📄 4.26.kishore.ps
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%%EndProlog
%%BeginSetup
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TeXDict begin
%%PaperSize: Letter
%%EndSetup
%%Page: 1 1
1 0 bop 257 262 a Ff(Solution)12 b(to)j(4.26)g(b)o(y)g(Shalinee)e
(Kishore)h(4/2/99)40 b Fe(\(a\))14 b(W)m(e)f(kno)o(w)g(from)f
(\(3.115\))257 311 y(that)i(the)h(w)o(orst)f(asymptotic)f(e\013ectiv)o
(e)i(energy)g(of)e(a)g(detector)j(is)e(de\014ned)h(as)702
403 y Fd(!)q Fe(\()p Fd(A)776 409 y Fc(1)795 403 y Fd(;)7
b(A)845 409 y Fc(2)863 403 y Fe(\))12 b(=)g(min)1011
369 y Fb(\000)1030 403 y Fd(A)1061 386 y Fc(2)1061 413
y(1)1079 403 y Fd(\021)1100 409 y Fc(1)1119 403 y Fd(;)7
b(A)1169 386 y Fc(2)1169 413 y(2)1187 403 y Fd(\021)1208
409 y Fc(2)1226 369 y Fb(\001)1636 403 y Fe(\(1\))257
494 y(Also,)22 b(from)d(the)i(analysis)e(of)h(the)h(optim)o(um)c
(detector)22 b(of)e(a)h(t)o(w)o(o-user)g(sync)o(hronous)257
544 y(CDMA)d(system,)h(w)o(e)f(kno)o(w)g(that)g Fd(\021)854
550 y Fc(1)890 544 y Fe(and)g Fd(\021)996 550 y Fc(2)1015
544 y Fe(,)g(the)h(optim)o(um)c(asymptotic)i(m)o(ultiuser)257
594 y(e\016ciency)e(for)f(users)h(1)f(and)f(2,)g(resp)q(ectiv)o(ely)m
(,)i(are)651 709 y Fd(\021)672 715 y Fc(1)732 709 y Fe(=)42
b(min)882 651 y Fb(\022)912 709 y Fe(1)p Fd(;)7 b Fe(1)h(+)1028
681 y Fd(A)1059 666 y Fc(2)1059 691 y(2)p 1028 700 50
2 v 1028 738 a Fd(A)1059 723 y Fc(2)1059 749 y(1)1092
709 y Fa(\000)h Fe(2)p Fa(j)p Fd(\032)p Fa(j)1204 681
y Fd(A)1235 687 y Fc(2)p 1204 700 V 1204 738 a Fd(A)1235
744 y Fc(1)1258 651 y Fb(\023)1636 709 y Fe(\(2\))651
827 y Fd(\021)672 833 y Fc(2)732 827 y Fe(=)42 b(min)882
769 y Fb(\022)912 827 y Fe(1)p Fd(;)7 b Fe(1)h(+)1028
799 y Fd(A)1059 784 y Fc(2)1059 809 y(1)p 1028 818 V
1028 856 a Fd(A)1059 841 y Fc(2)1059 867 y(2)1092 827
y Fa(\000)h Fe(2)p Fa(j)p Fd(\032)p Fa(j)1204 799 y Fd(A)1235
805 y Fc(1)p 1204 818 V 1204 856 a Fd(A)1235 862 y Fc(2)1258
769 y Fb(\023)1636 827 y Fe(\(3\))257 941 y(The)15 b(w)o(orst)f
(asymptotic)e(e\013ectiv)o(e)k(energy)e(is)g(then)257
1033 y Fd(!)q Fe(\()p Fd(A)331 1039 y Fc(1)351 1033 y
Fd(;)7 b(A)401 1039 y Fc(2)419 1033 y Fe(\))k(=)h(min)566
999 y Fb(\000)585 1033 y Fe(min)662 999 y Fb(\000)681
1033 y Fd(A)712 1015 y Fc(2)712 1043 y(1)730 1033 y Fd(;)7
b(A)780 1015 y Fc(2)780 1043 y(1)808 1033 y Fe(+)i Fd(A)880
1015 y Fc(2)880 1043 y(2)908 1033 y Fa(\000)h Fe(2)p
Fa(j)p Fd(\032)p Fa(j)p Fd(A)1047 1039 y Fc(2)1065 1033
y Fd(A)1096 1039 y Fc(1)1115 999 y Fb(\001)1140 1033
y Fd(;)d Fe(min)1235 999 y Fb(\000)1254 1033 y Fd(A)1285
1015 y Fc(2)1285 1043 y(2)1304 1033 y Fd(;)g(A)1354 1015
