4.10.kishore.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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%%EndProlog
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1 0 bop 257 262 a Fj(Solution)18 b(to)j(4.10)g(b)o(y)g(Shalinee)e
(Kishore)h(\(3/9/99\))40 b Fi(W)m(e)18 b(wish)h(to)f(sho)o(w)h(the)257
311 y(equiv)n(alence)11 b(in)f(maximi)o(zing)e(\012\()p
Fj(b)p Fi(\))i(o)o(v)o(er)h Fj(b)g Fh(2)g(f\000)p Fi(1)p
Fg(;)c Fi(1)p Fh(g)1135 296 y Ff(K)1176 311 y Fi(and)k(minim)o(izing)c
Fj(x)1490 296 y Ff(T)1516 311 y Fj(Gx)s Fi(+)s Fj(z)1638
296 y Ff(T)1664 311 y Fj(x)257 361 y Fi(o)o(v)o(er)14
b Fj(x)e Fh(2)f(f)p Fi(0)p Fg(;)c Fi(1)p Fh(g)525 346
y Ff(K)556 361 y Fi(.)18 b(W)m(e)13 b(b)q(egin)h(b)o(y)g(noting)f
(that:)878 462 y Fj(x)e Fi(=)963 434 y Fj(b)e Fi(+)g
Fj(1)p 963 452 102 2 v 1003 490 a Fi(2)1636 462 y(\(1\))257
552 y(where)15 b Fj(1)f Fi(is)g(a)f Fg(K)18 b Fi(column)12
b(v)o(ector)i(of)g(ones.)k(This)c(means)872 631 y Fj(b)d
Fi(=)h(2)p Fj(x)d Fi(+)h Fj(1)561 b Fi(\(2\))257 710
y(W)m(e)14 b(kno)o(w)f(that)750 760 y(\012\()p Fj(b)p
Fi(\))e(=)h(2)p Fj(b)942 743 y Ff(T)968 760 y Fj(Ay)e
Fh(\000)f Fj(b)1107 743 y Ff(T)1133 760 y Fj(Hb)439 b
Fi(\(3\))257 828 y(W)m(e)14 b(can)g(then)h(substitute)g(the)f
(expression)h(for)f Fj(b)f Fi(in)h(terms)f(of)h Fj(x)g
Fi(in)o(to)f(\012:)415 907 y(\012\()p Fj(x)p Fi(\))42
b(=)g(2\(2)p Fj(x)9 b Fh(\000)h Fj(1)p Fi(\))792 889
y Ff(T)818 907 y Fj(Ay)g Fh(\000)f Fi(\(2)p Fj(x)h Fh(\000)f
Fj(1)p Fi(\))1083 889 y Ff(T)1109 907 y Fj(H)p Fi(\(2)p
Fj(x)h Fh(\000)g Fj(1)p Fi(\))336 b(\(4\))544 969 y(=)42
b(2\(2)p Fj(x)701 952 y Ff(T)727 969 y Fj(Ay)10 b Fh(\000)g
Fj(1)864 952 y Ff(T)890 969 y Fj(Ay)q Fi(\))f Fh(\000)h
Fi(\(4)p Fj(x)1081 952 y Ff(T)1107 969 y Fj(Hx)g Fh(\000)f
Fi(2)p Fj(x)1266 952 y Ff(T)1292 969 y Fj(H2)h Fi(+)f
Fj(1)1428 952 y Ff(T)1454 969 y Fj(H1)p Fi(\))105 b(\(5\))544
1031 y(=)42 b Fj(x)643 1014 y Ff(T)669 1031 y Fi(\(4)p
Fj(Ay)10 b Fi(+)g Fj(H4)p Fi(\))g Fh(\000)f Fj(x)972
1014 y Ff(T)998 1031 y Fi(\(4)p Fj(H)p Fi(\))p Fj(x)h
Fh(\000)g Fj(2)1189 1014 y Ff(T)1215 1031 y Fj(Ay)g Fh(\000)f
Fj(1)1351 1014 y Ff(T)1377 1031 y Fj(H1)198 b Fi(\(6\))257
1110 y(Maximizing)11 b(\012\()p Fj(x)p Fi(\))k(o)o(v)o(er)f
Fj(x)g Fi(is)f(then)i(equiv)n(alen)o(t)e(to:)683 1189
y(max)649 1219 y Fe(x)p Fd(2f)p Fc(0)p Ff(;)p Fc(1)p
Fd(g)769 1210 y Fb(K)802 1189 y Fj(x)827 1172 y Ff(T)853
1189 y Fi(\(4)p Fj(Ay)d Fi(+)g Fj(H4)p Fi(\))g Fh(\000)f
Fj(x)1156 1172 y Ff(T)1182 1189 y Fi(\(4)p Fj(H)p Fi(\))p
