www.pudn.com > SolutionsforMultiuserDetection.rar > 3.24.huaidai.ps

%!PS (but not EPSF; comments have been disabled)
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%DVIPSSource:  TeX output 1999.02.25:2314
/TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N
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ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false
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D E /Fm 6 122 df<020FB712FEF1FFE04A17F8869126000FFEC7EA1FFF6F4802037F07
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D101
D107 D109 D111
D114
D119 D
E end
TeXDict begin

1 0 bop 1146 569 a Fn(ELE533)52 b(Homew)l(ork)f(4)1586
951 y Fm(Huaiyu)40 b(Dai)1574 1271 y Fl(25)f(F)-10 b(eb)39
b(1999)-186 1714 y Fk(Problem)51 b(3.24.)83 b Fj(Sho)m(w\(3.64\):)70
b(If)45 b Fk(X)g Fj(is)h(a)f(complexed-v)-5 b(alued,)48
b(zero-mean)d(Gaussian)h Fi(L)p Fj(-v)m(ector)g(with)-186
1834 y(indep)s(enden)m(t)38 b(real)d(and)i(imaginary)d(parts)i(and)h
(if)e(the)i(eigen)m(v)-5 b(alues)37 b Fi(\025)2498 1849
y Fh(1)2537 1834 y Fi(;)17 b(:)g(:)g(:)f(;)h(\025)2813
1849 y Fg(L)2901 1834 y Fj(of)36 b(the)h(co)m(v)-5 b(ariance)36
b(matrix)-186 1955 y Fi(E)6 b Fj([)p Fk(XX)89 1918 y
Ff(\003)128 1955 y Fj(])32 b(are)h(distinct,)f(then)1154
2149 y Fi(E)6 b Fj([)p Fi(Q)p Fj(\()p Fe(k)p Fk(X)p Fe(k)p
Fj(\)])28 b(=)1791 2041 y Fg(L)1755 2066 y Fd(X)1758
2249 y Fg(i)p Fh(=1)1902 2082 y Fi(\013)1964 2097 y Fg(i)p
1902 2126 91 4 v 1922 2218 a Fj(2)2002 2149 y(\(1)22
b Fe(\000)2366 2082 y Fj(1)p 2220 2126 340 4 v 2220 2143
a Fd(q)p 2303 2143 257 4 v 94 x Fj(1)g(+)2498 2197 y
Fh(2)p 2482 2213 68 4 v 2482 2271 a Fg(\025)2523 2281
y Fc(i)2570 2149 y Fj(\))-186 2414 y(where)1475 2584
y Fi(\013)1537 2599 y Fg(j)1601 2584 y Fj(=)1803 2476
y Fg(L)1774 2501 y Fd(Y)1705 2685 y Fg(i)p Fh(=1)p Fg(;i)p
Ff(6)p Fh(=)p Fg(j)2080 2516 y Fi(\025)2137 2531 y Fg(j)p
1977 2561 301 4 v 1977 2652 a Fi(\025)2034 2667 y Fg(j)2092
2652 y Fe(\000)h Fi(\025)2249 2667 y Fg(i)-186 2917 y
Fk(Solution:)-186 3124 y Fj(Denote)33 b(the)g(co)m(v)-5
b(ariance)32 b(matrix)f(as)1430 3244 y Fk(\006)d Fj(=)f
