Turbo coding (CH 16)
Parallel concatenated codes Distance properties Not exceptionally high minimum distance ut !"e#! code#ords o" !lo#! #eight Trellis complexity $sually extremely high trellis complexity Decoding %uboptimum (but close to &') iterati(e (turbo) decoding Per"ormance 'o# error probability at %N)s close to the %hannon limit
History
%hannon (1+,-). The channel/s %N) (012N channel) determines the capacity C3 in the sense that "or code rates R 4 C #e can ha(e error5"ree transmission 6or each code rate R #e can compute the %hannon limit Di""icult to approach the %hannon limit by classical methods ut777 2allager (1+61) and Tanner (1+-1) errou3 2la(ieux3 and Thitima8shima in(ented turbo codes in 1++9
:ncoding
:ncode in"ormation by a systematic encoder $sually a recursi(e systematic rate ; con(olutional encoder )eorder in"ormation bits :ncode permuted in"ormation bits again3 using a recursi(e systematic encoder (may be the same)7 Delete the systematic bits this time
:xample3 more detailed
%tarting #ith rate ; component codes #e get approximately rate 1>9 Can be punctured (parity or in"ormation bits) to ad8ust the rate Can add more interlea(ers and component codes to lo#er the rate 'arge in"ormation bloc=s gi(e etter distance properties etter #or=ing decoding algorithm %imple component codes (?,@) are best "or moderate :)s Anterlea(er design is di""icult3 and there is no =no#n techniBue to design the best one7 Design criteria are. Amplementation complexity Per"ormance at lo# %N) (pseudorandom5li=e) Per"ormance at high %N) (high minimum distance) Disadvantage: Delay in decoding <
)emar=s
:xample
1ater"all region
:rror "loor region
? ,3 K ? 6<<96
6
Distance properties o" turbo codes
Classical coding approach is to maximiDe minimum distance Ne# approach. 6e# code#ords #ith lo# #eights )ecall. An a "eed"or#ard encoder3 a lo#5#eight code#ord is usually generated by a lo#5#eight input seBuence An a "eedbac= encoder3 a lo#5#eight code#ord is usually generated by an input in"ormation seBuence that is a multiple o" the "eedbac= polynomial7 E"ten higher input #eights %pectral thinning
C
%pectral thinning. :xample
%pectral thinning. :xample
%pectral thinning. :xample
1F
%pectral thinning. )emar=s
)eBuires "eedbac= encoder %ingle one input in "eed"or#ard encoder. 'ocal #eight gain e""ect %ingle one input in "eedbac= encoder. 2ains #eight (at least) until next input one is seen )eBuires an interlea(er to ma=e the code time5(arying %tronger e""ect "or longer bloc= lengthsG similar #eight spectrum as random codes &oderate e""ect on minimum distance
11
Anterlea(ers "or turbo codes
2oal. Anput patterns #hich produce lo#5#eight #ords in one component code should map through the interlea(er to patterns #hich produce high5#eight #ords in the other component code Anterlea(ers #ith !traditional! structure is usually bad "or turbo codes Anterlea(ers #ith a randomli=e structure achie(e the abo(e goal to a larger extent Anterlea(ers #hich are pseudorandom #ith constraints on spreading properties3 and #ith additional constraints based on the particular component encoders3 ha(e pro(ided good results ut such !randomli=e! interlea(ers may be hard to implement in an e""icient manner Dithered relati(e prime (D)P) and Buadratic permutation polynomial (HPP) interlea(ers are easy to implement and ha(e 1* (ery good properties as #ellI
loc= interlea(er. :xample
Critical input seBuence is (1JD<)Dl
19
:""ects o" bloc= interlea(er
1,
Pseudorandom interlea(ers
Kour "a(ourite (pseudo)random generator together #ith table loo=up Huadratic congruence cm km(mJ1)>* (mod K)3 F m 4 K3 to generate an index mapping "unction cm cmJ1 (mod K)3 k is an odd integer :xample #ith K ? , and k ? 1. (cm) ? (F31393*) and interlea(er is de"ined by (1393F3*)7 This pattern can also be shi"ted cyclically %tatistical properties are similar to random interlea(ers #hen K is a po#er o" * 1<
Turbo decoding
Channel
16
Turbo decoding
Lc ? ,Es >NF L(1)(ul) = ln(P(ul ? J1Lr13 La(1)) > P(ul ? 51Lr13 La(1)))
%A%E 1
%A%E *
0priori
:xtrinsic
1C