Fj(x)339 b Fi(\(7\))257 1294 y(since)18 b(the)f(constan)o(t)g(terms)f
(in)g(\(6\))h(do)f(not)g(dep)q(end)i(on)e Fj(x)p Fi(.)26
b(W)m(e)16 b(can)h(further)g(simplify)257 1344 y(this:)704
1394 y(max)670 1423 y Fe(x)p Fd(2f)p Fc(0)p Ff(;)p Fc(1)p
Fd(g)790 1415 y Fb(K)823 1394 y Fj(x)848 1377 y Ff(T)874
1394 y Fi(\()p Fj(Ay)10 b Fi(+)g Fj(H1)p Fi(\))g Fh(\000)f
Fj(x)1156 1377 y Ff(T)1182 1394 y Fi(\()p Fj(H)p Fi(\))p
Fj(x)360 b Fi(\(8\))257 1485 y(This)14 b(is)g(then)g(equiv)n(alen)o(t)g
(to)f(the)i(follo)o(wing)c(minim)o(ization)692 1564 y(min)653
1594 y Fe(x)p Fd(2f)p Fc(0)p Ff(;)p Fc(1)p Fd(g)773 1585
y Fb(K)807 1564 y Fj(x)832 1547 y Ff(T)858 1564 y Fi(\()p
Fj(H)p Fi(\))p Fj(x)f Fi(+)g Fj(x)1029 1547 y Ff(T)1055
1564 y Fi(\()p Fh(\000)p Fj(Ay)h Fh(\000)e Fj(H1)p Fi(\))343
b(\(9\))257 1673 y(F)m(or)16 b Fj(x)f Fh(2)g(f)p Fi(0)p
Fg(;)7 b Fi(1)p Fh(g)520 1658 y Ff(K)551 1673 y Fi(,)16
b(w)o(e)g(note)g(that)g(for)g(an)o(y)f Fg(i)p Fi(-th)i(elemen)o(t)e(of)
h Fj(x)g Fi(is)g(suc)o(h)h(that)f Fg(x)1560 1679 y Ff(i)1588
1673 y Fi(=)g Fg(x)1660 1658 y Fc(2)1660 1684 y Ff(i)1678
1673 y Fi(.)257 1723 y(Therefore,)f(for)f(an)f(arbitrary)h
Fg(K)j Fi(v)o(ector)e Fg(\036)p Fi(,)e(w)o(e)h(see)h(that)773
1841 y Fj(x)798 1823 y Ff(T)824 1841 y Fg(\036)41 b Fi(=)979
1789 y Ff(K)964 1801 y Fa(X)967 1890 y Ff(i)p Fc(=1)1031
1841 y Fg(x)1055 1847 y Ff(i)1069 1841 y Fg(\036)1094
1847 y Ff(i)1616 1841 y Fi(\(10\))890 1986 y(=)979 1934
y Ff(K)964 1947 y Fa(X)967 2035 y Ff(i)p Fc(=1)1031 1986
y Fg(x)1055 1969 y Fc(2)1055 1996 y Ff(i)1074 1986 y
Fg(\036)1099 1992 y Ff(i)1616 1986 y Fi(\(11\))890 2093
y(=)h Fj(x)989 2076 y Ff(T)1016 2093 y Fi(\(diag)o Fg(\036)p
Fi(\))p Fj(x)442 b Fi(\(12\))257 2172 y(W)m(e)14 b(can)g(then)h
(rewrite)f(\(9\))g(as)540 2251 y(min)502 2280 y Fe(x)p
Fd(2f)p Fc(0)p Ff(;)p Fc(1)p Fd(g)622 2272 y Fb(K)655
2251 y Fj(x)680 2234 y Ff(T)706 2251 y Fi(\()p Fj(H)p
Fi(\))p Fj(x)c Fh(\000)g Fj(x)877 2234 y Ff(T)903 2251
y Fi(\(diag)o Fg(\036)p Fi(\))p Fj(x)g Fi(+)f Fj(x)1137
2234 y Ff(T)1164 2251 y Fi(\()p Fg(\036)g Fh(\000)g Fj(Ay)h
Fh(\000)g Fj(H1)p Fi(\))171 b(\(13\))257 2354 y(This)14
b(is)g(then)817 2404 y(min)778 2433 y Fe(x)p Fd(2f)p
Fc(0)p Ff(;)p Fc(1)p Fd(g)898 2425 y Fb(K)931 2404 y
Fj(x)956 2387 y Ff(T)983 2404 y Fj(Gx)9 b Fi(+)g Fj(z)1117
2387 y Ff(T)1144 2404 y Fj(x)447 b Fi(\(14\))257 2495
y(where)15 b Fj(G)f Fi(and)f Fj(z)i Fi(are)f(as)g(de\014ned)h(in)e(the)
i(problem)d(statemen)o(t.)963 2628 y(1)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF