6.6.pramod.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300dpi
TeXDict begin

%%EndSetup
%%Page: 1 1
1 0 bop -30 72 1305 2 v 1275 72 a Fl(Problem)11 b(6.6,)f(Autho)o(r:)15
b(Pramo)q(d)d(Visw)o(anath,)-30 175 y Fk(6.6\(a\):)-30
263 y(The)22 b(estimator)f(that)h(minimi)o(zes)d(the)i(mean-square)h
(error)g(is)f(the)h(conditional)g(exp)q(ectation)f(of)i
Fj(b)1901 270 y Fi(1)-30 323 y Fk(giv)o(en)15 b Fj(y)r
Fk(\()p Fj(t)p Fk(\))h(and)h(is)524 397 y(~)516 410 y
Fj(b)537 417 y Fi(1)598 410 y Fk(=)42 b Fh(E)13 b Fk([)o
Fj(b)h Fg(j)f Fj(y)r Fk(\()p Fj(t)p Fk(\)])598 483 y(=)42
b Fh(P)o Fk([)p Fj(b)13 b Fk(=)h(1)g Fg(j)g Fj(y)r Fk(\()p
Fj(t)p Fk(\)])c Fg(\000)h Fh(P)o Fk([)p Fj(b)i Fk(=)h
Fg(\000)p Fk(1)g Fg(j)g Fj(y)r Fk(\()p Fj(t)p Fk(\)])598
578 y(=)775 544 y(2)p 683 566 209 2 v 683 619 a(1)d(+)772
596 y Ff(L)796 601 y Fe(1)813 596 y Fi(\()p Fd(\000)p
Fi(1\))p 772 607 114 2 v 772 636 a Ff(L)796 641 y Fe(1)813
636 y Fi(\(+1\))907 578 y Fg(\000)g Fk(1)58 b(from)15
b(\(4.18\))598 727 y(=)878 693 y(2)p 683 715 416 2 v
683 814 a(1)c(+)772 770 y(exp)847 726 y Fc(\020)871 770
y Fd(\000)903 745 y Fe(2)p Fb(A)941 736 y Fe(2)941 759
y(1)959 745 y Fb(y)975 752 y Fe(1)p 904 762 89 2 v 922
787 a(2)p Fb(\033)956 780 y Fe(2)997 770 y Fd(\000)p
Ff(A)1050 758 y Fe(2)1050 781 y(1)1069 726 y Fc(\021)p
772 802 322 2 v 792 839 a Fk(exp)866 843 y(\()890 821
y Fe(2)p Fb(A)928 828 y Fe(1)946 821 y Fb(y)962 828 y
Fe(1)p 890 832 89 2 v 909 857 a(2)p Fb(\033)943 850 y
Fe(2)984 839 y Fd(\000)p Ff(A)1037 828 y Fe(2)1037 850
y(1)1055 843 y Fk(\))1114 727 y Fg(\000)g Fk(1)598 953
y(=)683 917 y(1)g Fg(\000)g Fk(exp)847 896 y Fd(\000)p
Fi(2)p Ff(A)918 901 y Fe(1)936 896 y Ff(y)953 901 y Fe(1)p
847 906 124 2 v 890 934 a Ff(\033)911 925 y Fe(2)p 683
941 294 2 v 683 990 a Fk(1)g(+)g(exp)847 969 y Fd(\000)p
Fi(2)p Ff(A)918 974 y Fe(1)936 969 y Ff(y)953 974 y Fe(1)p
847 978 124 2 v 889 1007 a Ff(\033)910 998 y Fe(2)598
1092 y Fk(=)42 b(tanh)783 1031 y Fc(\022)819 1058 y Fj(A)856
1065 y Fi(1)875 1058 y Fj(y)899 1065 y Fi(1)p 819 1080
100 2 v 854 1126 a Fj(\033)924 1031 y Fc(\023)-30 1195
y Fk(and)17 b(the)f(corresp)q(onding)h(mean-square)f(error)g(is)-30
1282 y(6.6\(b\):)386 1457 y Fj(c)8 b Fk(\()p Fj(A)471
1464 y Fi(1)491 1457 y Fj(;)g(\033)r Fk(\))40 b(=)i(argmin)831
1469 y Ff(\013)p Fd(2)p Fa(R)900 1457 y Fh(E)936 1409
y Fc(h)956 1457 y Fk(\()p Fj(b)996 1464 y Fi(1)1026 1457
y Fg(\000)11 b Fj(\013y)1131 1464 y Fi(1)1151 1457 y
Fk(\))1170 1433 y Fi(2)1190 1409 y Fc(i)602 1549 y Fk(=)42
b(argmin)831 1561 y Ff(\013)p Fd(2)p Fa(R)900 1549 y
Fh(E)936 1501 y Fc(h)956 1549 y Fk(\(\(1)11 b Fg(\000)g
Fj(A)1116 1556 y Fi(1)1135 1549 y Fj(\013)p Fk(\))e Fj(b)1215
