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with(plot s):" }}}{SECT 0 {PARA 201 "" 0 "" {TEXT 201 37 "Understanding the bino mial expansion " }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 73 "term : = (n, t, lp) -> log[10] (binomial(n,t)*10^(lp*t)*(1-10^lp)^(n-t));" } {MPLTEXT 1 0 1 "\n" }}}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 " > " 0 "" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 202 12 "Fix n vary t" }} {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 8 "n := 10;" }{MPLTEXT 1 0 9 " \nt1 := 3;" }{MPLTEXT 1 0 90 "\nf1 := lp -> term(n, t1 , -lp); \+ G1 := plot(f1(x), x = 0..5, y=0..-14, color=blue):" }{MPLTEXT 1 0 91 "\nf2 := lp -> term(n, t1 + 1, -lp); G2 := plot(f2(x), x = 0..5, \+ y=0..-14, color=green):" }{MPLTEXT 1 0 91 "\nf3 := lp -> term(n, t1 + \+ 2, -lp); G3 := plot(f3(x), x = 0..5, y=0..-14,color=yellow):" } {MPLTEXT 1 0 19 "\ndisplay(G1,G2,G3);" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 10 "n := 1000;" }{MPLTEXT 1 0 10 "\nt1 := 10;" }{MPLTEXT 1 0 41 "\nxl := 0; xh := 5; yl := 0; yh := -15;" }{MPLTEXT 1 0 92 " \nf1 := lp -> term(n, t1 , -lp); G1 := plot(f1(x), x = xl..xh, y=yl..yh, color=blue):" }{MPLTEXT 1 0 93 "\nf2 := lp -> term(n, t1 + \+ 1, -lp); G2 := plot(f2(x), x = xl..xh, y=yl..yh, color=green):" } {MPLTEXT 1 0 93 "\nf3 := lp -> term(n, t1 + 2, -lp); G3 := plot(f 3(x), x = xl..xh, y=yl..yh,color=yellow):" }{MPLTEXT 1 0 67 "\nf4 := l p -> log[10]( sum( 10^ (term(n, t1 + j, -lp)) , j=0..10) );" } {MPLTEXT 1 0 61 "\n G4 := plot(f4(x), x = xl..xh, y=yl..yh,co lor=red):" }{MPLTEXT 1 0 23 "\ndisplay(G1,G2,G3, G4);" }}}}{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 202 12 "Fix t va ry n" }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 7 "t := 3;" }{MPLTEXT 1 0 11 "\nn1 := 10; " }{MPLTEXT 1 0 89 "\nf1 := lp -> term(n1 , t,- lp); G1 := plot(f1(x), x = 0..5, y=0..-14, color=blue):" } {MPLTEXT 1 0 90 "\nf2 := lp -> term(10*n1, t,-lp); G2 := plot(f2 (x), x = 0..5, y=0..-14, color=green):" }{MPLTEXT 1 0 91 "\nf3 := lp - > term(100*n1, t,-lp); G3 := plot(f3(x), x = 0..5, y=0..-14, colo r=yellow):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "\ndisplay(G1,G2,G3) ;" }}}}{PARA 203 "" 0 "" }{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 202 14 "Fix t/n vary n" }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 12 "tau := 1/10;" }{MPLTEXT 1 0 22 "\nn := [10, 100, 100 0];" }{MPLTEXT 1 0 101 "\nf1 := lp -> term(n[1],floor(tau*n[1]), -lp); G1 := plot(f1(x), x = 0..5, y=0..-14, color=blue):" }{MPLTEXT 1 0 102 "\nf2 := lp -> term(n[2],floor(tau*n[2]), -lp); G2 := plot( f2(x), x = 0..5, y=0..-14, color=green):" }{MPLTEXT 1 0 103 "\nf3 := l p -> term(n[3],floor(tau*n[3]), -lp); G3 := plot(f3(x), x = 0..5, y=0..-14, color=yellow):" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 19 "\ndi splay(G1,G2,G3);" }}}}}{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 201 "" 0 "" {TEXT 201 31 "Performance of repetition codes" }}{EXCHG {PARA 203 "" 0 "" {TEXT 200 54 " A repetition code has generator matrix [1,1 ,...,1]. " }{TEXT 200 136 "\nWe use the majority-vote decoding algori thm. Provided more than half of the received symbols are unchanged, w e can correct the errors." }{TEXT 200 91 "\nThe correction capability \+ grows linearly with the length, but the rate of the code is 1/n." } {TEXT 200 57 "\nWe'll use odd length, n= 2t+1. We can correct t error s." }{TEXT 203 1 " " }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 19 "t \+ := [1,2,3,4,5,6];" }{MPLTEXT 1 0 16 "\nnum := nops(t);" }{MPLTEXT 1 0 36 "\nn := [seq(2*t[i] + 1, i= 1..num) ];" }{MPLTEXT 1 0 57 "\nf := [s eq(lp -> term(n[i], t[i] +1, -lp), i = 1..num) ];" }{MPLTEXT 1 0 51 " \ncol := [blue, green, yellow, blue, green, yellow];" }{MPLTEXT 1 0 74 "\nG := [seq(plot(f[i](x), x = 0..5, y=0..-14, color=col[i]), i= 1 ..num )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 24 "t := [1,4, 16, 64, 256];" }{MPLTEXT 1 0 16 "\nnu m := nops(t);" }{MPLTEXT 1 0 36 "\nn := [seq(2*t[i] + 1, i= 1..num) ]; " }{MPLTEXT 1 0 54 "\nf := [seq(lp -> term(n[i], t[i], -lp), i = 1..nu m) ];" }{MPLTEXT 1 0 51 "\ncol := [blue, green, yellow, blue, green, y ellow];" }{MPLTEXT 1 0 74 "\nG := [seq(plot(f[i](x), x = 0..5, y=0..-1 4, color=col[i]), i= 1..num )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }} }{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 201 "" 0 "" {TEXT 201 28 "Performance of Hamming codes" }} {EXCHG {PARA 203 "" 0 "" {TEXT 200 59 "Hamming codes have parameters [ 2^r -1, 2^r - 1 - r, 3]. " }{TEXT 200 71 "\nThey correct only one e rror, and the rate approaches 1 as r increases." }}}{EXCHG {PARA 200 " > " 0 "" {MPLTEXT 1 0 7 "t := 1;" }{MPLTEXT 1 0 25 "\nr := [3, 4, 5, 6 , 7, 8];" }{MPLTEXT 1 0 16 "\nnum := nops(r);" }{MPLTEXT 1 0 37 "\nn : = [seq(2^ r[i] - 1, i= 1..num) ];" }{MPLTEXT 1 0 51 "\nf := [seq(lp -> term(n[i], 2, -lp), i = 1..num) ];" }{MPLTEXT 1 0 51 "\ncol := [blue, green, yellow, blue, green, yellow];" }{MPLTEXT 1 0 74 "\nG := [seq(p lot(f[i](x), x = 0..5, y=0..-14, color=col[i]), i= 1..num )]:" } {MPLTEXT 1 0 13 "\ndisplay(G); " }}}{EXCHG {PARA 200 "> " 0 "" }} {EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 201 "" 0 "" {TEXT 201 29 "Reed-Mul ler codes of rate 1/2" }}{EXCHG {PARA 203 "" 0 "" {TEXT 200 141 "Reed- Muller codes are another large family of codes. The length of the bin ary codes are a power of 2 (though one can shorten, puncture etc.)" }} {PARA 203 "" 0 "" }{PARA 203 "" 0 "" {TEXT 200 85 "Here we compare thr ee codes of rate 1/2 and parameters [ 2^m, 2^(m-1), 2^((m+1)/2) ]" }} {PARA 203 "" 0 "" {TEXT 200 35 "so it corrects 2^((m-1)/2) errors " } }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 9 "num := 6;" }{MPLTEXT 1 0 42 "\nm := [seq ( 2*i+1, i = 1..num)];" }{MPLTEXT 1 0 42 "\nn := [seq ( 2^m[i], i = 1..num)];" }{MPLTEXT 1 0 42 "\nd := [se q ( 2^((m[i]+1)/2), i = 1..num)];" }{MPLTEXT 1 0 42 "\nt := [seq ( 2^( (m[i]-1)/2), i = 1..num)];" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 54 "\nf := [seq(lp -> term(n[i], t[i], -lp), i = 1..num) ];" }{MPLTEXT 1 0 51 "\ncol := [blue, green, yellow, blue, green, yellow];" }{MPLTEXT 1 0 74 "\nG := [seq(plot(f[i](x), x = 0..5, y=0..-14, color=col[i]), i= 1..num )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }}}{EXCHG {PARA 200 "> \+ " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }} {EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 201 "" 0 "" {TEXT 201 24 "Performa nce of BCH codes" }}{EXCHG {PARA 203 "" 0 "" {TEXT 200 94 "Here we loo k at the performance of four BCH codes of rates roughly 3/4 and of var ying lengths." }}{PARA 203 "" 0 "" }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 18 "r := [5, 6, 7, 8];" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 34 "n := [seq( 2^r[i] -1, i = 1..4)];" }{MPLTEXT 1 0 24 "\nk := [21, 45, 92, 191];" }{MPLTEXT 1 0 46 "\nrate := [seq( evalf(k [j]/n[j]), j= 1..4) ];" }{MPLTEXT 1 0 21 "\nd := [5, 7, 11, 17];" } {MPLTEXT 1 0 19 "\nt := [2, 3, 5, 8];" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 15 "num := nops(t);" }{MPLTEXT 1 0 57 "\nf := [seq(lp -> \+ term(n[i], t[i] +1, -lp), i = 1..num) ];" }{MPLTEXT 1 0 53 "\ncol := [ blue, green, yellow, orange, green, yellow];" }{MPLTEXT 1 0 74 "\nG := [seq(plot(f[i](x), x = 0..5, y=0..-14, color=col[i]), i= 1..num )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }}}{EXCHG {PARA 200 "> " 0 "" }}} {PARA 204 "" 0 "" }{PARA 204 "" 0 "" }{EXCHG {PARA 200 "> " 0 "" }} {SECT 0 {PARA 201 "" 0 "" {TEXT 201 18 "Reed-Solomon codes" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 202 24 "Symbol error probability" }}{EXCHG {PARA 203 "" 0 "" {TEXT 204 102 "We compute the symbol error from the \+ bit error probability, pb and the mumber of bits in a symbol, s." } {TEXT 200 1 "\n" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 20 "psymb \+ := proc(pb, s)" }{MPLTEXT 1 0 10 "\nlocal ps;" }{MPLTEXT 1 0 1 "\n" } {MPLTEXT 1 0 63 "\nps := sum(binomial(s, j) * pb^j * (1-pb) ^ (s-j) , \+ j = 1..s );" }{MPLTEXT 1 0 1 "\n" }{MPLTEXT 1 0 12 "\nreturn(ps);" } {MPLTEXT 1 0 10 "\nend proc;" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 14 "psymb(.01, 5);" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 40 "lpsymb := (lpb, s) -> lpb + log[10](s);" }{MPLTEXT 1 0 1 "\n" }}} {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 66 "F:= [lpb -> log[10]( psymb (10^lpb, 8) ), lpb -> lpsymb(lpb, 8) ];" }}}{EXCHG {PARA 200 "> " 0 " " {MPLTEXT 1 0 15 "psymb(.001, 8);" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 52 "G := [seq(plot (F[i](-lpb), lpb = 1..5 ), i=1..2 )]:" }}}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 11 "display(G);" }}}} {SECT 0 {PARA 202 "" 0 "" {TEXT 202 75 "Comparison of two Reed-Solomon codes of different lengths but the same rate" }}{EXCHG {PARA 203 "" 0 "" {TEXT 200 88 "We compare tgwo codes of the same rate with the same alphabet, but of different lengths:" }{TEXT 200 65 "\nThe [255, 127, \+ 129] RS code over F_256 which corrects 64 errors;" }{TEXT 200 68 "\nTh e [32,15,17] shortened RS code over F_256 which corrects8 errors" }}} {EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 15 "n := [32, 255];" } {MPLTEXT 1 0 14 "\nt := [9, 65];" }{MPLTEXT 1 0 10 "\nnum := 2;" } {MPLTEXT 1 0 95 "\nf := [seq(lp -> term(n[i], t[i], lpsymb(-lp,8)), i \+ = 1..num) ]; #lpsymb(lp,8)), i = 1..num) ];" }{MPLTEXT 1 0 51 "\ncol : = [blue, green, yellow, blue, green, yellow];" }{MPLTEXT 1 0 73 "\nG : = [seq(plot(f[i](x), x = 1..5, y=-1..-14, color=col[i]), i= 1..2 )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }}}{EXCHG {PARA 200 "> " 0 "" }}} {EXCHG {PARA 200 "> " 0 "" }}{SECT 0 {PARA 202 "" 0 "" {TEXT 202 46 "C omparison with the same correction capability" }}{EXCHG {PARA 203 "" 0 "" {TEXT 200 34 "Two codes both correcting 9 errors" }{TEXT 200 64 " \nThe [255, 239, 17] RS code over F_256 which corrects 64 errors;" } {TEXT 200 68 "\nThe [32,15,17] shortened RS code over F_256 which cor rects8 errors" }}{PARA 203 "" 0 "" }}{EXCHG {PARA 200 "> " 0 "" {MPLTEXT 1 0 15 "n := [32, 255];" }{MPLTEXT 1 0 13 "\nt := [9, 9];" } {MPLTEXT 1 0 10 "\nnum := 2;" }{MPLTEXT 1 0 95 "\nf := [seq(lp -> term (n[i], t[i], lpsymb(-lp,8)), i = 1..num) ]; #lpsymb(lp,8)), i = 1..num ) ];" }{MPLTEXT 1 0 51 "\ncol := [blue, green, yellow, blue, green, ye llow];" }{MPLTEXT 1 0 73 "\nG := [seq(plot(f[i](x), x = 1..5, y=-1..-1 4, color=col[i]), i= 1..2 )]:" }{MPLTEXT 1 0 13 "\ndisplay(G); " }}} {EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> " 0 "" }} {SECT 0 {PARA 202 "" 0 "" {TEXT 202 46 "Comparison with a binary code \+ of the same rate" }}{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> \+ " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}}{EXCHG {PARA 200 "> " 0 "" }} {EXCHG {PARA 200 "> " 0 "" }}{EXCHG {PARA 200 "> " 0 "" }}{PARA 204 "" 0 "" }{PARA 204 "" 0 "" }{PARA 204 "" 0 "" }{PARA 204 "" 0 "" }} {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }