Module 6. Channel Coding. Version 2 ECE IIT, Kharagpur

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Clifford Burns
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1 Module 6 Chael Codg

2 Lesso 36 Coded Modulato Schemes

3 After readg ths lesso, you wll lear about Trells Code Modulato; Set parttog TCM; Decodg TCM; The modulated waveform a covetoal ucoded carrer modulato scheme cotas all formato about the modulatg message sgal. As we have dscussed earler Module #5, the modulatg sgal uses the quadrature carrer susods PSK ad QAM modulatos. The modulator accepts the message sgal dscrete tme dscrete ampltude from (bary or multlevel) ad processes t followg the chose modulato scheme to satsfy the trasmsso ad recepto requremets. The modulatg sgal s geerally treated as a radom sgal ad the modulatg symbols are vewed as statstcally depedet ad havg equal probablty of occurrece. Ths tradtoal approach has several merts such as, () smplfed system desg ad aalyss because of modular approach, () desg of modulato ad trasmsso schemes whch are depedet of the behavor of sgal source, () avalablty of well developed theory ad practce for the desg of recevers whch are optmum (or ear optmum). To put t smply, the modular approach to system desg allows oe to desg a modulato scheme almost depedet of the precedg error cotrol ecoder ad the desg of a ecoder largely depedet of the modulato format. Hece, the ed-toed system performace s made up of the gas cotrbuted by the ecoder, modulator ad other modules separately. However, t s terestg to ote that the demarcato betwee a coder ad a modulator s deed artfcal ad oe may very well mage a combed coded modulato scheme. The tradtoal approach s based more towards optmzato of a ecoder depedet of the modulato format ad o optmzato of a modulato scheme depedet of the codg strategy. I fact, such approach of optmzato at the subsystem level may ot esure a ed-to-ed optmzed system desg. A systems approach towards a combed codg ad modulato scheme s meagful order to obta better system performace. Sgfcat progress has take place over the last two decades o the cocepts of combed codg ad modulato (also referred as coded modulato schemes). Several schemes have bee suggested the lterature ad advaced modems have bee successfully developed explotg the ew cocepts. Amogst the several strateges, whch have bee popular, trells coded modulato (TCM) s a promet oe. A trells coded modulato scheme mproves the relablty of a dgtal trasmsso system wthout badwdth expaso or reducto of data rate whe compared to a ucoded trasmsso scheme usg the same trasmsso badwdth. We dscuss the basc features of TCM ths secto after troducg some cocepts of dstace measure etc., commo to all coded modulato schemes. A TCM scheme uses the cocept of tree or trells codg ad hece s the ame TCM. However

4 some terestg combed codg ad modulato schemes have bee suggested recetly followg the cocepts of block codg as well. Dstace Measure Let us cosder a -dmesoal sgal space of Fg showg a set of sgal pots. The two dmesos are defed by two orthoormal bass fuctos correspodg to formato symbols, whch are to be modulated. The dmesos ad hece the x- ad y-axes may also be terpreted as real ad magary axes whe complex low pass equvalet represetato of arrowbad modulated sgals s cosdered. The pots the costellato are dstgushable from oe aother as ther locatos are separate. Several possble subsets have bee dcated Fg capturg multple modulato formats. Subset Subset Subset 3 Subset 4 Fg Some two-dmesoal sgal costellatos (M = m, m=,,8) of TCM codes.

5 Let there be N vald sgal pots, deoted by (x + jy ), N, the two dmesoal sgal space such that each pot sgfes a formato bearg symbol (or sgal) dfferet from oe aother. Now, the Eucldea dstace d j betwee ay two sgal pots, say (x, y ) ad (x j, y j ) ths D Cartesa sgal space may be expressed as, d j =(x -x j ) +(y -y j ). Note that for detectg a receved symbol at the demodulator presece of ose etc., t s mportat to maxmze the mmum dstace (say, d m ) betwee ay two adjacet sgal pots the space. Ths mples that the N sgal pots should be well dstrbuted the sgal space such that the d m amogst them s the largest. Ths strategy esures good performace of a demodulato scheme especally whe all the symbols are equally lkely to occur. Suppose we wsh to trasmt data from a source emttg two formato bts every T secods. Oe ca desg a system several ways to accomplsh the task such as the followg: () use ucoded QPSK modulato, wth oe sgal carryg two formato bts trasmtted every T secods. () use a covolutoal code of rate r = /3 ad same QPSK modulato. Each QPSK symbol ow carres 4/3 formato bts ad hece, the symbol durato should be reduced to T/3 secods. Ths mples that the requred trasmsso badwdth s 50% more compared to the scheme (). () use a covolutoal code of rate r = /3 ad 8-Phase Shft Keyg (8PSK) modulato scheme to esure a symbol durato of T sec. Each symbol, ow cosstg of 3 bts, carres two bts of formato ad o expaso trasmsso badwdth s ecessary. Ths s the basc cocept of TCM. Now, a M-ary PSK modulato scheme s kow to be more ad more power effcet for larger values of M. That s, to esure a average BER of, say, 0-5, 8-PSKmodulato scheme eeds more E b /N o compared to QPSK ad 6-PSK scheme eeds eve more of E b /N o. So, oe may apprehed that the scheme () may be power effcet but actually ths s ot true as the assocated covolutoal code esures a cosderable mprovemet the symbol detecto process. It has bee foud that a mpressve overall codg ga to the tue of 3 6dB may be acheved at a average BER of 0-5. So, the et result of ths approach of combed codg ad modulato s some codg ga at o extra badwdth. The complexty of such a scheme s comparable to that of a scheme employg codg (wth soft decso decodg) ad demodulato schemes separately. TCM s extesvely used hgh bt rate modems usg telephoe cables. The addtoal codg ga due to trells-coded modulato has made t possble to crease the speed of the trasmsso.

6 Set Parttog The cetral feature of TCM s based o the cocept of sgal-set parttog that creates scope of redudacy for codg the sgal space. The mmum Eucldea dstace (d m ) of a TCM scheme s maxmzed through set parttog. The cocept of set parttog s show Fg for a 6-QAM sgal costellato. The costellato cossts of 6 sgal pots where each pot s represeted by four formato bts. The sgal set s successvely dvded to smaller sets wth hgher values of mmum tra-set dstace. The smallest sgal costellatos fally obtaed are labeled as D0, D,, D7 Fg The followg basc rules are followed for set-parttog: Rule #: Members of the same partto are assged to parallel trastos. Rule #: Members of the ext larger partto are assged to adjacet trastos,.e. trastos stemmg from, or mergg the same ode. Assumpto: All the sgal pots are equally lkely to occur. Z 0 = 0 A Z = 0 B 0 Z = 0 B Z = 0 C 0 C C C 3 Z = 0 Z = 0 Z = 0 D 0 D D D 4 D 6 D 3 D 5 D 7 Fg Set parttog of the 6QAM costellato A TCM ecoder cossts of a covolutoal ecoder cascaded to a sgal mapper. Fg shows the geeral structure of a TCM ecoder. As show the fgure, a group of m-bts are cosdered at a tme. The rate /(+) covolutoal ecoder codes formato bts to a codeword of (+) bts whle the remag (m-) bts are ot ecoded. However, the ew group of (++m-) = m+ bts are used to select oe of the m+ pots from the sgal space followg the techque of set-parttog.

7 m b m m m + m+ Z Sgal Mappg m b m Z Ucoded Bts b + b b Covolutoal Ecoder Rate = + Z Z Z Select Sgal From Subsets Coded Bts Select Subsets X Fg Geeral structure of TCM ecoder The covolutoal ecoder may be oe of several possble types such a lear o-systematc feed forward type or a feedback type etc. Fg shows a ecoder of r = /3, sutable for use wth a 8-pot sgal costellato. x : x h = Z Z 0 h 3 = T + T + T 0 h 0 = 0 Z Fg Structure of a TCM ecoder based o r=/3 covolutoal codg ad 8-PSK modulato format Decodg TCM The cocept of code trells, as troduced Lesso #35, s expaded to descrbe the operato of a TCM decoder. Dstace propertes of a TCM scheme ca be studed through ts trells dagram the same way as for covolutoal codes. The TCM decoder-demodulator searches the most lkely path through the trells from the receved

8 sequece of osy sgal pots. Because of ose, the chose path may ot always cocde wth the correct path. The Vterb algorthm s also used the TCM decoder. Note that there s a oe-to-oe correspodece betwee sgal sequeces ad the paths a trells. Hece, the maxmum-lkelhood (ML) decso o a sequece of receved symbols cossts of searchg for the trells path wth the mmum Eucldea dstace to the receved sequece. The resultat trells for TCM looks somewhat dfferet compared to the trells of a covolutoal code oly. There wll be multple parallel braches betwee two odes all of whch wll correspod to the same bt patter for the frst (+) bts as obtaed from the covolutoal ecoder.

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions.

K-Map 1. In contrast, Karnaugh map (K-map) method provides a straightforward procedure for simplifying Boolean functions. K-Map Lesso Objectves: Eve though Boolea expressos ca be smplfed by algebrac mapulato, such a approach lacks clear regular rules for each succeedg step ad t s dffcult to determe whether the smplest expresso