5.12.kishore.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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00161A7F9119>I E end
%%EndProlog
%%BeginSetup
%%Feature: *Resolution 300
TeXDict begin @letter
%%EndSetup
%%Page: 1 1
bop 262 307 a Fh(Solutio)o(n)9 b(to)i(5.12)h(b)o(y)g(Shalinee)d(Kishore)i
(4/9/99)40 b Fg(\(a\))11 b(Let)g(us)g(de\014ne)g(the)g(matrix)262
357 y Fh(S)i Fg(as)816 430 y Fh(S)e Fg(=)911 378 y Ff(1)898
390 y Fe(X)899 479 y Fd(j)r Fc(=0)958 430 y Fg(\()p Fh(I)e
Fb(\000)h Fa(\013)p Fh(R)p Fg(\))1122 413 y Fd(j)1641 430 y
Fg(\(1\))262 546 y(No)o(w)j(observ)o(e)520 659 y Fh(S)c Fb(\000)g
Fg(\()p Fh(I)g Fb(\000)h Fa(\013)p Fh(R)p Fg(\))p Fh(S)41 b
Fg(=)916 607 y Ff(1)902 619 y Fe(X)904 708 y Fd(j)r Fc(=0)962
659 y Fg(\()p Fh(I)10 b Fb(\000)f Fa(\013)p Fh(R)p Fg(\))1126
641 y Fd(j)1153 659 y Fb(\000)1208 607 y Ff(1)1194 619 y Fe(X)1196
708 y Fd(j)r Fc(=1)1254 659 y Fg(\()p Fh(I)h Fb(\000)f Fa(\013)p
Fh(R)p Fg(\))1418 641 y Fd(j)1641 659 y Fg(\(2\))829 770 y(=)41
b(\()p Fh(I)10 b Fb(\000)f Fa(\013)p Fh(R)p Fg(\))1066 753
y Fc(0)1641 770 y Fg(\(3\))829 832 y(=)41 b Fh(I)721 b Fg(\(4\))262
922 y(W)m(e)13 b(can)h(then)h(write)765 1012 y Fh(I)42 b Fg(=)g
Fh(S)8 b Fb(\000)i Fg(\()p Fh(I)f Fb(\000)h Fa(\013)p Fh(R)p
Fg(\))p Fh(S)474 b Fg(\(5\))825 1074 y(=)42 b(\()p Fh(I)9 b
Fb(\000)h Fg(\()p Fh(I)f Fb(\000)h Fa(\013)p Fh(R)p Fg(\)\))p
Fh(S)450 b Fg(\(6\))825 1137 y(=)42 b Fa(\013)p Fh(RS)652 b
Fg(\(7\))262 1227 y(Th)o(us,)718 1308 y Fh(R)754 1290 y Ff(\000)p
Fc(1)810 1308 y Fg(=)12 b Fa(\013)p Fh(S)f Fg(=)g Fa(\013)1009
1256 y Ff(1)996 1268 y Fe(X)997 1357 y Fd(j)r Fc(=0)1056 1308
y Fg(\()p Fh(I)e Fb(\000)h Fa(\013)p Fh(R)p Fg(\))1220 1290
y Fd(j)1641 1308 y Fg(\(8\))262 1423 y(Note,)18 b(ho)o(w)o(ev)o(er,)h(that)f
(there)h(are)f(v)n(alues)f(of)h Fa(\013)f Fg(for)g(whic)o(h)h(the)h(series)g
(simply)c(cannot)262 1473 y(appro)o(ximate)8 b Fh(R)534 1458
y Ff(\000)p Fc(1)579 1473 y Fg(.)17 b(One)11 b(example)f(is)g(if)g(w)o(e)h
(select)h Fa(\013)g Fg(=)f(1)p Fa(=\025)p Fg(,)g(where)h Fa(\025)f
Fg(is)g(an)f(eigen)o(v)n(alue)262 1523 y(of)j Fh(R)p Fg(.)18
b(W)m(e)c(kno)o(w)f(that)h Fh(Rx)e Fg(=)g Fa(\025)p Fh(x)j
Fg(\(where)g Fh(x)f Fg(is)g(an)g(eigen)o(v)o(ector)h(of)e Fh(R)p
Fg(\).)18 b(Therefore,)d(w)o(e)262 1573 y(see)h(that)g Fh(I)p
