Spring Ammar Abu-Hudrouss Islamic University Gaza

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1 Chaper 7 Reed-Solomon Code Spring 9 Ammar Abu-Hudrouss Islamic Universiy Gaza ١ Inroducion A Reed Solomon code is a special case of a BCH code in which he lengh of he code is one less han he size of he field over which he symbols are defined I consiss of sequences of lengh q whose roos include consecuive powers of he primiive elemen of GF(q) The Fourier ransform over GF(q) will conain consecuive zeros Noe ha because boh he roos and he symbols are specified in GF(q), he generaor polynomial will have only he specified roos; (NO Conjugaes) Slide ٢ ١

2 Inroducion To consruc he generaor for a Reed Solomon code, and we decide ha he roos will be from i o i+-, he generaor polynomial will be g i i i i x Example: Suppose we wish o consruc a double-error correcing, lengh 7 RS code; we firs consruc GF(8) using he primiive polynomial + + as shown We decide o choose i =, placing he roos from o The generaor polynomial is x g g x Slide ٣ Time Domain Encoding The encoding of a Reed Solomon code can be done by a long division mehod similar o ha of Chaper Example For he Reed Solomon code above, encode he daa sequence The daa maps o he symbols Four zeros are appended, corresponding o he four pariy checks o be generaed, and he divisor is he generaor sequence Slide ٤ ٢

3 Time Domain Encoding The remainder is so ha he codeword is Expressed as a binary sequence his is Slide ٥ Time Domain Encoding The encoder circui for a Reed Solomon code is shown g n-k- g g g U() Slide ٦ ٣

4 Time Domain Encoding For our example he encoder will be U() Slide ٧ Decoding Reed Solomon Code The frequency domain algebraic decoding mehod explained in Chaper Error value calculaion can be done using he Forney algorihm Where we found ha A z E z f z z z k Where (z) is he connecion polynomial, E (z) is syndrome in frequency domain and (z) is - order evaluaor polynomial Slide ٨ ٤

5 ٥ Slide ٩ Decoding Reed Solomon Code We calculae he error evaluaor polynomial and also '(z), he formal derivaive of he connecion polynomial This is found o be In oher words, ge rid of he zero coefficien of A and hen se all he odd erms in he resuling series o zero The error value in posiion m is now evaluaed a z = m The parameer i is he saring locaion of he roos of he generaor polynomial even z z odd z z z z z e i m ' / Slide ١٠ Decoding Reed Solomon Code Example Consider he codeword previously generaed for he double-error correcing (7, ) RS code We creae errors in posiions and, assuming ha we receive The frequency domain syndrome of his sequence is S S S S

6 Decoding Reed Solomon Code Now we form he key equaion for which he soluion is Which leads o And he connecion polynomial is z z z The roos for he connecion polynomial is and Slide ١١ Decoding Reed Solomon Code Having found wo roos for a connecion polynomial of degree indicaes successful error correcion wih errors locaed a posiions - and -, ie posiions and We now calculae he error evaluaor polynomial, aking he powers from o of S(z)A(z) Therefore e m z S S z S z z z z ' z m z Evaluaing a m = and z z m z Slide ١٢ ٦

7 Decoding Reed Solomon Code This leads o e e The received symbol a posiion is herefore correced o The received symbol a posiion is correced o This successfully complees he decoding Slide ١٣ Frequency domain encoding of RS Code As he Fourier ransform of a Reed Solomon code word conains n k consecuive zeros, i is possible o encode by considering he informaion o be a frequency domain vecor, appending he appropriae zeros and inverse ransforming Example We will choose again a (7, ) double-error correcing RS code over GF( ) Le he informaion be,, The frequency domain code word is, and he inverse is We now creae a wo-symbol error, say in posiion and in posiion, as in our previous example The received sequence is Slide ١٤ ٧

8 Frequency domain encoding of RS Code The decoding proceed wih finding he Fourier ransform of he sequence which gives wih syndrome being We now form he key equaion wih = and A = : The soluion is Slide ١٥ Frequency domain encoding of RS Code We use he shif regiser shown o generae he code sequence in he frequency domain In our example =, =, and = E =, E = Cyclic he shif regiser generaes, which are he frequency domain error a locaions, and Slide ١٦ ٨

9 Frequency domain encoding of RS Code The nex wo values generaes which are he syndrome componens S and S so he code will repea The complee error sequence in he frequency domain is Adding his o he Fourier ransform of he received sequence which is The resul is, which is he original codeword wihou errors wih four zeros syndrome Slide ١٧ Frequency domain encoding of RS Code Example For he previous example consider hree errors have occurred, and he received sequence is The Fourier ransform is R z z z z z z z The key equaions are found o be Or =, =, and = Slide ١٨ ٩

10 Frequency domain encoding of RS Code The shif regiser become Afer iniializing he shif regiser wih The oupu is followed by which is no S his means ha here is an error which we canno correc Slide ١٩ Frequency domain encoding of RS Code Example For he previous example consider single errors have occurred, and he received sequence is The Fourier ransform is R z z z z z z z The key equaions are found o be Or =, =, and = Slide ٢٠ ١٠

11 Frequency domain encoding of RS Code The shif regiser become Afer iniializing he shif regiser wih The oupu is followed by which is he value of s The complee error sequence is Adding his o The resul is, Slide ٢١ Erasure Decoding Reed-Solomon has he abiliy o recover even known as erasure Erasure is a symbol which is more likely o be in error Example Consider he same previous example bu wih one error a posiion and wo erasures in posiions and The received sequence will be aken as The Fourier ransform is R z z z z z z z Slide ٢٢ ١١

12 Erasure Decoding The lower erms of R forms he syndrome The erasure polynomial is z z z z z The error locaor polynomial is and he produc is z z zz z z z By convoluion wih he syndrome polynomial Slide ٢٣ Erasure Decoding Which means ha = which leads o zz z z The following circui can be used for recursive exension Slide ٢٤ ١٢

13 Erasure Decoding Loading wih values and shifing gives he sequence and hen regeneraing he syndrome The decoding is successful The erms are added o componen from he Fourier ransform of he received sequence yo give he recovered informaion If we choose Forney algorihm o correc errors in ime domain z S zz z mod z z z We need also he formal derivaive of (z)(z) which is z Slide ٢٥ Erasure Decoding Therefore a posiion i he error value is z z ei z i z Evaluaing his a - and - which are he roos of (z) and - which is roo of (z) and found by Chien search The resuls are e e,, e The received sequence hen is, corresponding o he original codeword Slide ٢٦ ١٣

14 ١٤ Slide ٢٧ Welch-Berlekamp Algorihm Consider he previous double error correcion examples we ransmi a codeword from a (7,) RS code wih roos,,, and To sar wih, we need o calculae some values needed as inpu for he algorihm Therefore g =, g =, g =, and g = For error value calculaion, we compue g x g k n j k n i i k n k n C Slide ٢٨ Welch-Berlekamp Algorihm For our example C = we now need for each daa locaion o find he value hi = C /g( i ) The values are found o be Assume now we receive, The firs sep is o compue he syndrome by long division This found o be s =, s =, s =, and s = The inpu o WB algorihm is he se of poin (S j, j ) where S j = s j /g j / / / h h h

15 Welch-Berlekamp Algorihm The inpu poins are herefore (, ),(, ),(, ),(, ) We need o find wo polynomials Q() and N() for which Q j j S N for j n k j And he lengh L[Q(),N()], defined as he maximum of deg[q()] and deg[n()]+ has he minimum possible value Slide ٢٩ Welch-Berlekamp Algorihm The seps of he algorihms are now Se Q ( ) =, N ( ) =, W ( ) =, V ( ) = and d = Evaluae D = Q d ( d )Sd + N d ( d ) If D =, se W d+ = W d (+ d ), V d+ = V d (+ d ) and go o sep oherwise, se D = W d ( d )S d + V d ( d ) Se Q d+ = Q d (+ d ), N d+ = N d (+ d ), W d+ = W d + Q d D /D, V d+ = V d + N d D /D, Check wheher L[W d, V d ] was less han or equal o L[Q d, N d ]; if i was hen swap Q d+, N d+ wih W d+, V d+ incremen d 7 If d < n k, reurn o sep ; oherwise, Q() = Q d (), N()= N d () Slide ٣٠ ١٥

16 Welch-Berlekamp Algorihm For our example The error locaor polynomial is + + which has roos and The error values in he daa posiions are k N ek hk k Q' Slide ٣١ Welch-Berlekamp Algorihm Where Q () is he formal derivaive of Q() In his case Q () = e / The received informaion herefore is Slide ٣٢ ١٦