3.5.huaidai.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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TeXDict begin

1 0 bop 1146 569 a Fl(ELE533)52 b(Homew)l(ork)f(3)1586
951 y Fk(Huaiyu)40 b(Dai)1574 1271 y Fj(19)f(F)-10 b(eb)39
b(1999)-40 1688 y Fi(Problem)h(3.5.)53 b Fh(Consider)36
b(the)g(follo)m(wing)d(m-ary)i(h)m(yp)s(othesis)i(testing)e(problem)g
(with)g Fg(L)p Fh(-dimensional)-186 1808 y(observ)-5
b(ations:)1486 1901 y Fg(H)1567 1916 y Ff(1)1634 1901
y Fh(:)28 b Fi(y)h Fh(=)f Fi(Cx)2021 1916 y Fe(1)2088
1901 y Fh(+)22 b Fi(d)p Fg(;)1867 1996 y Fh(.)1867 2030
y(.)1867 2063 y(.)1459 2183 y Fg(H)1540 2198 y Fd(m)1634
2183 y Fh(:)28 b Fi(y)h Fh(=)f Fi(Cx)2021 2198 y Fe(m)2115
2183 y Fh(+)22 b Fi(d)q Fg(;)-186 2334 y Fh(where)37
b Fi(C)g Fh(is)e(an)h(in)m(v)m(ertible)g Fg(L)25 b Fc(\002)g
Fg(L)36 b Fh(matrix.)53 b(F)-8 b(urthermore,)37 b(w)m(e)g(assume)g
(that)f Fi(C)g Fh(and)g Fi(d)g Fh(are)h(indep)s(enden)m(t)-186
2455 y(of)31 b(the)i(random)d(v)m(ector)j Fi(x)p Fh(,)f(and)g(their)g
(join)m(t)e(distribution)g(is)i(indep)s(en)m(tden)m(t)h(of)e(the)h
(true)h(h)m(yp)s(othesis.)44 b(Sho)m(w)-186 2575 y(that)33
b(if)g Fi(C)g Fh(and)h Fi(d)g Fh(are)g(kno)m(wn)h(to)e(the)h(detector,)
h(then)f(the)g(mxim)m(um-lik)m(eliho)s(o)s(d)28 b(decision)33
b(rule)h(is)f(the)h Fi(C)f Fh(is)-186 2696 y(the)g(mxim)m(um-lik)m
(eliho)s(o)s(d)27 b(decision)32 b(rule)g(for)1624 2850
y Fg(H)1705 2865 y Ff(1)1771 2850 y Fh(:)c Fi(z)g Fh(=)f
Fi(x)2066 2865 y Fe(1)2111 2850 y Fg(;)1867 2946 y Fh(.)1867
2979 y(.)1867 3012 y(.)1596 3132 y Fg(H)1677 3147 y Fd(m)1771
3132 y Fh(:)h Fi(z)g Fh(=)f Fi(x)2066 3147 y Fe(m)2138
3132 y Fg(;)-186 3308 y Fh(applied)k(to)1526 3428 y Fi(z)d
Fh(=)f Fi(C)1788 3387 y Fb(\000)p Fe(1)1888 3428 y Fh(\()p
Fi(y)d Fc(\000)e Fi(d)p Fh(\))q Fg(:)-40 3672 y Fi(Solution:)-40
3878 y Fh(F)-8 b(rom)31 b(Prop)s(osition)g(3.1,)h(the)h(mxim)m(um-lik)m
(eliho)s(o)s(d)27 b(decision)32 b(rule)h(is)f(giv)m(en)g(b)m(y)692
4128 y(\012)762 4143 y Fd(i)818 4128 y Fh(=)c Fc(f)p
Fg(z)k Fh(:)c Fg(f)1152 4144 y Fd(Z)5 b Fb(j)p Fd(i)1252
4128 y Fh(\()p Fg(z)t Fh(\))29 b(=)78 b(max)1509 4186
y Fd(j)t Ff(=1)p Fd(;:::)n(;m)1809 4128 y Fg(f)1857 4144
y Fd(Z)5 b Fb(j)p Fd(j)1966 4128 y Fh(\()p Fg(z)t Fh(\))p
Fc(g)22 b(\000)2267 4020 y Fd(i)p Fb(\000)p Ff(1)2278
4045 y Fa([)2263 4227 y Fd(j)t Ff(=1)2402 4128 y Fh(\012)2472
4143 y Fd(j)2509 4128 y Fg(;)17 b(i)28 b Fh(=)f(1)p Fg(;)17
b(:)g(:)g(:)f(;)h(m)753 b Fh(\(1\))-40 4474 y(Then)34
b(for)e(original)d(observ)-5 b(ations,)33 b(the)g(ML)g(decision)f(rule)
g(is)665 4723 y(\012)735 4738 y Fd(i)791 4723 y Fh(=)c
Fc(f)p Fi(y)h Fh(:)f Fg(f)1136 4739 y Fe(Y)r Fb(j)p Fd(i)1247
4723 y Fh(\()p Fi(y)r Fh(\))f(=)78 b(max)1515 4781 y
Fd(j)t Ff(=1)p Fd(;:::)n(;m)1814 4723 y Fg(f)1862 4739
y Fe(Y)r Fb(j)p Fd(j)1982 4723 y Fh(\()p Fi(y)r Fh(\))p
Fc(g)21 b(\000)2294 4615 y Fd(i)p Fb(\000)p Ff(1)2305
4640 y Fa([)2290 4823 y Fd(j)t Ff(=1)2429 4723 y Fh(\012)2499
4738 y Fd(j)2536 4723 y Fg(;)c(i)28 b Fh(=)f(1)p Fg(;)17
b(:)g(:)g(:)f(;)h(m)726 b Fh(\(2\))-40 5069 y(F)-8 b(or)32
b(new)i(data,)e(the)h(ML)g(decision)f(rule)g(is)695 5319
y(\012)765 5334 y Fd(i)821 5319 y Fh(=)c Fc(f)p Fi(z)f
Fh(:)h Fg(f)1155 5334 y Fe(Z)p Fb(j)p Fd(i)1252 5319
y Fh(\()p Fi(z)p Fh(\))g(=)78 b(max)1509 5376 y Fd(j)t
Ff(=1)p Fd(;:::)n(;m)1809 5319 y Fg(f)1857 5334 y Fe(Z)p
Fb(j)p Fd(j)1963 5319 y Fh(\()p Fi(z)p Fh(\))p Fc(g)21
b(\000)2264 5211 y Fd(i)p Fb(\000)p Ff(1)2275 5235 y
Fa([)2260 5418 y Fd(j)t Ff(=1)2399 5319 y Fh(\012)2469
5334 y Fd(j)2506 5319 y Fg(;)c(i)28 b Fh(=)f(1)p Fg(;)17
b(:)g(:)g(:)f(;)h(m)756 b Fh(\(3\))1857 5655 y(1)p eop
2 1 bop -40 167 a Fh(Since)53 b(the)f Fi(C)g Fh(and)h
Fi(d)f Fh(are)g(indep)s(enden)m(t)i(of)d(the)i(true)f(h)m(yp)s
(othesis,)59 b(and)52 b(kno)m(wn)h(to)f(the)h(detector,)-186
287 y Fi(z)27 b Fh(=)h Fi(C)76 251 y Fb(\000)p Fe(1)176
287 y Fh(\()p Fi(y)23 b Fc(\000)g Fi(d)p Fh(\))41 b(is)f(just)i(linear)
d(transformation)f(of)j(the)g(original)c(observ)-5 b(ations.)69
b(b)s(ecause)42 b Fi(C)e Fh(is)h(in)m(v)m(ert-)-186 408
y(ible,)36 b(the)h(transformation)e(is)h(one-to-one)g(mapping.)54
b(So)36 b Fi(z)h Fh(is)f(an)g(equiv)-5 b(alen)m(t)37
b(observ)-5 b(ation)36 b(to)g Fi(y)p Fh(.)56 b(F)-8 b(urther)-186
528 y(more,)32 b(w)m(e)i(ha)m(v)m(e)1020 648 y Fg(f)1068
664 y Fe(Y)r Fb(j)p Fd(i)1179 648 y Fh(\()p Fi(y)r Fh(\))27
b(=)g Fg(f)1494 664 y Fe(Z)p Fb(j)p Fd(i)1592 648 y Fh(\()p
Fi(z)g Fh(=)h Fi(C)1892 607 y Fb(\000)p Fe(1)1992 648
y Fh(\()p Fi(y)23 b Fc(\000)g Fi(d)p Fh(\)\))p Fg(=)p
Fc(j)p Fg(det)p Fh(\()p Fi(C)p Fh(\))p Fc(j)1080 b Fh(\(4\))-186
823 y(Note)33 b(that)f Fc(j)p Fg(det)p Fh(\()p Fi(C)p
Fh(\))p Fc(j)g Fh(is)g(constan)m(t,)i(So)e(decision)g(rule)g(\(3\))g
(has)h(the)g(same)g(e\013ect)g(as)g(decision)f(rule)g(\(2\).)1857
5655 y(2)p eop end
userdict /end-hook known{end-hook}if