y Fc(2)1354 1043 y(2)1381 1033 y Fe(+)j Fd(A)1454 1015
y Fc(2)1454 1043 y(1)1482 1033 y Fa(\000)f Fe(2)p Fa(j)p
Fd(\032)p Fa(j)p Fd(A)1620 1039 y Fc(2)1638 1033 y Fd(A)1669
1039 y Fc(1)1688 999 y Fb(\001\001)1636 1082 y Fe(\(4\))257
1132 y(This)14 b(e\013ectiv)o(ely)h(reduces)h(to)549
1224 y Fd(!)q Fe(\()p Fd(A)623 1230 y Fc(1)642 1224 y
Fd(;)7 b(A)692 1230 y Fc(2)710 1224 y Fe(\))12 b(=)g(min)858
1190 y Fb(\000)877 1224 y Fd(A)908 1206 y Fc(2)908 1234
y(1)927 1224 y Fd(;)7 b(A)977 1206 y Fc(2)977 1234 y(2)995
1224 y Fd(;)g(A)1045 1206 y Fc(2)1045 1234 y(1)1072 1224
y Fe(+)j Fd(A)1145 1206 y Fc(2)1145 1234 y(2)1173 1224
y Fa(\000)f Fe(2)p Fa(j)p Fd(\032)p Fa(j)p Fd(A)1311
1230 y Fc(2)1329 1224 y Fd(A)1360 1230 y Fc(1)1379 1190
y Fb(\001)1636 1224 y Fe(\(5\))320 1315 y(\(b\))14 b(The)g(equal)g(p)q
(o)o(w)o(er)g(p)q(enalt)o(y)g(is)f(de\014ned)i(as)893
1398 y(1)p 878 1416 V 878 1454 a Fd(A)909 1442 y Fc(2)933
1426 y Fd(!)q Fe(\()p Fd(A;)7 b(A)p Fe(\))563 b(\(6\))257
1529 y(In)14 b(this)g(particular)g(system,)f(the)i(equal)e(p)q(o)o(w)o
(er)h(p)q(enalt)o(y)g(reduces)i(to)723 1609 y(1)p 709
1628 V 709 1666 a Fd(A)740 1654 y Fc(2)763 1637 y Fd(!)q
Fe(\()p Fd(A;)7 b(A)p Fe(\))12 b(=)g(min)5 b(\(1)p Fd(;)i
Fe(2)h Fa(\000)i Fe(2)p Fa(j)p Fd(\032)p Fa(j)p Fe(\))392
b(\(7\))257 1740 y(This)14 b(p)q(enalt)o(y)g(is)g(0)f(dB)i(or)e(simply)
f(equal)i(to)g(1,)f(if:)867 1832 y(2)c Fa(\000)g Fe(2)p
Fa(j)p Fd(\032)p Fa(j)i(\025)h Fe(1)556 b(\(8\))257 1923
y(This)14 b(then)h(reduces)h(to)897 1973 y Fa(j)p Fd(\032)p
Fa(j)11 b(\024)h Fe(0)p Fd(:)p Fe(5)585 b(\(9\))257 2047
y(\(c\))19 b(In)e(the)h(case)h(when)e Fa(j)p Fd(\032)p
Fa(j)g Fd(<)h(:)p Fe(5,)f(w)o(e)h(see)h(that)e(the)h(p)q(o)o(w)o(er)g
(p)q(enalt)o(y)f(for)h(using)f(equal)257 2097 y(p)q(o)o(w)o(ers)c(is)e
(0)h(dB.)f(This)h(is)g(equiv)n(alen)o(t)f(to)g(sa)o(ying)g(that)h(the)g
(w)o(orst)g(asymptotic)f(m)o(ultiuser)257 2147 y(e\016ciency)k(among)c
(the)j(users)h(is)e(1.)18 b(In)13 b(other)i(w)o(ords,)e(in)g(this)g
(high)g(SNR)g(region)h(for)f(the)257 2197 y(sp)q(eci\014ed)18
b(correlation,)d(the)i(t)o(w)o(o)e(users)i(are)g(op)q(erating)e(as)h
(if)f(in)g(a)h(single-user)g(c)o(hannel)257 2247 y(with)f(additiv)o(e)f
(white)h(Guassian)g(noise.)21 b(The)16 b(users)g(are)f(op)q(erating)g
(as)g(if)g(in)f(t)o(w)o(o)h(inde-)257 2296 y(p)q(enden)o(t)f(systems)f
(that)g(can)g(obtain)f(the)h(desired)h(bit)e(error)i(rate)f(as)g(long)f
(as)h(their)g(SNR)257 2346 y(is)h(greater)h(than)f(or)g(equal)f(to)h
(12)f(dB.)963 2628 y(1)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF
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