In TCM it is easier to deal with {-1,+1} than {0,1} initially.

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1 Repreentation or TCM Ungeroec repreentation: (conidered earlier A inary convolutional encoder A modulator part. Analytical repreentation: The eect o the trelli encoder i to tranorm a equence o input ymol to a equence o modulated- channel ignal. I = (a I, a I-,, a I-L The unction (. mut e nonlinear. panding the unction : Let a, a,., a n tae on value in the et {,}. (a, a,, a n = ( Σ(i I ( a I Σ (j>i ij ( a I a j Σ (h>j>i hji ( a i a j a h..n (n a a..a n where the coeicient (l are a et o contant that may e determined rom the n value the unction can aume. In TCM it i eaier to deal with {-,} than {,} initially. I = a I (,,, n = d ( Σ(i d I ( I Σ (j>i d ij ( I j Σ (h>j>i d hji ( i j h. d.n (n.. n (****

2 Let denote a column vector whoe n component are all the value that can tae. Let d denote the column vector o the (unnown coeicient. Finally let B e a n n matri where each row repreent the n value taen y all the product o the I in (**** or each - peciic n-tuple,,.., n or the correponding value o. Then (**** can e given in the matri orm = B d We ee that B ha orthogonal row (and column, and thu B i a hadamard Matri. B B T = n I B = ( / n B T Thereore the vector o the unnown coeicient can e computed eaily y d = ( / n B T We can ay that d i the Hadamard tranorm o the vector.

3 ample: AM ignal A rate ½ TCM cheme with -AM ignal with memory v=. The value along the tranition etween tate repreent the output ymol (,,, that i going rom preent tate (, to the net tate (, in repone to the input ymol. For n= y chooing d ( = (,, = d ( d ( d ( d ( d ( d ( d ( From the trelli diagram, we have (,, = - (-,,= (,-,=- (-,-,= (,,-= (-,,-=- (-,-,-=- (,-,= We have T = [ ] d T = [ d ( d ( d ( d ( d ( d ( d ( ] and

4 = = = = = ( ( ( ( ( ( (, (,,, (,, ( (,,,, (, (,,, (,, ( B We get (,, =

5 Implementation: comined modulation / coding proce = - - Conider now z and z a repreenting output MSB and LSB repectively. Output MSB z - - LSB z - - ignal - - = -z z From aove = y comparing z = MSB z = LSB The correponding relation or and variale are y = a a (mod - MSB y = a LSB 5

6 Thi will give the convolutional encoder / mapper realization. y = a a (mod - MSB y = a LSB - - y y (chec!! Similar analytical epreion can e otained or PS, QAM cheme and alo or MTCM cheme. Reerence or thi ection:. R. Calderan and J.. Mazo, A new decription o trelli code, I Tranaction Inormation Theory, Vol. IT-, pp , ov TCM Tet: Chapter 5 Further Reading: Multidimenional Modulation Rotationally Invariant Trelli Code (RIC Channel Coding or Telecommunication: Chapter on Coded Modulation (Martin Boert 6

7 Perormance Analyi o TCM The error event o length d i the event that the decoder chooe the incorrect equence intead o the tranmitted equence correponding to the incorrect trelli path that leave rom the correct path and remerge with it eactly d dicrete time later. The union ound tate that the um o the proailitie o the individual event i greater than or equal to the proaility o the union o the event. Let u aume that the code i ha uniorm error property, i.e., the error proaility doen t depend on the tranmitted equence. Ungeroec TCM i uually the cae. The upper ound o the it error proaility i uually otained through the union ound a P = ad (, Pd (, ( d = where i the numer o it per encoding interval, a d (, i the numer o error it correponding to the error event (, o length d, P d ( i the pairwie error proaility, i.e., the proaility o decoder chooing the equence intead o the tranmitted equence. From ( we can aume that the tranmitted equence i all zero equence. The incorrect path i a path leaving rom the all zero equence and merging with it later.

8 TCM or Frequency-Flat Fading Channel The deign criterion o TCM in AWG channel i to maimize the ree uclidean ditance ( ree d []. However, TCM with optimum ( ree d might not perorm well in requency-lat ading channel. Divalar and Simon derived the perormance ound o TCM or requency-lat ading channel []. Aumption: Frequency-lat Rician ading channel. Ideal interleaving (uncorrelated ading Perect channel etimation η P ep ( ( where i the ratio o power in the direct plu pecular component to that in the diue component, i the ymol energy, i the one-ided noie power pectral denity, η i the et o time uch that. q. ( can e written a ep ( d P ( with = η d ln ( d d

9 ote that or = (no ading d = = and d (5 and d i merely the um o the uclidean ditance along the error event path For = (Rayleigh ading d = and d = ln (6 and or large / value, the pairwie error proaility ecome ( P (7 η i.e., it i inverely proportional to the product o the quared uclidean ditance along the error event path. Sutituting (7 into (, we have P a(, d = η The it error proaility will e dominated y the term that ha the lowet rate o decent with /. Thi correpond to the error event path with the mallet numer o ymol in η. Thi path i reerred to hortet error event path. Deine the length o the hortet error event path y the numer o nonzero pairwie ditance etween ymol along the error event. Thereore, the deign criterion o TCM in the requency-lat independent ading channel i to maimize the length o the hortet error event path and the product o the ranch ditance along that path. ote that the length o the hortet error event path i alo called eective code length (CL. ample TCM ( Four-tate Ungeroec TCM [] TCM ( Four-tate Wilon/Leung TCM [] (8

10 6 v u u D D v v 8-PS Mapping 57 6 Four-tate Ungeroec' TCM v 75 6 u u D D v v 8-PS Mapping 57 Four-tate non-parallel tranition TCM 6 57 Code TCM ( TCM ( ective Code Length Minimum uclidean ditance..59 Minimum Product ditance..7. Perormance o TCM cheme over AWG channel. Perormance o TCM Scheme over Frequency-Flat Rayleigh Fading Channel.-.- BR tate Ungeroec' TCM -tate Wilon/Leung' TCM 6 8 / BR.-.- -tate Ungeroec' TCM.- -tate Wilon/Leung' TCM /

11 TCM or Frequency-Selective Fading Channel In requency-elective ading channel, the perormance o code deteriorate ecaue there i overlapping etween adjacent ymol or ISI. qualization can mitigate the eect o ISI. Joint MLS equalization and decoding o TCM wa dicued y Chiu and Chao []. Viteri decoding i perormed on the comined trelli o the channel and decoder. The deign criteria o TCM in requency-elective ading channel with joint MLS equalization and decoder were concluded in [5]. The parameter that aect the perormance are the length o the hortet error event path and the product o the ISI ditance deined a: jth, L ISI ditance = σ (9 j= gj j j where L i the channel memory, σ i hal o the variance o the channel impule repone o path gj However, Hamied and Stuer [6] ound that uncoded ytem outperorm the trelli-coded ytem when joint MLS equalization and decoding i ued. Thi i ecaue the interleaving cannot e ued. A more appropriate technique i to ue a eparate equalizer and a decoder with a deinterleaver etween them. Variou ot-output equalizer have een employed with TCM in [7]. The reult howed the eneit o ot-output over hard-output igniicantly. Chen and Chuang [8] tudied the perormance o TCM without equalization. They concluded that oth uilt-in time diverity (length o the hortet error event path and uclidean ditance are important or TCM cheme in requency-elective ading channel. The new TCM code deign in requency-elective ading channel remain open or variou type o receiver. 5

12 Reerence. G. Ungeroec, Channel Coding with Multilevel/Phae Signal, I Tran. Inorm. Theory, vol. IT-8, no., pp , Jan.98.. D. Divalar and M.. Simon, The Deign o Trelli Coded MPS or Fading Channel: Perormance Criteria, I Tran. Commun., vol. 6, no. 9, pp. -, Sept S.G. Wilon and Y.S. Leung, Trelli-Coded Phae Modulation on Rayleigh Channel, Proceeding o the 987 Int. Con. on Commun.(ICC 87, pp. 79-7, M. Chiu and C. Chao, Perormance o Joint qualization and Trelli-Coded Modulation on Multipath Fading Channel, I Tran. Commun., vol., no. //, pp. -, Fe./Mar./Apr P. Taraa and. Rajatheva, Upper ound on the it error proaility o TCM over requency-elective Rayleigh ading channel, lectronic Letter, vol. 5, no., pp. 8-8, Oct th, A. Hamied and G. L. Stuer, Perormance o Trelli Coded Modulation or qualized Multipath Fading ISI Channel, I Tran. Vehicular Tech., vol., no., pp. 5-58, Fe H. Ha and R.M.A.P. Rajatheva, Perormance o Trelli-Coded Modulation on Frequency-Selective Rayleigh Fading Channel with Sot-output Algorithm, in VTC 98 Con. Rec., 8 th I Vehicular Technology Conerence 998, vol..,