Fa(\025)10 b Fb(\000)h Fh(R)j Fg(=)g(0)h(\(since)i Fh(x)d Fb(6)p
Fg(=)h(0\).)22 b(So)15 b(if)g(w)o(e)h(select)g Fa(\013)e Fg(=)h(1)p
Fa(=\025)p Fg(,)g(then)h Fh(I)10 b Fb(\000)h Fa(\013)p Fh(R)i
Fg(=)i(0)262 1623 y(and)d(th)o(us)h(\()p Fh(I)6 b Fb(\000)g
Fa(\013)p Fh(R)p Fg(\))588 1608 y Fd(j)606 1623 y Fg(,)12 b(for)g(all)f
Fa(j)r Fg(.)18 b(Th)o(us,)12 b(b)o(y)h(selecting)g Fa(\013)p
Fg('s)f(suc)o(h)h(that,)f(0)f Fa(<)h(\013)f(<)h Fg(1)p Fa(=\025)1619
1629 y Fc(max)1682 1623 y Fg(,)262 1673 y(w)o(e)i(a)o(v)o(oid)e(suc)o(h)j
(degenerate)g(case)g(and)f(the)h(series)g(ab)q(o)o(v)o(e)e(holds.)324
1722 y(\(b\))i(If)f(w)o(e)h(appro)o(ximate)e Fh(R)775 1707
y Ff(\000)p Fc(1)834 1722 y Fg(b)o(y)i(neglecting)g(all)e(but)i(the)g
(\014rst)h(t)o(w)o(o)e(terms)h(in)f(\(8\),)262 1772 y(the)g(error)h(matrix,)c
Fh(E)p Fg(,)j(is)767 1862 y Fh(E)42 b Fg(=)g Fh(S)8 b Fb(\000)i
Fh(R)1027 1845 y Ff(\000)p Fc(1)1641 1862 y Fg(\(9\))840 1958
y(=)42 b Fa(\013)961 1906 y Ff(1)947 1918 y Fe(X)948 2007 y
Fd(j)r Fc(=2)1007 1958 y Fg(\()p Fh(I)10 b Fb(\000)f Fa(\013)p
Fh(R)p Fg(\))1171 1941 y Fd(j)1620 1958 y Fg(\(10\))262 2095
y(where)15 b Fh(R)418 2080 y Ff(\000)p Fc(1)473 2095 y Fg(=)d
Fa(\013)p Fh(I)d Fg(+)h Fa(\013)p Fg(\()p Fh(I)f Fb(\000)h
Fa(\013)p Fh(R)p Fg(\).)17 b(W)m(e)d(can)g(rewrite)h(this)f(as)g(the)g(follo)
o(wing:)596 2185 y Fh(R)632 2168 y Ff(\000)p Fc(1)718 2185
y Fg(=)42 b Fh(S)9 b Fb(\000)g Fh(E)720 b Fg(\(11\))718 2280
y(=)42 b Fa(\013)839 2228 y Ff(1)826 2241 y Fe(X)827 2329 y
Fd(j)r Fc(=0)886 2280 y Fg(\()p Fh(I)9 b Fb(\000)h Fa(\013)p
Fh(R)p Fg(\))1050 2263 y Fd(j)1076 2280 y Fb(\000)1131 2228
y Ff(1)1118 2241 y Fe(X)1119 2329 y Fd(j)r Fc(=2)1178 2280
y Fg(\()p Fh(I)f Fb(\000)h Fa(\013)p Fh(R)p Fg(\))1342 2263
y Fd(j)1620 2280 y Fg(\(12\))718 2387 y(=)42 b(\()p Fa(\013)p
Fg(\()p Fh(I)9 b Fb(\000)h Fa(\013)p Fh(R)p Fg(\))f(+)g Fa(\013)p
Fh(I)p Fg(\))510 b(\(13\))718 2449 y(=)42 b(2)p Fa(\013)p Fh(I)9
b Fb(\000)g Fa(\013)935 2432 y Fc(2)954 2449 y Fh(R)630 b Fg(\(14\))967
2574 y(1)p eop
%%Page: 2 2
bop 262 307 a Fg(Minimizing)8 b(the)j(trace)h(of)e(the)i(error)g(matrix)d(is)
i(equiv)n(alen)o(t)f(to)h(maxim)o(izing)d(the)j(trace)h(of)262
357 y(the)g(appro)o(ximate)e Fh(R)605 342 y Ff(\000)p Fc(1)661
357 y Fg(matrix.)15 b(The)d(diagonal)e(elemen)o(ts)h(of)h(2)p
Fa(\013)p Fh(I)f Fg(are)h(simply)d(2)p Fa(\013)p Fg(.)17 b(The)262
407 y(diagonal)11 b(elemen)o(ts)i(of)g Fa(\013)670 392 y Fc(2)688
407 y Fh(R)g Fg(are)h Fa(\013)833 392 y Fc(2)864 407 y Fg(\(since)g
Fa(\032)1002 413 y Fd(ii)1040 407 y Fg(=)e(1\).)17 b(Th)o(us)d(the)g
(diagonal)d(elemen)o(ts)i(of)262 457 y(the)d(error)h(matrix,)e
Fh(R)607 442 y Ff(\000)p Fc(1)651 457 y Fg(,)i(are)f(then)h(2)p
Fa(\013)r Fb(\000)r Fa(\013)941 442 y Fc(2)959 457 y Fg(.)17
b(The)11 b(trace)g(therefore)g(is)f(simply)f Fa(n)p Fg(\(2)p
Fa(\013)r Fb(\000)r Fa(\013)1660 442 y Fc(2)1678 457 y Fg(\))262
506 y(where)17 b Fa(n)g Fg(is)g(the)g(dimensionalit)o(y)d(of)i(the)i(matrix)d
Fh(R)p Fg(.)26 b(T)m(o)16 b(maxim)o(ize)f(trace\()p Fh(R)1556
491 y Ff(\000)p Fc(1)1601 506 y Fg(\),)i(w)o(e)262 556 y(\014nd)d(the)g
Fa(\013)g Fg(that)g(solv)o(es)789 631 y Fa(d)p 769 649 56 2
v 776 687 a(d\013)829 659 y Fg(\(2)p Fa(n\013)9 b Fb(\000)h
Fa(n\013)1021 642 y Fc(2)1039 659 y Fg(\))41 b(=)h(0)429 b(\(15\))886
735 y(2)p Fa(n)9 b Fb(\000)h Fg(2)p Fa(n\013)40 b Fg(=)i(0)429
b(\(16\))1028 797 y Fa(\013)41 b Fg(=)h(1)429 b(\(17\))262
889 y(Note)13 b(that)f(since)i(2)p Fa(n\013)7 b Fb(\000)g Fa(n\013)721
874 y Fc(2)751 889 y Fg(is)12 b(a)g(conca)o(v)o(e)h(function)g(\(a)f(conca)o
(v)o(e)h(parab)q(ola\),)f(the)h(solu-)262 938 y(tion)g Fa(\013)e
Fg(=)h(1)i(do)q(es)g(in)g(fact)g(maxim)o(ize)d(the)k(trace)g(of)e
Fh(R)1132 923 y Ff(\000)p Fc(1)1176 938 y Fg(.)18 b(This)c(in)g(turn)g(minim)
o(izes)e(the)262 988 y(trace)k(of)e Fh(E)p Fg(.)23 b(\(c\))15
b(W)m(e)g(can)h(then)f(use)h(the)g(appro)o(ximate)d(decorrelator,)k(whic)o(h)
e(appro)o(xi-)262 1038 y(mates)10 b Fh(R)415 1023 y Ff(\000)p
Fc(1)471 1038 y Fg(=)i(2)p Fh(I)5 b Fb(\000)g Fh(R)p Fg(,)11
b(for)g(not)g(only)g(those)h(correlation)g(matricies)e(whic)o(h)i(are)g
(strongly)262 1088 y(diagonal)e(but)i(for)g(those)g(matricies)f(whose)i
(maxim)n(um)8 b(eigen)o(v)n(alue)j(is)h(less)h(than)f(1.)17
b(This)262 1138 y(w)o(a)o(y)m(,)12 b(w)o(e)i(select)h Fa(\013)c
Fg(=)h(1)f Fa(<)h Fg(1)p Fa(=\025)752 1144 y Fc(max)829 1138
y Fg(and)i(appro)o(ximate)e(the)i Fh(R)1257 1123 y Ff(\000)p
Fc(1)1316 1138 y Fg(with)f(2)p Fh(I)c Fb(\000)h Fh(R)p Fg(.)967
2574 y(2)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF