www.pudn.com > SolutionsforMultiuserDetection.rar > 1.6.yyao.ps

%!PS-Adobe-2.0
%%Creator: dvips, version 5.395 (C) 1986-90 Radical Eye Software
%%Title: hw1.dvi
%%Pages: 3 1
%%BoundingBox: 0 0 612 792
%%EndComments
%%BeginProcSet: tex.pro
/TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch
load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{
isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale
Resolution VResolution vsize neg mul TR}B /@letter{/vsize 10 N}B /@landscape{
/isls true N /vsize -1 N}B /@a4{/vsize 10.6929133858 N}B /@legal{/vsize 13 N}
B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[
1 0 0 -1 0 0]N /FBB[0 0 0 0]N /df{/sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf
X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /df-tail{/nn 8 dict N nn begin
/FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array
/BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2
array copy cvx N load 0 nn put /ctr 0 N[}B /E{pop nn dup definefont setfont}B
/ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B
/ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub
get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup
length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ctr 0 N
/CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get
/ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width
add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub
ch-yoff .1 add]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{
]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1
sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D}B
/bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto}B /eop{
clear SI restore showpage userdict /eop-hook known{eop-hook}if}B /@start{
userdict /start-hook known{start-hook}if /VResolution X /Resolution X 1000 div
/DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for}
B /p /show load N /RMat[1 0 0 -1 0 0]N /BDot 8 string N /v{/ruley X /rulex X V
}B /V{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask
grestore}B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta
add tail}B /b{S p tail}B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{
1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w
}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /w{0 rmoveto}B /x{0 S rmoveto}
B /y{3 2 roll p a}B /bos{/SS save N}B /eos{clear SS restore}B end
%%EndProcSet
TeXDict begin 1000 300 300 @start /Fa 2 49 df0
D<0E0E1E1C1C1C383838707060E0E0C0070F7F8F0A>48 D E /Fb 2 51
df<18F8F81818181818181818FFFF080D7D8C0E>49 D<3F00FF80E3C0C0C000C001C001800780
0E003CC070C0FF80FF800A0D7E8C0E>I E /Fc 5 99 df<07C0000FF1803C7380303B00701B00
601F00601E00E01C00C03C00E07C0071FC007FCF803F0F80110D7E8C16>11
D<07E03FC007F03FC001F00E0001F00C0001F80C0003B81C0003BC1800031C1800031E1800070E
3800070E3000060F3000060730000E07F0000E03E0000C03E0000C01E0001C01E000FF01C000FF
00C0001A147F931A>78 D<07FFE007FFF001E03801C01801C01C01C01C03C03803803803807003
80E007FFC007FF000700000F00000F00000E00000E00001E0000FF8000FF800016147F9315>80
D<07B01FF03CF030E07060606060E0E0C0C0CCE1CC63DC7FF83E780E0D7E8C12>97
D<7C007C001C0018001800380038003F803F8079C070C070C060C061C0E180C180C380E7007E00
3C000A147E930F>I E /Fd 1 68 df<001FF06000FFFCE003FC1FE00FE007E01FC003E01F8001
E03F0000E07F0000E07F0000E07E000060FE000060FE000000FE000000FE000000FE000000FE00
0000FE000000FE0000007E0000607F0000607F0000603F0000E01F8000C01FC001C00FE0078003
FC1F0000FFFC00001FF0001B1C7D9B22>67 D E /Fe 6 104 df0 D<000001800000078000001F0000007C000001F0000007C000001F0000007C000001F000
0007C000001F000000FC000000F0000000F80000003E0000000F80000003E0000000F80000003E
0000000F80000003E0000000F80000003E0000000F800000038000000000000000000000000000
000000000000000000000000000000FFFFFF80FFFFFF8019227D9920>20
DI<007FF801FFF807C0000F0000
1C0000380000300000700000600000E00000C00000C00000FFFFF8FFFFF8C00000C00000E00000
6000007000003000003800001C00000F000007C00001FFF8007FF8151A7D961C>50
D<001F007F00F801E001C001C001C001C001C001C001C001C001C001C001C001C001C003C00F80
FF00FC00FF000F8003C001C001C001C001C001C001C001C001C001C001C001C001C001C001E000
F8007F001F10297E9E15>102 DI E /Ff 5 51
df<070E0E1C383870707060E0E0E0E0E0E0E0E0E0E06070707038381C0E0E07081E7E950D>40
DI<00
6000006000006000006000006000006000006000006000006000006000FFFFF0FFFFF000600000
600000600000600000600000600000600000600000600000600014167E9119>43
D<06001E00FE00EE000E000E000E000E000E000E000E000E000E000E000E000E000E00FFE0FFE0
0B137D9211>49 D<1F807FC0E1E0E0F0E070E0700070007000E001E003C007800F001E303C3078
70FFE0FFE0FFE00C137E9211>I E /Fg 12 121 df<00FC0003FE000F8F061E078E3C038C3C03
8C7803DC7803D8F801F8F001F0F001E0F001E0F001C0F003C0700FC07C3FEE3FF8FC0FE0781712
7F911B>11 D<003E0000FF0001E38003C00003800003000007000007800007C00003E00003F000
01F80003FC000FBC001E1E003C0E003C0E00780E00780E00F80C00F00C00F00C00F01C00F01800
F03800F030007870007FE0001FC000020000111E7F9C12>14 D<70F8F8F87005057D840C>58
D<70F8F8F878181818383070E060050D7D840C>I<03FFFF0003FFFFC0003E01E0003C00F0003C
00F0003C0070007C0070007800F0007800F0007801E000F803C000F0078000FFFF0000FFFE0001
F01F0001E0078001E003C001E003C003E003C003C003C003C003C003C003C007C0078007800F00
07801E000F807C00FFFFF800FFFFE0001C1C7E9B1F>66 D<0003F818001FFE38007E077801F803
F003E001F007C000F00F8000F01F0000E01E0000E03C0000E03C0000E078000000780000007800
0000F8000000F0000000F0000000F0000000F0000300F0000300F00007007800060078000E003C
001C003E0078001FC1F00007FFC00001FE00001D1C7E9B1E>I<03FE03FF8003FE03FF80003F00
7C00003F003000003F803000003F80700000778070000067C060000067C060000063C0E00000E3
E0E00000C3E0C00000C1E0C00000C1F1C00001C1F1C0000180F180000180F9800001807B800003
807B800003007F000003003F000003003F000007003F000006001E000006001E00001F001E0000
FFE00E0000FFE00C0000211C7E9B21>78 D<03FFFF0003FFFFC0003E03E0003C00F0003C00F000
7C00F0007C00F0007800F0007800F000F800E000F801E000F001C000F0038001F00F0001FFFE00
01FFF00001E0000003E0000003E0000003C0000003C0000007C0000007C0000007800000078000
000F800000FFF80000FFF800001C1C7E9B1B>80 D<03FFFE0003FFFF80003E03E0003C00E0003C
00F0003C00F0007C00F0007800F0007800F0007801E000F801C000F0038000F00F0000FFFC0001
FFFC0001E07E0001E01E0001E01E0003E01E0003C01E0003C01E0003C03E0007C03E0007803C00
07803C180F803E38FFF81FF0FFF80FE01D1C7E9B20>82 D<3E1F803F3FE07771E067E0E0E7C0E0
C780E00700E00F01E00F01C00E01C00E03C01E038E1E038E1C078C1C071C3C07B83C03F03801E0
17127F9119>110 D<01C001C001C003C003C0038003800780FFF0FFF007000F000F000E000E00
1E001E001C001C003C383C38383038703CE03FC00F800D1A80990F>116
D<0F8F803FDFC038F9C070F1C0E0F3C0C0E3C000E3C001E00001E00001C00001C00073C0E0F3C0
E0F380C0F781C0E7C780FFFF007C7C0013127E9118>120 D E /Fh 51 122
df<003F1F8001FFFFC003C3F3C00783E3C00F03E3C00E01C0000E01C0000E01C0000E01C0000E
01C0000E01C000FFFFFC00FFFFFC000E01C0000E01C0000E01C0000E01C0000E01C0000E01C000
0E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0000E01C0007F87FC007F87FC
001A1D809C18>11 D<003F0001FF8003C3C00783C00F03C00E03C00E00000E00000E00000E0000
0E0000FFFFC0FFFFC00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0
0E01C00E01C00E01C00E01C07F87F87F87F8151D809C17>I<01C00380038007000E000C001C00
1800380038007000700070007000E000E000E000E000E000E000E000E000E000E000E000E000E0
00E00070007000700070003800380018001C000C000E0007000380038001C00A2A7D9E10>40
DI<00060000000600000006000000060000000600000006000000
060000000600000006000000060000000600000006000000060000FFFFFFE0FFFFFFE000060000
000600000006000000060000000600000006000000060000000600000006000000060000000600
0000060000000600001B1C7E9720>43 D<70F8F8F878181818383070E060050D7D840C>II<70F8F8F87005057D840C>I<07E00FF01C38381C781E700E700EF00FF0
0FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00FF00F700E700E781E381C1C380FF007E0101B
7E9A15>48 D<030007003F00FF00C7000700070007000700070007000700070007000700070007
0007000700070007000700070007000700FFF8FFF80D1B7C9A15>I<0FE03FF878FC603EF01EF8
1FF80FF80F700F000F001F001E003E003C007800F001E001C0038007000E031C0338037006FFFE
FFFEFFFE101B7E9A15>I<0FE03FF8387C783E7C1E781E781E001E003C003C00F807F007E00078
003C001E000F000F000F700FF80FF80FF81EF01E787C3FF80FE0101B7E9A15>I<70F8F8F87000
0000000000000070F8F8F87005127D910C>58 D61
D<003FC18001FFF18003F07B800FC01F801F000F801E0007803C0003807C000380780003807800
0180F0000180F0000000F0000000F0000000F0000000F0000000F0000000F00000007800018078
0001807C0001803C0003801E0003001F0007000FC00E0003F03C0001FFF000003FC000191C7E9B
1E>67 D70 D72 DI76 D78 D<07F1801FFD803C1F80700780700380E0
0380E00180E00180F00000F80000FE00007FE0003FFC001FFE000FFF0000FF80000F800007C000
03C00001C0C001C0C001C0E001C0E00380F00780FE0F00DFFE00C7F800121C7E9B17>83
D<7FFFFFC07FFFFFC0780F03C0700F01C0600F00C0E00F00E0C00F0060C00F0060C00F0060C00F
0060000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F000000
0F0000000F0000000F0000000F0000000F0000000F0000000F000003FFFC0003FFFC001B1C7F9B
1E>II87
D89 D91 D93 D<1FE0003FF8003C3C003C1E00180E00
000E00001E0007FE003FFE007E0E00F80E00F80E00F00E60F00E60F81E607C7E607FFFC01FC780
13127F9115>97 DI<07F80FFC3E3C3C3C78187800F000F0
00F000F000F000F000780078063C0E3F1C0FF807F00F127F9112>I<001F80001F800003800003
8000038000038000038000038000038000038000038007F3801FFF803E1F807C0780780380F803
80F00380F00380F00380F00380F00380F00380F003807807807C0F803E1F801FFBF007E3F0141D
7F9C17>I<07E01FF83E7C781C781EF01EFFFEFFFEF000F000F000F000780078063C0E3F1C0FF8
07F00F127F9112>I<00FC03FE079E071E0F1E0E000E000E000E000E000E00FFE0FFE00E000E00
0E000E000E000E000E000E000E000E000E000E000E000E007FE07FE00F1D809C0D>I<07E7C01F
FFC03C3DC0781E00781E00781E00781E00781E00781E003C3C003FF80037E00070000070000078
00003FFC003FFF007FFF807807C0F003C0E001C0E001C0F003C0F807C07C0F801FFE0007F80012
1B7F9115>II<3C007C007C007C003C0000000000000000
0000000000FC00FC001C001C001C001C001C001C001C001C001C001C001C001C001C001C00FF80
FF80091D7F9C0C>I107 DIII<03F0000FFC001E1E00380700780780700380F003C0F003
C0F003C0F003C0F003C0F003C07003807807803807001E1E000FFC0003F00012127F9115>II<07F1801FF9803F1F803C0F80780780780380F00380F00380F00380F00380F00380
F00380F803807807807C0F803E1F801FFB8007E380000380000380000380000380000380000380
001FF0001FF0141A7F9116>II<1FB07FF0F0F0E070E030F030F8007FC07FE01FF000F8
C078C038E038F078F8F0FFF0CFC00D127F9110>I<0C000C000C000C000C001C001C003C00FFE0
FFE01C001C001C001C001C001C001C001C001C301C301C301C301C301E700FE007C00C1A7F9910
>IIII<7F8FF07F8FF00F0F80070F00038E0001DC0001D80000F00000700000780000
F80001DC00038E00030E000707001F0780FF8FF8FF8FF81512809116>II
E /Fi 11 118 df<7CFEFEFEFEFE7C07077C8610>46 D<00180000780001F800FFF800FFF80001
F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001
F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8007FFFE07F
FFE013207C9F1C>49 D<003FC001FFE003F07007C0F80F81F81F01F83E01F83E01F87E00F07C00
007C0000FC0800FCFFC0FDFFF0FF81F8FF00F8FE007CFE007CFE007EFC007EFC007EFC007EFC00
7E7C007E7C007E7E007E3E007C3E00FC1F01F80FC3F007FFC000FF0017207E9F1C>54
D<03FE0C0FFF9C1F03FC3E00FC7C007C78003CF8001CF8001CF8000CFC000CFE0000FF0000FFF0
007FFF007FFFE03FFFF01FFFF80FFFFC03FFFE007FFE0003FF0000FF00007F00003FC0001FC000
1FC0001FE0001FE0001EF0003EFC007CFF80F8E7FFF0C0FFC018227DA11F>83
D<003F8001FFC003F7E007E7E007E7E00FC7E00FC3C00FC0000FC0000FC0000FC0000FC0000FC0
00FFFC00FFFC000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0
000FC0000FC0000FC0000FC0000FC0000FC0000FC0007FFC007FFC0013237FA211>102
D<1F003F803F803F803F803F801F000000000000000000000000000000FF80FF801F801F801F80
1F801F801F801F801F801F801F801F801F801F801F801F801F801F801F80FFF0FFF00C247FA30F
>105 D108 D110 D<00FE0007FFC00F83E01E00F03E00F8
7C007C7C007C7C007CFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C3E00F8
1F01F00F83E007FFC000FE0017167E951C>I<00C00000C00000C00000C00001C00001C00003C0
0007C0000FC0001FC000FFFF00FFFF000FC0000FC0000FC0000FC0000FC0000FC0000FC0000FC0
000FC0000FC0000FC0000FC1800FC1800FC1800FC1800FC1800FE38007E70003FF0000FC001120
7F9F16>116 DI E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300
TeXDict begin @letter
%%EndSetup
%%Page: 1 1
bop 815 307 a Fi(Solution)20 b(of)f(1.6)866 357 y Fh(Yingw)o(ei)13
b(Y)m(ao)262 457 y(\(a\))h Fg(x)353 463 y Ff(1)372 457 y Fh(\()p
Fg(t)p Fh(\))h(and)g Fg(x)540 463 y Ff(2)558 457 y Fh(\()p
Fg(t)p Fh(\))g(are)g(indep)q(enden)o(t)h(to)f(noise)g Fg(n)p
Fh(\()p Fg(t)p Fh(\),)g(so)g(for)f(user)i(1)f(the)g(addition)f(of)262
506 y Fg(x)286 512 y Ff(2)304 506 y Fh(\()p Fg(t)p Fh(\))g(will)e(not)i
(decrease)i(the)f(noise)f(lev)o(el)f(in)h(the)g(c)o(hannel,)g
Fg(R)1266 512 y Ff(1)1298 506 y Fh(m)o(ust)f(satisfy)827 617
y(0)e Fe(\024)h Fg(R)935 623 y Ff(1)964 617 y Fe(\024)g Fg(C)s
Fh(\()1062 589 y Fg(P)1089 595 y Ff(1)p 1062 607 46 2 v 1066
645 a Fg(N)1112 617 y Fh(\))262 699 y(Similarl)o(y)m(,)827
768 y(0)f Fe(\024)h Fg(R)935 774 y Ff(2)964 768 y Fe(\024)g
Fg(C)s Fh(\()1062 740 y Fg(P)1089 746 y Ff(2)p 1062 758 V 1066
796 a Fg(N)1112 768 y Fh(\))262 853 y(Since)19 b(t)o(w)o(o)f(enco)q(ders)i
(op)q(erate)g(indep)q(eden)o(tly)m(,)f(the)g(p)q(o)o(w)o(er)g(of)f
Fg(x)p Fh(\()p Fg(t)p Fh(\))h(m)o(ust)e(not)i(exceed)262 903
y Fg(P)289 909 y Ff(1)316 903 y Fh(+)10 b Fg(P)385 909 y Ff(2)403
903 y Fh(,)j(so)h(the)h(rate)f(of)f Fg(x)p Fh(\()p Fg(t)p Fh(\))h(should)g
(satisfy)820 1009 y Fg(R)e Fe(\024)f Fg(C)s Fh(\()961 980 y
Fg(P)988 986 y Ff(1)1016 980 y Fh(+)e Fg(P)1084 986 y Ff(2)p
961 999 142 2 v 1013 1037 a Fg(N)1107 1009 y Fh(\))p Fg(;)262
1105 y Fh(but)14 b Fg(R)d Fh(=)h Fg(R)457 1111 y Ff(1)484 1105
y Fh(+)e Fg(R)558 1111 y Ff(2)576 1105 y Fh(,)j(so)761 1174
y Fg(R)793 1180 y Ff(1)820 1174 y Fh(+)c Fg(R)893 1180 y Ff(2)923
1174 y Fe(\024)j Fg(C)s Fh(\()1021 1146 y Fg(P)1048 1152 y
Ff(1)1075 1146 y Fh(+)e Fg(P)1144 1152 y Ff(2)p 1021 1164 V
1072 1202 a Fg(N)1167 1174 y Fh(\))p Fg(:)262 1256 y Fh(So)j(the)i(necessary)
h(condition)d(for)g(reliable)h(comm)o(unicati)o(on)d(is)854
1341 y(\()p Fg(R)902 1347 y Ff(1)920 1341 y Fg(;)c(R)971 1347
y Ff(2)989 1341 y Fh(\))k Fe(2)g Fd(C)p Fg(:)262 1426 y Fh(\(b\))16
b(Using)f(successiv)o(e)k(deco)q(ding)d(,)g(assume)f(w)o(e)h(deco)q(de)i
Fg(x)1222 1432 y Ff(1)1240 1426 y Fh(\()p Fg(t)p Fh(\))e(\014rst)h(and)f
(then)g(deco)q(de)262 1475 y Fg(x)286 1481 y Ff(2)304 1475
y Fh(\()p Fg(t)p Fh(\).)26 b(When)17 b(deco)q(ding)f Fg(x)713
1481 y Ff(1)732 1475 y Fh(\()p Fg(t)p Fh(\),)h(w)o(e)g(can)f(treat)h
Fg(x)1078 1481 y Ff(2)1097 1475 y Fh(\()p Fg(t)p Fh(\))g(as)f(noise.)26
b(Since)17 b Fg(x)1477 1481 y Ff(2)1496 1475 y Fh(\()p Fg(t)p
Fh(\))f(is)h(inde-)262 1525 y(p)q(enden)o(t)d(of)f Fg(n)p Fh(\()p
Fg(t)p Fh(\),)h(the)g(total)f(noise)h(p)q(o)o(w)o(er)g(is)f
Fg(P)1028 1531 y Ff(2)1055 1525 y Fh(+)c Fg(N)c Fh(,)13 b(so)h(the)g(highest)
g(rate)g(user)h(1)e(can)262 1575 y(attain)g(is)g Fg(C)s Fh(\()507
1558 y Fc(P)528 1562 y Fb(1)p 479 1565 93 2 v 479 1589 a Fc(P)500
1593 y Fb(2)517 1589 y Ff(+)p Fc(N)576 1575 y Fh(\).)262 1635
y(Since)i(when)g Fg(R)512 1641 y Ff(1)543 1635 y Fh(=)e Fg(C)s
Fh(\()669 1618 y Fc(P)690 1622 y Fb(1)p 641 1625 V 641 1649
a Fc(P)662 1653 y Fb(2)679 1649 y Ff(+)p Fc(N)738 1635 y Fh(\),)i(the)g(data)
f(of)g(user)i(1)e(can)h(b)q(e)g(deco)q(ded)h(reliably)m(,)d(the)i(in-)262
1684 y(teference)h(brough)o(t)f(b)o(y)f Fg(x)675 1690 y Ff(1)694
1684 y Fh(\()p Fg(t)p Fh(\))h(is)f(completely)g(suppressed)j(in)d(our)h
(successiv)o(e)h(deco)q(ding)262 1734 y(sc)o(heme)f(and)g(user)h(2)f(can)h
(transmit)e(at)h(the)h(rate)f Fg(C)s Fh(\()1125 1717 y Fc(P)1146
1721 y Fb(2)p 1125 1724 38 2 v 1129 1749 a Fc(N)1168 1734 y
Fh(\))g(reliably)m(.)21 b(Hence)16 b(the)g(reliable)262 1784
y(comm)o(uni)o(cation)11 b(is)j(p)q(ossible)g(at)g(\()p Fg(R)848
1769 y Fc(b)848 1794 y Ff(1)866 1784 y Fg(;)7 b(R)917 1769
y Fc(b)917 1794 y Ff(2)934 1784 y Fh(\).)262 1834 y(Using)13
b(similar)e(argumen)o(t,)h(w)o(e)h(can)h(sho)o(w)f(that)h(the)g(rate)g(pair)e
(\()p Fg(R)1327 1819 y Fc(a)1327 1844 y Ff(1)1348 1834 y Fg(;)7
b(R)1399 1819 y Fc(a)1399 1844 y Ff(2)1418 1834 y Fh(\))13
b(is)h(also)e(attain-)262 1884 y(able.)262 1933 y(\(c\))i(If)g(\()p
384 1900 32 2 v Fg(R)416 1939 y Ff(1)434 1933 y Fg(;)p 453
1900 V 7 w(R)484 1939 y Ff(2)503 1933 y Fh(\))g(is)g(a)f(con)o(v)o(ex)h(com)o
(bination)e(of)h(\()p Fg(R)1077 1918 y Fc(a)1077 1944 y Ff(1)1097
1933 y Fg(;)7 b(R)1148 1918 y Fc(a)1148 1944 y Ff(2)1168 1933
y Fh(\))14 b(and)f(\()p Fg(R)1326 1918 y Fc(b)1326 1944 y Ff(1)1345
1933 y Fg(;)7 b(R)1396 1918 y Fc(b)1396 1944 y Ff(2)1413 1933
y Fh(\),)14 b(that)g(is)p 656 2006 V 656 2039 a Fg(R)688 2045
y Ff(1)718 2039 y Fh(=)e Fg(\013C)s Fh(\()843 2011 y Fg(P)870
2017 y Ff(1)p 843 2030 46 2 v 847 2068 a Fg(N)893 2039 y Fh(\))d(+)h(\(1)f
Fe(\000)g Fg(\013)p Fh(\))p Fg(C)s Fh(\()1188 2011 y Fg(P)1215
2017 y Ff(1)p 1144 2030 134 2 v 1144 2068 a Fg(P)1171 2074
y Ff(2)1198 2068 y Fh(+)h Fg(N)1283 2039 y Fh(\))p 656 2136
32 2 v 656 2170 a Fg(R)688 2176 y Ff(2)718 2170 y Fh(=)i Fg(\013C)s
Fh(\()887 2142 y Fg(P)914 2148 y Ff(2)p 843 2160 134 2 v 843
2198 a Fg(P)870 2204 y Ff(1)897 2198 y Fh(+)e Fg(N)981 2170
y Fh(\))g(+)f(\(1)g Fe(\000)h Fg(\013)p Fh(\))p Fg(C)s Fh(\()1233
2142 y Fg(P)1260 2148 y Ff(2)p 1233 2160 46 2 v 1237 2198 a
Fg(N)1283 2170 y Fh(\))262 2266 y(where)15 b(0)c Fe(\024)h
Fg(\013)f Fe(\024)h Fh(1,)h(then)p 680 2233 32 2 v 14 w Fg(R)712
2272 y Ff(1)740 2266 y Fh(+)p 782 2233 V 10 w Fg(R)813 2272
y Ff(2)844 2266 y Fh(=)e Fg(\013C)s Fh(\()968 2249 y Fc(P)989
2253 y Fb(1)1005 2249 y Ff(+)p Fc(P)1051 2253 y Fb(2)p 968
2256 101 2 v 1003 2281 a Fc(N)1073 2266 y Fh(\))e(+)h(\(1)f
Fe(\000)h Fg(\013)p Fh(\))p Fg(C)s Fh(\()1325 2249 y Fc(P)1346
2253 y Fb(1)1361 2249 y Ff(+)p Fc(P)1407 2253 y Fb(2)p 1324
2256 V 1360 2281 a Fc(N)1429 2266 y Fh(\))i(=)g Fg(C)s Fh(\()1555
2249 y Fc(P)1576 2253 y Fb(1)1592 2249 y Ff(+)p Fc(P)1638 2253
y Fb(2)p 1555 2256 V 1590 2281 a Fc(N)1660 2266 y Fh(\).)262
2321 y(Since)i(the)g(function)f Fg(C)s Fh(\()659 2305 y Fc(P)p
657 2312 30 2 v 657 2335 a(N)691 2321 y Fh(\))f(=)g Fg(B)d
Fh(log)857 2331 y Ff(2)876 2321 y Fh(\(1)f(+)969 2305 y Fc(P)p
967 2312 V 967 2335 a(N)1002 2321 y Fh(\))14 b(increases)h(when)1320
2305 y Fc(P)p 1318 2312 V 1318 2335 a(N)1366 2321 y Fh(increases,)g(w)o(e)f
(ha)o(v)o(e)727 2431 y Fg(C)s Fh(\()825 2403 y Fg(P)852 2409
y Ff(2)p 781 2422 134 2 v 781 2460 a Fg(P)808 2466 y Ff(1)835
2460 y Fh(+)c Fg(N)919 2431 y Fh(\))i Fe(\024)p 991 2398 32
2 v 12 w Fg(R)1023 2437 y Ff(2)1053 2431 y Fe(\024)g Fg(C)s
Fh(\()1151 2403 y Fg(P)1178 2409 y Ff(2)p 1150 2422 46 2 v
1154 2460 a Fg(N)1201 2431 y Fh(\))p Fg(:)967 2574 y Fh(1)p
eop
%%Page: 2 2
bop 262 307 a Fh(Note)15 b(that)g Fg(C)s Fh(\()p Fg(x)p Fh(\))g(is)f(a)h(con)
o(tin)o(uous)g(function)g(of)f Fg(x)p Fh(,)h(so)g(there)h(exists)g(0)d
Fe(\024)h Fg(\016)h Fe(\024)f Fg(P)1569 313 y Ff(1)1587 307
y Fh(,)h(suc)o(h)262 357 y(that)p 827 384 32 2 v 827 418 a
Fg(R)859 424 y Ff(2)889 418 y Fh(=)d Fg(C)s Fh(\()1018 390
y Fg(P)1045 396 y Ff(2)p 987 408 109 2 v 987 446 a Fg(N)i Fh(+)9
b Fg(\016)1100 418 y Fh(\))p Fg(:)262 514 y Fh(Hence)15 b(an)o(y)e(con)o(v)o
(ex)h(com)o(bination)e(of)h(\()p Fg(R)931 499 y Fc(a)931 524
y Ff(1)951 514 y Fg(;)7 b(R)1002 499 y Fc(a)1002 524 y Ff(2)1021
514 y Fh(\),\()p Fg(R)1097 499 y Fc(b)1097 524 y Ff(1)1115
514 y Fg(;)g(R)1166 499 y Fc(b)1166 524 y Ff(2)1184 514 y Fh(\))14
b(m)o(ust)f(satisfy)p 833 593 32 2 v 833 626 a Fg(R)865 632
y Ff(2)895 626 y Fh(=)f Fg(C)s Fh(\()1024 598 y Fg(P)1051 604
y Ff(2)p 992 617 109 2 v 992 655 a Fg(N)j Fh(+)9 b Fg(\016)1106
626 y Fh(\))p 766 727 32 2 v 766 760 a Fg(R)798 766 y Ff(1)826
760 y Fh(+)p 867 727 V 9 w Fg(R)899 766 y Ff(2)929 760 y Fh(=)j
Fg(C)s Fh(\()1027 732 y Fg(P)1054 738 y Ff(1)1081 732 y Fh(+)e
Fg(P)1150 738 y Ff(2)p 1027 751 142 2 v 1078 789 a Fg(N)1173
760 y Fh(\))262 847 y(for)j(some)g(0)e Fe(\024)h Fg(\016)i
Fe(\024)e Fg(P)608 853 y Ff(1)626 847 y Fh(.)262 896 y(Consider)18
b(an)f(enco)q(der)j(for)d(user)i(1)e(that)h(m)o(ultiplexes)e(its)i(input)g
(binary)f(stream)g(in)o(to)262 946 y(t)o(w)o(o)c(indep)q(enden)o(t)i(enco)q
(ders)h(with)d(p)q(o)o(w)o(ers)h Fg(\016)i Fh(and)e Fg(P)1118
952 y Ff(1)1145 946 y Fe(\000)9 b Fg(\016)16 b Fh(resp)q(ectiv)o(ely)m(.)j(W)
m(e)13 b(can)h(treat)262 996 y(the)i(outputs)g(of)g(these)h(t)o(w)o(o)e(enco)
q(ders)j(as)e(from)e(t)o(w)o(o)i(di\013eren)o(t)h(users.)25
b(Lab)q(el)16 b(them)f(b)o(y)262 1046 y(user)h(1)372 1052 y
Fc(a)407 1046 y Fh(and)g(user)g(1)600 1052 y Fc(b)632 1046
y Fh(for)g(outputs)g(with)f(p)q(o)o(w)o(ers)h Fg(\016)h Fh(and)f
Fg(P)1232 1052 y Ff(1)1260 1046 y Fe(\000)11 b Fg(\016)18 b
Fh(resp)q(ectiv)o(ely)m(.)23 b(In)16 b(the)262 1096 y(receiv)o(er)i(side)f(w)
o(e)f(use)i(a)e(3-stage)h(successiv)o(e)i(deco)q(der:)25 b(the)17
b(\014rst)g(stage)g(deco)q(des)i(user)262 1145 y(1)283 1151
y Fc(b)299 1145 y Fh(,)14 b(the)h(second)h(stage)f(deco)q(des)h(user)f(2,)f
(the)h(third)g(stage)g(deco)q(des)h(user)f(1)1475 1151 y Fc(a)1495
1145 y Fh(.)20 b(Then)15 b(the)262 1195 y(follo)o(wing)c(rates)k(can)f(b)q(e)
g(attained)g(in)f(reliable)h(comm)o(unicatio)o(n:)854 1306
y Fg(R)886 1312 y Ff(1)p Fc(a)933 1306 y Fh(=)e Fg(C)s Fh(\()1040
1278 y Fg(\016)p 1031 1296 38 2 v 1031 1334 a(N)1074 1306 y
Fh(\))p Fg(;)772 1437 y(R)804 1443 y Ff(1)p Fc(b)848 1437 y
Fh(=)g Fg(C)s Fh(\()990 1409 y Fg(P)1017 1415 y Ff(1)1045 1409
y Fe(\000)d Fg(\016)p 946 1427 205 2 v 946 1465 a(P)973 1471
y Ff(2)1000 1465 y Fh(+)h Fg(N)k Fh(+)9 b Fg(\016)1155 1437
y Fh(\))p Fg(;)827 1557 y(R)859 1563 y Ff(2)889 1557 y Fh(=)j
Fg(C)s Fh(\()1018 1529 y Fg(P)1045 1535 y Ff(2)p 987 1548 109
2 v 987 1586 a Fg(N)i Fh(+)9 b Fg(\016)1100 1557 y Fh(\))p
Fg(:)262 1655 y Fh(It)k(is)h(straigh)o(tforw)o(ard)g(to)f(v)o(erify)h(that)g
Fg(R)929 1661 y Ff(1)p Fc(a)974 1655 y Fh(+)c Fg(R)1048 1661
y Ff(1)p Fc(b)1089 1655 y Fh(+)g Fg(R)1163 1661 y Ff(2)1193
1655 y Fh(=)h Fg(C)s Fh(\()1290 1638 y Fc(P)1311 1642 y Fb(1)1328
1638 y Ff(+)p Fc(P)1374 1642 y Fb(2)p 1290 1645 101 2 v 1326
1669 a Fc(N)1395 1655 y Fh(\).)262 1705 y(The)h(ab)q(o)o(v)o(e)g(argumen)o
(ts)g(sho)o(w)g(that)h(reliable)f(comm)o(unicatio)o(n)e(is)i(p)q(ossible)h
(for)f(an)o(y)g(con-)262 1755 y(v)o(ex)h(com)o(bination)f(of)h(\()p
Fg(R)668 1740 y Fc(a)668 1765 y Ff(1)688 1755 y Fg(;)7 b(R)739
1740 y Fc(a)739 1765 y Ff(2)759 1755 y Fh(\))14 b(and)f(\()p
Fg(R)917 1740 y Fc(b)917 1765 y Ff(1)936 1755 y Fg(;)7 b(R)987
1740 y Fc(b)987 1765 y Ff(2)1004 1755 y Fh(\),)14 b(that)g(is)f(for)h(an)o(y)
f(\()p 1335 1721 32 2 v Fg(R)1367 1761 y Ff(1)1386 1755 y Fg(;)p
1405 1721 V 7 w(R)1436 1761 y Ff(2)1455 1755 y Fh(\))h(suc)o(h)g(that)p
827 1834 V 827 1867 a Fg(R)859 1873 y Ff(2)889 1867 y Fh(=)e
Fg(C)s Fh(\()1018 1839 y Fg(P)1045 1845 y Ff(2)p 987 1857 109
2 v 987 1896 a Fg(N)i Fh(+)9 b Fg(\016)1100 1867 y Fh(\))p
Fg(;)p 760 1968 32 2 v 760 2001 a(R)792 2007 y Ff(1)820 2001
y Fh(+)p 861 1968 V 9 w Fg(R)893 2007 y Ff(2)923 2001 y Fh(=)j
Fg(C)s Fh(\()1021 1973 y Fg(P)1048 1979 y Ff(1)1075 1973 y
Fh(+)e Fg(P)1144 1979 y Ff(2)p 1021 1992 142 2 v 1073 2030
a Fg(N)1167 2001 y Fh(\))p Fg(:)262 2090 y Fh(F)m(or)j(an)o(y)g(\()p
Fg(R)462 2096 y Ff(1)481 2090 y Fg(;)7 b(R)532 2096 y Ff(2)549
2090 y Fh(\))12 b Fe(2)f Fd(C)p Fh(,)j(let)888 2140 y Fg(R)920
2146 y Ff(1)938 2121 y Fa(0)961 2140 y Fh(=)e Fg(R)1037 2146
y Ff(1)1055 2140 y Fg(;)630 2233 y(R)662 2239 y Ff(2)680 2215
y Fa(0)703 2233 y Fh(=)g(min)n Fe(f)p Fg(C)s Fh(\()891 2205
y Fg(P)918 2211 y Ff(2)p 891 2224 46 2 v 895 2262 a Fg(N)941
2233 y Fh(\))p Fg(;)7 b(C)s Fh(\()1030 2205 y Fg(P)1057 2211
y Ff(1)1084 2205 y Fh(+)j Fg(P)1153 2211 y Ff(2)p 1030 2224
142 2 v 1081 2262 a Fg(N)1176 2233 y Fh(\))f Fe(\000)h Fg(R)1275
2239 y Ff(1)1293 2233 y Fe(g)p Fg(:)262 2325 y Fh(It)j(is)h(easy)h(to)e(see)i
(that)f(\()p Fg(R)695 2331 y Ff(1)714 2307 y Fa(0)725 2325
y Fg(;)7 b(R)776 2331 y Ff(2)794 2307 y Fa(0)806 2325 y Fh(\))13
b(satisfy)749 2437 y Fg(R)781 2443 y Ff(1)799 2419 y Fa(0)820
2437 y Fh(+)c Fg(R)893 2443 y Ff(2)912 2419 y Fa(0)935 2437
y Fe(\024)j Fg(C)s Fh(\()1033 2409 y Fg(P)1060 2415 y Ff(1)1087
2409 y Fh(+)d Fg(P)1155 2415 y Ff(2)p 1032 2428 V 1084 2466
a Fg(N)1179 2437 y Fh(\))p Fg(;)967 2574 y Fh(2)p eop
%%Page: 3 3
bop 821 322 a Fg(R)853 328 y Ff(2)872 304 y Fa(0)895 322 y
Fh(=)12 b Fg(C)s Fh(\()1024 294 y Fg(P)1051 300 y Ff(2)p 992
313 109 2 v 992 351 a Fg(N)i Fh(+)c Fg(\016)1106 322 y Fh(\))p
Fg(;)262 412 y Fh(for)j(some)g(0)e Fe(\024)h Fg(\016)i Fe(\024)e
Fg(P)608 418 y Ff(1)626 412 y Fh(.)262 462 y(So)h(\()p Fg(R)367
468 y Ff(1)385 444 y Fa(0)397 462 y Fg(;)7 b(R)448 468 y Ff(2)466
444 y Fa(0)477 462 y Fh(\))14 b(is)g(attainable)f(and)h(since)790
553 y Fg(R)822 559 y Ff(1)840 535 y Fa(0)863 553 y Fh(=)e Fg(R)939
559 y Ff(1)957 553 y Fg(;)18 b(R)1019 559 y Ff(2)1037 535 y
Fa(0)1060 553 y Fe(\025)12 b Fg(R)1136 559 y Ff(2)1154 553
y Fg(;)262 644 y Fh(\()p Fg(R)310 650 y Ff(1)328 644 y Fg(;)7
b(R)379 650 y Ff(2)396 644 y Fh(\))14 b(is)g(also)f(attainable.)262
694 y(Con)o(v)o(ersely)m(,)e(from)g(part)h(\(a\))g(w)o(e)g(kno)o(w)g(that)g
(a)g(necessary)i(condition)d(for)h(\()p Fg(R)1487 700 y Ff(1)1505
694 y Fg(;)7 b(R)1556 700 y Ff(2)1574 694 y Fh(\))12 b(to)g(b)q(e)262
744 y(attainable)h(is)g(\()p Fg(R)547 750 y Ff(1)566 744 y
Fg(;)7 b(R)617 750 y Ff(2)634 744 y Fh(\))12 b Fe(2)f Fd(C)p
Fh(.)262 794 y(Hence)k(reliable)e(comm)o(unication)e(is)i(p)q(ossible)i(if)e
(and)g(only)g(if)h(\()p Fg(R)1314 800 y Ff(1)1332 794 y Fg(;)7
b(R)1383 800 y Ff(2)1400 794 y Fh(\))12 b Fe(2)f Fd(C)p Fh(.)262
844 y(\(d\))j(The)g(rate)g(pairs)g(\()p Fg(R)651 850 y Ff(1)670
844 y Fg(;)7 b(R)721 850 y Ff(2)738 844 y Fh(\))14 b(should)g(satisfy)722
956 y(0)d Fe(\024)h Fg(R)830 962 y Ff(1)860 956 y Fe(\024)g
Fg(\013B)d Fh(log)1025 966 y Ff(2)1043 956 y Fh(\(1)g(+)1145
928 y Fg(P)1172 934 y Ff(1)p 1136 947 65 2 v 1136 985 a Fg(N)c(\013)1205
956 y Fh(\))p Fg(;)618 1087 y Fh(0)12 b Fe(\024)f Fg(R)726
1093 y Ff(2)756 1087 y Fe(\024)h Fh(\(1)d Fe(\000)h Fg(\013)p
Fh(\))p Fg(B)f Fh(log)1025 1097 y Ff(2)1043 1087 y Fh(\(1)g(+)1197
1059 y Fg(P)1224 1065 y Ff(2)p 1136 1077 169 2 v 1136 1115
a Fg(N)c Fh(\(1)k Fe(\000)g Fg(\013)p Fh(\))1309 1087 y(\))p
Fg(:)262 1183 y Fh(So)k(the)i(highest)f(rate)g(w)o(e)g(can)h(get)f(while)f
(reliable)h(comm)o(uni)o(cation)d(is)j(p)q(ossible)g(is)262
1294 y Fg(B)9 b Fh(log)356 1304 y Ff(2)374 1294 y Fh([\(1+)470
1266 y Fg(P)497 1272 y Ff(1)p 460 1284 65 2 v 460 1322 a Fg(N)c(\013)530
1294 y Fh(\))546 1277 y Fc(\013)569 1294 y Fh(\(1+)705 1266
y Fg(P)732 1272 y Ff(2)p 643 1284 169 2 v 643 1322 a Fg(N)g
Fh(\(1)k Fe(\000)h Fg(\013)p Fh(\))817 1294 y(\))833 1277 y
Ff(1)p Fa(\000)p Fc(\013)899 1294 y Fh(])h Fe(\024)h Fg(B)d
Fh(log)1060 1304 y Ff(2)1079 1294 y Fh([)p Fg(\013)p Fh(\(1+)1201
1266 y Fg(P)1228 1272 y Ff(1)p 1191 1284 65 2 v 1191 1322 a
Fg(N)c(\013)1261 1294 y Fh(\)+\(1)p Fe(\000)p Fg(\013)p Fh(\)\(1+)1557
1266 y Fg(P)1584 1272 y Ff(2)p 1495 1284 169 2 v 1495 1322
a Fg(N)g Fh(\(1)k Fe(\000)h Fg(\013)p Fh(\))1669 1294 y(])h(=)h
Fg(B)d Fh(log)1830 1304 y Ff(2)1848 1294 y Fh(\(1+)1922 1266
y Fg(P)1949 1272 y Ff(1)1977 1266 y Fh(+)g Fg(P)2045 1272 y
Ff(2)p 1922 1284 142 2 v 1974 1322 a Fg(N)2069 1294 y Fh(\))262
1415 y(The)14 b(equalit)o(y)f(holds)g(if)g(and)h(only)f(if)g(1)c(+)945
1398 y Fc(P)966 1402 y Fb(1)p 938 1405 51 2 v 938 1429 a Fc(N)s(\013)1005
1415 y Fh(=)j(1)d(+)1167 1398 y Fc(P)1188 1402 y Fb(2)p 1126
1405 120 2 v 1126 1429 a Fc(N)s Ff(\(1)p Fa(\000)p Fc(\013)p
Ff(\))1250 1415 y Fh(,)k(that)h(is)g Fg(\013)d Fh(=)1526 1398
y Fc(P)1547 1402 y Fb(1)p 1494 1405 101 2 v 1494 1429 a Fc(P)1515
1433 y Fb(1)1531 1429 y Ff(+)p Fc(P)1577 1433 y Fb(2)1599 1415
y Fh(.)967 2574 y(3)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF