signl Flow Grph: A signl flow grph onsists of noes n irete rnhes n it opertes y the following rules: A rnh multiplies the signl t its input noe y the trnsmittne hrterizing tht rnh. A noe with inoming rnhes sums the signls proue y ll of these rnhes. 3 The signl t noe is pplie eqully to ll the rnhes outgoing from tht noe. 4 The trnsfer funtion of the grph is the rtio of the output of the signl to the input signl The rnhes of the grph is lele s D =, D, D, D 3 where the exponent of D enotes the Hmming istne etween the sequene of output its orresponing to eh rnh n the sequene of output its orresponing to the ll-zero rnh. Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 57 / 8

100 Struturl Properties of Convolutionl Coes Signl Flow Grph: There is one pth whih eprts from ll zeros t time t n merges with zeros t time t 4 with istne of 5. There re two pths of istne 6 one eprts from ll zeros t time t n merges with zeros t time t 5 n nother eprts from ll zeros t time t n merges with zeros t time t 6. t t t 3 t 4 t 5 t 6 = = = = Stey Stte Figure: Trellis igrm = = = = t t t 3 t 4 t 5 t 6 Figure: Trellis igrm showing istne from ll zeros pth Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 58 / 8

101 Struturl Properties of Convolutionl Coes Signl Flow Grph: Brnhes of the stte igrm re lele s D, D or D where the D enotes the Hmming istne from the rnh wor of tht rnh to the ll zeros rnh. Noe is split into two noes lele s n e one whih represents the input n the other output of the stte igrm. All pths originting t = n terminting terminting t e = Output it () Input it () = () D () = = () () = () () = D = D D = D = D e ()= D = Figure: Signl flow grph Figure: Stte igrm X = D X + X X = DX + DX X = DX + DX X e = D X Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 59 / 8

102 Struturl Properties of Convolutionl Coes Signl Flow Grph: D Output it () () = () D = D Input it () = = () = D = D = D e ()= D = () = () Figure: Signl flow grph () Figure: Stte igrm There re two forwr pths Gin of the forwr pth (e) is: P = (D D D ) = (D 5 ) Gin of the forwr pth (e) is:p = (D D D D ) = (D 6 ) D D D D D = D Figure: Loops in signl flow grph = (L + L + L 3 ) + (L L ) = (D + D + D 3 ) + (D ) = D D 3 + D 3 = D = (L ) = D = G = = G k k k T (D) = P + P D 5 ( D) + D 6 () D = D 5 D 6 + D 6 = D5 D D There re three loops in grph: Gin of the self loop t () is: L = D Gin of the loop for () is:l = (D ) = D Gin of the loop for () is:l 3 = (D D D) = D 3 T (D) = D 5 + D 6 + 4D l D l+5 [Use Binomil expnsion: ( x) = + x + x + x 3 + x 4 +..] Mnjunth. P (JNNCE) Coing Tehniques June 8, 3 6 / 8