1556 y Fi(1)1245 1549 y Fg(\000)i Fj(\013\033)r(n)1385
1556 y Fi(1)1405 1549 y Fk(\))1424 1525 y Fi(2)1444 1501
y Fc(i)602 1640 y Fk(=)42 b(argmin)831 1652 y Ff(\013)p
Fd(2)p Fa(R)908 1592 y Fc(n\020)961 1640 y Fj(A)998 1620
y Fi(2)998 1652 y(1)1028 1640 y Fk(+)11 b Fj(\033)1107
1620 y Fi(2)1126 1592 y Fc(\021)1160 1640 y Fj(\013)1191
1620 y Fi(2)1222 1640 y Fk(+)g(1)g Fg(\000)g Fk(2)p Fj(\013A)1448
1647 y Fi(1)1468 1592 y Fc(o)602 1749 y Fk(=)741 1716
y Fj(A)778 1723 y Fi(1)p 687 1738 166 2 v 687 1783 a
Fj(A)724 1766 y Fi(2)724 1794 y(1)754 1783 y Fk(+)g Fj(\033)833
1769 y Fi(2)-30 1873 y Fk(and)17 b(the)f(corresp)q(onding)h(error)f(is)
671 1853 y Ff(\033)692 1841 y Fe(2)p 635 1861 110 2 v
635 1890 a Ff(\033)656 1881 y Fe(2)674 1890 y Fi(+)p
Ff(A)727 1879 y Fe(2)727 1902 y(1)750 1873 y Fk(.)-30
1971 y(6.6\(c\):)43 2059 y(F)l(or)h(an)o(y)f(estimator)438
2046 y(^)440 2059 y Fj(b)461 2066 y Fi(1)496 2059 y Fk(that)h(is)f
(constrained)h(to)f(b)q(e)h(binary)f(v)m(alue,)f(the)h(error)g(is)645
2162 y Fh(E)681 2101 y Fc(\024)703 2113 y(\020)728 2162
y Fj(b)749 2169 y Fi(1)779 2162 y Fg(\000)827 2148 y
Fk(^)829 2162 y Fj(b)850 2169 y Fi(1)869 2113 y Fc(\021)894
2125 y Fi(2)914 2101 y Fc(\025)950 2162 y Fk(=)d(4)p
Fh(P)1059 2113 y Fc(h)1079 2162 y Fj(b)1100 2169 y Fi(1)1133
2162 y Fg(6)p Fk(=)1183 2148 y(^)1185 2162 y Fj(b)1206
2169 y Fi(1)1226 2113 y Fc(i)-30 2265 y Fk(and)g(hence)g(the)f(MMSE)h
(estimator)e(constrained)i(to)h(b)q(e)f(binary)f(v)m(alued)h(is)g(just)
g(the)f(minim)o(um)c(bit-rate)-30 2325 y(detector,)15
b(namely)g(sgn)9 b(\()p Fj(y)465 2332 y Fi(1)484 2325
y Fk(\))16 b(and)h(the)f(corresp)q(onding)h(error)g(is)f(4)p
Fj(Q)1251 2277 y Fc(\020)1281 2305 y Ff(A)1307 2310 y
Fe(1)p 1281 2313 44 2 v 1292 2342 a Ff(\033)1329 2277
y Fc(\021)1354 2325 y Fk(.)-30 2419 y(6.6\(d\):)43 2507
y(I)g(ha)o(v)o(e)f(not)i(y)o(et)e(got)i(a)g(satisfactory)f(closed)g
(form)f(expression)h(for)g(the)g(rate)h(of)f(deca)o(y)f(of)i(estimate)
-30 2567 y(error)g(using)h(estimator)d(in)i(6.6\(a\).)25
b(My)16 b(conjecture)g(is)h(\()p Fj(f)5 b(r)q(ac\033)r(A)1214
2574 y Fi(1)1233 2567 y Fk(\))1252 2543 y Fi(2)1272 2567
y Fk(.)24 b(The)17 b(rates)g(of)g(deca)o(y)g(using)g(the)-30
2656 y(other)f(t)o(w)o(o)g(estimators)g(are)g(clearly)734
2636 y Fi(1)p 668 2644 151 2 v 668 2687 a(1+)713 2691
y Fk(\()737 2669 y Fb(A)760 2676 y Fe(1)p 737 2680 41
2 v 748 2700 a Fb(\033)782 2691 y Fk(\))801 2662 y Fe(2)840
2656 y Fk(and)h Fj(e)958 2633 y Fd(\000)995 2610 y Fb(A)1018
2600 y Fe(2)1018 2622 y(1)p 990 2626 52 2 v 990 2651
a(2)p Fb(\033)1024 2644 y Fe(2)1048 2656 y Fk(.)933 2828
y(1)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF