1、s | )U D CI|I 0 S/v V 5 “:DVBSRSpI/ !9 LC() Design and Implemention of RS Error CorrectionEncoder/Decoder Based on DVB Standard 3 d = a q# h S V Sa b S E j 2009 M417|S xSE92010 M1 E j b S E j 2009 M417|S xSE92010 M1 E j lE /B/ byNPS B q ll ii% 7TS 5 q . i% bTS bb. pa+b = b+a;ab = ba.c. p(a+b)+c = a+
2、(b+c);a(bc) = (ab)cd. s pa(b+c) = ab+ac;(c+b)a = ca+bae.a+0 = af.a1 = ag.F ia,iFE I -a, Pa+(a) = 0h.F ia 6= 0,iE I(a1), Paa1 = 1x3.1.2KGF(p) QKGF(n)+ Q4:1.KGF(p) T“ GF(p) T T. 9 cRS I # E - f(x) = an1 xn1 +an2 xn2 +an3 xn3 +:a1 x+a0“ an1 an2 an3 :a1 a0 GF(2) = Tb2. :Fnx V U“ |1GF(p) T“b T iB Fnxa(x)
3、b(x)6=0iBB Tq(x)r(x) :a(x) = q(x)b(x)+r(x)5q(x) , r(x) :b3.!f(x) Fnx B TFnx Tg(x)h(x) Tg(x)ah(x) ?$f(x)“5g(x)h(x)1f(x)b:g(x) h(x)(modf(x)4.; T!f(x) Q v , T “ # f(x) ?$GF(p) T“f(x) GF(p) ; Tb5. TGF(p) mQ; T ?$ “Ke nB T(xn +1)Q n pm15 T T(primitive-polynomial) ( T TKQ“ 1 T nB TbcKQ 1 O T(xn +1) nB TKe
4、 nB T)b6. TB TKv b f(x) = x7 +1, 7 x3.1.3KGF(p) KGF(p)l 012l,p-1“M p b Ta,b2GF(p),5a+b r(modp)ab s(modp)“V UbYV MFMT KGF(p) =bI K =GF(2)# ZGF(2m)b=GF(2), MF sN M b7 ZGF(2m) MF M 5Mt/ b 10 cRS I # Ex3.1.4= TV U# iBn= VB vn-1 T V Ubf(x) = an1 xn1 +an2 xn2 +an3 xn3 +an4 xn4 +:+a1 x+a0V Un= (an1 an2 an3
5、 :a1 a0)b xn1 V U - “an1bV UB= 10101010 TV U /f(x) = x7 +x5 +x3 +x TE Y T i T“ MFhs T“ MF s Mh bp T H P TE1 p TGF(2)MGF(2m) Hi Tf(x) = x2m1 +1,|f(x) = x2m1 +1y Ts TB f(x)y0 OKQ mbp(x) = x3 +x+1 TGF(23) T B Tf(x) =x7 +1y0b Tf(x) = x7 +1 6 y0x3 +x2 +1x+1 V T(3-1-1) V Ub(x+1)(x3 +x2 +1)(x3 +x+1) = (x4
6、+x3 +x+x3 +x2 +1)(x3 +x+1)= (x4 +x2 +x+1)(x3 +x+1)= (x7 +x5 +x4 +x5 +x3 +x2 +x4 +x2+x+x3 +x+1)= x7 +1(3-1-1)|= T V ULC s= bx3.1.5 GF(2) ZGF(2m )V=GF(2) ZGF(2m )/ 4 2-1 f(x) GF(p) nQ TFnx V U“ |1GF(p) T“bFnx/ f(x) GF(p) Q ln T 8 Of(x) Fnx nQ; T H5Q ln T 8 Tb 2-2 f(x) GF(2) nQ; T,Fnx/ f(x) GF(2) Q ln
7、T 8,5Fnx/ f(x) 2Fa f(x) /B2nKGF(2) Z:GF (2m)GF(2) Zb:= T 2Fa f(x) = x2 +x+1 /B ZGF(2n)=0,1,x,x+1b GF(2)=01b 11 cRS I # EGF(2) , 0,1b ZGF(2m ) T,2m Vm B Z b:ZGF(22) 0,1,x,x+1b F00,01,10,11bal 7f(x) = xm +fm1 xm1 +fm2 xm2 +:+f1 x+f0 (3-1-2)(fi 2 GF(2);0 i m1)GF(2) Q m; TKGF(2m)GF(2) Q lm TF,GF(2m) = f
8、am1 xm1 +am2 xm2 +am3 xm3 +:+a1 x+a0g (3-1-3)i = 0;1;2:m1:ai 2f0;1g (am1 xm1 +am2 xm2 +am3 xm3 +: + a1 x+a0)b= VU5GF(2m) = fam1 am2 am3 :a1 a0g(i = 0;1;2:m1: ai 2 f0;1g)VUGF(2m)ZE TV Ub7GF(2m)V UGF(2m) d , “GF(2m) iB g, PGF(2m) id , VV Ug Tb gGF(2m):GF(2m) = fgij0 i 2m2g( 2-3) p(x) GF(Pm) mQ TGF(Pm)
9、 Q lmd , T 8 pm1 p(x) /B T b9 GF(Pm) iBa a LVBQ lmQ T9Bm= b Q a0;a1;a2 :a2m2 GF(Pm) d ,4b1=GF(2)# ZGF(2m) p2 VbZGF(2m) 3-1-4aiaj = ai+j;a2m1 = 1;ai = a2m1i ai ai I(3-1-4)!p(x) = x3 +x + 1 GF(23) T k 3 ZGF(23) d , bs 7ap(x) = x3 +x+1 = 05p(a) = a3 +a+1 = 0aQ fa0;a1;a2 :a2m2 :gGF(23) d , b fa0;a1;a2 :
10、a2m2 = a6g B bya7 = 1,1a v7,ak = ak7(k 7)b a9 = a7+2 = a7a2 = 1a2 a8 = a7+1 = a7a1 = 1a1 a3 = a+1,5 V:a4 = aa3 = a(a+1) = a2 +a 12 cRS I # Ea5 = aa4 = a(a2 +a) = a3 +a2 = a2 +a+1aa5 = a(a2 +a+1) = a6 = a3 +a2 +a = a+1+a2 +a = a2 +1Tnm3.1“ V TsKGF(2m) ba0a1a2aa3a4a5a6 aa7a8a9a10 a11a12a13r(x) a14a15a
11、7a16a17a18a19 0a 1 1 001 1a 1a 1x 010 2a 2a 2x 100 3a a +1. x +1. 011 4a 2a + 1a 2x + 1x 110 5a 2a + 1a +1 2x + 1x +1 111 6a 2a +1 2x +1 101 m3.1 GF(23) 100+011=111ba bm3.1, a2 + a3 =a2+a+1 = a5;T= 111b7(100)(011)=x2(x+1) = x3+x2 TM L= p(x)=x3 +x+1 b x3+x2mod p(x)=x2+x1+1=(111)b9 ?a M Tbm3-1: (100)(
12、011)=a2a3 = a5= (111)bm3-19 ?ZL p I pa2T, - a2=d0=a5,=V U 111 K MFaM V5sa T a T V Vsbx3.2 RS I E#qRS I “dd“dI “dI I “M7d“dI 5 M | Tbd“dI M | T4r 4yN b7“d r MeBt# “dI q LFSR(LQ 7i ) LCbx3.2.1 RSI ERS n,k d= 9 VV URS n,k,2tb km-bits F km-bits 32tm-bitsn 7 3nm- 13 cRS I # Ebits b /+1311.0kn2m+ 2.Yn=2m
13、-1.2.n-k=2tb3.Kl d0 = 2t+1 .4.l? Kv ? t.RSI d“d“dI L u(x), 3 T g(x),d“dIIz c(x),5c(x)=u(x).g(x)bI H VBE LC 4b“dI I 4B M bRS(n, k) /Z T u(x)=uk1 xk1 +uk1 xk2 +:+u0;3 Tg(x) = (x+1)(x+a)(x+a2):(x+ar1);(r 2t);5I c(x) = xr u(x) mod g(x)+xr u(x);a br=n-k, t pKv xr u(x) mod g(x)V Uxr u(x)“g(x) bx3.2.2 RSI
14、qRSI q T“E E nm 3-2 U45bL ! Q sYkQarQ= Ta(x)b(x)a(x) = ak xk +ak1 xk1 +:+a1 x+a0 (ai 2 GF(P);i = 0;1;2;:;k)(3-2-1)b(x) = br xr +br1 xr1 +:+b1 x+b0 (bj 2 GF(P);j = 0;1;2;:;r)(3-2-2)* TMc(x) = a(x):b(x)= ak br xk+r +(ak br1 +ak1 br)xk+r1 +:+(ak bri +ak1 br(i+1) +:+aki br)xk+ri +:+(a0 b1 +a1 b0)x+a0 b0
15、(3-2-3)qnm3.2 U E “ $ 9 5T b9 V “ H59 5T bD 7i - b ,b“E nm3.3 Uba(x),b(x)sY GF(p)nQrQ Ta(x) = a0 +a1 x+:+an xn (ai 2 GF(p);i = 0;1;2:;n) (3-2-4) 14 cRS I # Ea0a1a2a3a4a5a6a7a8a9a10a11a12a13a14a15a16a17a18 a19a20a21 a21 a21a22a23a24a25a26a14a27a21m3.2Em1b q(x) a (x) 1rb 1rb 2rb 0b D D D D m3.3“E mb(x
16、) = b0 +b1 x+:+br xr (bi 2 GF(p);i = 0;1;2:;r:n r) (3-2-5)a (x)=b(x)q(x)+r(x)bq(x)r(x)sY T Tb7i 5 , H 6 Mb bK rKH7i H 7 S K MBQ Bb “E =7i bB“E V nm3.4 U.L a(x) ,b(x)sY 3-2-r (x) a0a1a2a3a4a0a1a2a3a51rb q (x) 1b rb 2rb 0b D D D a (x) D m3.4V “E m4 3-2-5 TB“m3-4 LC xr a(x)“b(x)“E - M 7i -1 b , H 6 Mb
17、B 7 ST 7 S K MBQ Bb “E =7i b VRSI LCnm3.5 U: - I Mb u(x)=uk1 xk1 +uk1 xk2 +:+u03 Tg(x) = (x+1)(x+a)(x+a2):(x+ar1);(r 2t); 15 cRS I # Er (x) a0a1a2a3a4a0a1a2a3a51rb q (x) 1b rb 2rb 0b D D D a (x) D m3.5 RSI LCm5I c(x) = xr u(x) mod g(x)+xr u(x);I V V BK BQ “E H4 714 b=k 7 b D(? M H4 714K D(? T b“ - F
18、 S RSI bx3.3 RS E#qRS MRSI 4 p T#9 T p 1obRS 1 /:B lr(x) p T;=9 p T#9 T; pp; 9 p pb m3.6 V UError value ( )x ( )x S(x) a0a1 a0a2 FIFO a3 a4 a5 a6 a3a4a5a7a8a9a10a11a12a13a14a6a9a10a11 a3 a15 a16 a11 a3a4a5a7a8 m3.6 RS m(x)p T (x) 9 T S(x) Tb 16 cRS I # Ex3.3.1 p TSi! l r(x),5si = r(ai);i = 0;1;2:2t
19、1; ! u(x), 3 Tg(x), 3 3c(x)pm“ e(x)b - I :g(x) = (x+a0)(x+a1):(x+a2t1)t Vp bc(x) = xnk u(x) + p(x)p(x) = xnk u(x) mod (g(x)* Vr(x) = c(x)+e(x) (3-3-1)yp(x) = xnk u(x) mod (g(x) * xnk u(x) V/ T:xnk u(x) = q(x):g(x)+p(x) (3-3-2)q(x) Tb Vc(x) = xnk u(x)+p(x)= q(x):g(x)+p(x)+p(x)= q(x):g(x)(3-3-3)V 3-3-
20、1 Tr(x) = c(x)+e(x) Vr(x) = q(x):g(x)+e(x) (3-3-4)7g(x) = (x+a0)(x+a1):(x+a2t1)r(x)B V / T2031r(x) = h(x):(x+ai)+b(x)+b(x) (3-3-5)h(x) T,b(x) T. r(ai) = h(ai)(ai +ai)+b(ai)= b(ai)(3-3-6)9 VVY s T pb!r(x) = rn1 xn1 +rn2 xn2 +:r1 x+r0 (3-3-7)* Vr(ai) = rn1 ai(n1) +rn2 ai(n2) +:r1 ai +r0 (3-3-8) 17 cRS
21、 I # E|ai4 Vr(ai) = (rn1 ai(n2) +rn2 ai(n3) +:r1)ai +r0 (3-3-9)|ai4 Vr(ai) = (:(rn1 ai +rn2)ai +:r1):)ai +r0 (3-3-10) T0 3-3-6 3-3-10 V p 0qnm3.7R(X) ia D m3.7 p 0qm p T H| l GQ m3.7 U r(x) ,7i 1 pr(ai)b “iM VS0,S1,S2,.S2t1b T0 s(x) = s2t1 xn1 +:+s2 x2 +s1 x+s0 (3-3-11)t p bx3.3.29 p T#9 TB 1oB pp T
22、ZE1 BM EEuclid( x+ ) EbBM EBerlekamp1968 M4Massey1969 MLFSR LQ M7i LC BZEbBM Er LC1 x+ E4bSugiyama1975 M?C VEuclid( x+ ) pKv ZE9 V pp T BBZE LCM b 5 Euclid( x+)ZEb 1 Tpap1“ - si = r(ai);i = 0;1;2:2t1;L p Te(x) = ej1 xj1 +ej2 xj2 +ej3 xj3 +:+ejv xjv(v t) (3-3-12) 18 cRS I # Eejvpjvpb * Si=r(ai)=e(ai
23、)s0 = e(a0) = ej1 +ej2 +ej3 +:+ejvs1 = e(a1) = ej1 a1j1 +ej2 a1j2 +ej3 a1j3 +:+ejv a1jvs2t1 = e(a2t1) = ej1 a(2t1)j1 +ej2 a(2t1)j2 +:+ejv a(2t1)jv7xl = ajl5 T Vs0 = e(a0) = ej1 x0 +ej2 x0 +ej3 x0 +:+ejv x0s1 = e(a1) = ej1 x1 +ej2 x2 +ej3 x3 +:+ejv xvs2t1 = e(a2t1) = ej1 x1(2t1) +ej2 x2(2t1) +:+ejv x
24、jv(2t1)dB T0 sj =vXl=1eil xjl(j = 0;1;2:2t1) (3-3-13)V T(3-3-13)0 Vpap 0i “b Z 4A 6Eb 2 x+ E202130x+ E ? T9 p T9 T9 p T9 T p K9 9 /B9 p#p $b5lp T(x) =vYl=1(1xl x) = v xv +v1 xv1 +v2 xv2 +:+1 x+0(3-3-14)1o T(x) = s(x):(x)(modx2t) (3-3-15)(x)99 Ts(x) 0b T Vr(x) = s(x):(x)+(x)x2t(modx2t) (3-3-16)YV T0
25、pp T9 T51 x+ EbLg abKv V Ug = (a;b)b5Ai t, s Pg = at + bs, 19 cRS I # E T T9g(x)= a(x)b(x),9i/1“g(x)= p(x)a(x) + t(x)b(x).C1 pg(x)1 pt(x)ap(x)b E Z x+ Eb E1 “/ “ lt1 /B5a(x)“b(x) Tq1(x) Tr1(x) Va(x) = q1(x)b(x)+r1(x) (3-3-17)Tr1(x)lt * t(x)=q1(x),5 7t1(x) = q1(x)?/ b=b(x)“r1(x) Tq2(x) Tr2(x), Vb(x)
26、= q2(x)r1(x)+r2(x) (3-3-18)| 3-1-17a 3-1-18r1(x)h“q2(x)a(x) = q2(x)t1(x)+1b(x)+r2(x) (3-3-19)7t2(x) = q2(x)t1(x)+1 Tr2(x)lt * t(x)=t2(x)b5/Bb “ZEr1(x)“r2(x) Tq3(x) Tr3(x) Vr1(x) = q3(x)r2(x)+r3(x) (3-3-20) 3-1-17a 3-1-18 Th“r1(x)r2(x)1+q2(x)q3(x)a(x) = t1(x)+q3(x)t2(x)b(x)+r3(x) (3-3-21)7t3(x) = t1(x)+q3(x)t2(x) Tr3(x)lt * t(x)=t3(x)b5/Bb E1 s lt bV + Vw :ti(x) = qi(x)ti1(x)+ti2(x)ft0(x) = 1;t1(x) = 0gi = (1;2;:) (3-3-22)1o T Vx2tMa(x) 0s(x)Mb(x)K Hqti(x) p T(x)K Tri(x) (x)9 Tbx3.3.3 ppp T p + Chien ZEb ZE L