Fi(E)6 b Fj([)p Fk(XX)1917 3203 y Ff(\003)1956 3244 y
Fj(])28 b(=)g Fk(CC)2277 3203 y Fg(T)3823 3244 y Fj(\(1\))-186
3412 y(Whic)m(h)d(w)m(e)i(kno)m(w)f(as)f Fb(Cholesky)j(De)-5
b(c)g(omp)g(osition)23 b Fj(with)i Fi(C)32 b Fj(as)25
b(an)g(in)m(v)m(ertible)g(lo)m(w)m(er)g(triangular)e(matrix.)39
b(De\014ne)-186 3532 y(a)32 b(new)i(v)m(ector)f(from)f
Fi(X)40 b Fj(through)33 b(linear)e(transformation)1642
3737 y Fk(Y)f Fj(=)d Fk(C)1941 3696 y Ff(\000)p Fh(1)2036
3737 y Fk(X)1702 b Fj(\(2\))-186 3942 y(The)24 b(new)g(v)m(ector)g
Fk(Y)i Fj(is)c(a)h(Gaussian)f(v)m(ector)i(with)f(indep)s(enden)m(t)h
(real)f(and)g(imaginary)d(parts)j(and)h(indep)s(enden)m(t)-186
4063 y(co)s(e\016cien)m(ts)34 b(with)e(zero-mean)g(and)h(unit)f(v)-5
b(ariance)32 b(due)h(to)1146 4268 y Fi(E)6 b Fj([)p Fk(YY)1424
4227 y Ff(\003)1462 4268 y Fj(])28 b(=)g Fk(C)1702 4227
y Ff(\000)p Fh(1)1796 4268 y Fi(E)6 b Fj([)p Fk(XX)2071
4227 y Ff(\003)2110 4268 y Fj(]\()p Fk(C)2256 4227 y
Ff(\000)p Fh(1)2350 4268 y Fj(\))2388 4227 y Fg(T)2471
4268 y Fj(=)27 b Fk(I)1207 b Fj(\(3\))-186 4473 y(Supp)s(ose)34
b(co)m(v)-5 b(ariance)32 b(matrix)f Fk(\006)i Fj(is)f(symmetric)f(and)i
(p)s(ositiv)m(e)f(de\014nite)h(so)g(it)e(can)i(b)s(e)g(diagonalized)d
(as)1630 4678 y Fk(\006)e Fj(=)f Fk(T\003T)2076 4637
y Fg(T)3823 4678 y Fj(\(4\))-186 4883 y(Then)34 b(w)m(e)f(ha)m(v)m(e)
818 5053 y Fi(U)38 b Fj(=)28 b Fe(k)p Fk(X)p Fe(k)1211
5012 y Fh(2)1277 5053 y Fj(=)f Fk(X)1465 5012 y Ff(\003)1505
5053 y Fk(X)g Fj(=)g Fk(Y)1808 5012 y Ff(\003)1847 5053
y Fk(\006Y)j Fj(=)e Fk(Z)2215 5012 y Ff(\003)2255 5053
y Fk(\003Z)g Fj(=)2569 4945 y Fg(L)2533 4970 y Fd(X)2536
5152 y Fg(i)p Fh(=1)2669 5053 y Fi(\025)2726 5068 y Fg(i)2754
5053 y Fe(j)p Fi(Z)2849 5068 y Fg(i)2877 5053 y Fe(j)2905
5012 y Fh(2)3823 5053 y Fj(\(5\))-186 5285 y(Where)1671
5406 y Fk(Z)g Fj(=)g Fk(T)1949 5365 y Fg(T)2004 5406
y Fk(Y)1734 b Fj(\(6\))1857 5655 y(1)p eop
2 1 bop -186 167 a Fj(is)42 b(an)h(Unitary)f(transformation)e(so)j(v)m
(ector)h Fk(Z)f Fj(is)f(also)g(a)g(Gaussian)g(v)m(ector)i(with)e(indep)
s(enden)m(t)i(real)e(and)-186 287 y(imaginary)34 b(parts)i(and)h(indep)
s(enden)m(t)h(co)s(e\016cien)m(ts)f(with)f(zero-mean)g(and)h(unit)e(v)
-5 b(ariance.)55 b Fi(\025)3365 302 y Fg(i)3393 287 y
Fi(;)17 b(i)34 b Fj(=)g(1)p Fi(;)17 b(:)g(:)g(:)f(;)h(L)-186
408 y Fj(are)33 b(eigen)m(v)-5 b(alues)32 b(of)g(the)h(co)m(v)-5
b(ariance)33 b(matrix.)-186 614 y(Supp)s(ose)40 b(the)g(probabilit)m(y)
e(densit)m(y)i(function)f(and)g(c)m(haracteristic)g(function)g(of)g
Fi(U)50 b Fj(are)40 b Fi(f)3259 629 y Fg(U)3318 614 y
Fj(\()p Fi(u)p Fj(\))p Fi(;)17 b(u)37 b(>)j Fj(0)f(and)-186
734 y(\010\()p Fi(t)p Fj(\))28 b(=)g Fi(E)6 b Fj([)p
Fi(e)277 698 y Fg(itu)372 734 y Fj(],)32 b(then)i(the)f
Fb(L)-5 b(aplacian)33 b(T)-7 b(r)i(ansform)31 b Fj(of)h(the)h
Fb(p)-5 b(df)32 b Fj(is)898 850 y Fd(Z)981 877 y Ff(1)944
1039 y Fh(0)1072 967 y Fi(f)1120 982 y Fg(U)1179 967
y Fj(\()p Fi(u)p Fj(\))p Fi(e)1356 926 y Ff(\000)p Fg(su)1488
967 y Fi(du)27 b Fj(=)g Fi(\036)p Fj(\()p Fi(is)p Fj(\))83
b(=)g Fi(E)6 b Fj([)p Fi(e)2330 926 y Ff(\000)p Fg(su)2463
967 y Fj(])2021 1185 y(=)83 b Fi(E)6 b Fj([)p Fi(e)2330
1144 y Ff(\000)p Fg(s)2430 1086 y Fd(P)2517 1106 y Fc(L)2517
1173 y(i)p Fa(=1)2633 1144 y Fg(\025)2674 1154 y Fc(i)2700
1144 y Ff(j)p Fg(z)2753 1154 y Fc(i)2779 1144 y Ff(j)2799
1120 y Fa(2)2837 1185 y Fj(])2021 1405 y(=)2213 1297
y Fg(L)2184 1322 y Fd(Y)2180 1504 y Fg(i)p Fh(=1)2311
1405 y Fi(E)g Fj([)p Fi(e)2461 1363 y Ff(\000)p Fg(s\025)2590
1373 y Fc(i)2616 1363 y Ff(j)p Fg(z)2669 1373 y Fc(i)2695
1363 y Ff(j)2715 1340 y Fa(2)2754 1405 y Fj(])1042 b(\(7\))-186
1670 y(Where)34 b(the)f(last)e(step)j(is)e(obtained)g(b)m(y)h(indep)s
(endence)i(of)d Fi(Z)2102 1685 y Fg(i)2130 1670 y Fj(.)-186
1876 y(Because)40 b Fi(Z)260 1891 y Fg(i)326 1876 y Fj(is)d(a)h
(complexed-v)-5 b(alued)37 b(Gaussian)h(v)-5 b(ariable)36
b(with)i(unit)f(v)-5 b(ariance,)40 b(the)e(real)f(and)h(imaginary)-186
1997 y(part)43 b(of)h Fi(Z)226 2012 y Fg(i)297 1997 y
Fj(is)f(indep)s(enden)m(t)i(with)f(distribution)d Fe(N)15
b Fj(\(0)p Fi(;)i Fj(1)p Fi(=)p Fj(2\).)75 b(So)44 b
Fi(V)68 b Fj(=)47 b Fe(j)p Fi(Z)2751 2012 y Fg(i)2778
1997 y Fe(j)d Fj(has)g Fb(R)-5 b(ayleigh)44 b(Distribution)-186
2117 y Fi(f)-138 2132 y Fg(V)-77 2117 y Fj(\()p Fi(v)t
Fj(\))27 b(=)h(2)p Fi(v)20 b(e)342 2081 y Ff(\000)p Fg(v)433
2058 y Fa(2)473 2117 y Fi(;)d(v)31 b(>)c Fj(0.)43 b(Then)1118
2350 y Fi(E)6 b Fj([)p Fi(e)1268 2309 y Ff(\000)p Fg(s\025)1397
2319 y Fc(i)1423 2309 y Fg(v)1459 2286 y Fa(2)1499 2350
y Fj(])83 b(=)1768 2233 y Fd(Z)1851 2260 y Ff(1)1814
2422 y Fh(0)1942 2350 y Fi(e)1987 2309 y Ff(\000)p Fg(s\025)2116
2319 y Fc(i)2142 2309 y Fg(v)2178 2286 y Fa(2)2234 2350
y Fj(2)p Fi(v)20 b(e)2395 2309 y Ff(\000)p Fg(v)2486
2286 y Fa(2)2543 2350 y Fi(dv)1609 2575 y Fj(=)1768 2457
y Fd(Z)1851 2484 y Ff(1)1814 2646 y Fh(0)1942 2575 y
Fj(2)p Fi(v)g(e)2103 2533 y Ff(\000)p Fh(\()p Fg(s\025)2259
2543 y Fc(i)2286 2533 y Fh(+1\))p Fg(v)2439 2510 y Fa(2)1609
2811 y Fj(=)1903 2743 y(1)p 1778 2788 300 4 v 1778 2879
a Fi(s\025)1881 2894 y Fg(i)1931 2879 y Fj(+)i(1)3823
2811 y(\(8\))-186 3138 y(Plug)32 b(Eq.)h(8)g(in)m(to)f(Eq.)h(7,)f(w)m
(e)i(get)1141 3408 y Fi(E)6 b Fj([)p Fi(e)1291 3367 y
Ff(\000)p Fg(su)1424 3408 y Fj(])28 b(=)1615 3300 y Fg(L)1586
3325 y Fd(Y)1582 3507 y Fg(i)p Fh(=1)1849 3340 y Fj(1)p
1723 3385 V 1723 3476 a Fi(s\025)1826 3491 y Fg(i)1876
3476 y Fj(+)22 b(1)2061 3408 y(=)2200 3300 y Fg(L)2164
3325 y Fd(X)2167 3507 y Fg(i)p Fh(=1)2416 3340 y Fi(\013)2478
3355 y Fg(i)p 2311 3385 V 2311 3476 a Fi(s\025)2414 3491
y Fg(i)2464 3476 y Fj(+)g(1)3823 3408 y(\(9\))-186 3690
y(where)34 b(the)f(last)f(step)h(is)f(obtained)g(through)h(partial)d
(expansion)j(and)g Fi(\013)2542 3705 y Fg(i)2598 3690
y Fj(=)2701 3623 y Fd(Q)2780 3650 y Fg(L)2780 3714 y(j)t
Fh(=1)p Fg(;j)t Ff(6)p Fh(=)p Fg(i)3130 3650 y(\013)3175
3660 y Fc(i)p 3064 3667 205 4 v 3064 3724 a Fg(\013)3109
3734 y Fc(i)3135 3724 y Ff(\000)p Fg(\013)3235 3734 y
Fc(j)3278 3690 y Fj(.)-186 3907 y(With)f(in)m(v)m(erse)i
Fb(L)-5 b(aplacian)34 b(T)-7 b(r)i(ansform)31 b Fj(of)h(Eq.)h(9)f(w)m
(e)i(get)f(the)g Fb(p)-5 b(df)32 b Fj(of)g Fi(U)1301
4177 y(f)1349 4192 y Fg(U)1408 4177 y Fj(\()p Fi(u)p
Fj(\))27 b(=)1707 4069 y Fg(L)1671 4094 y Fd(X)1673 4276
y Fg(i)p Fh(=1)1817 4110 y Fi(\013)1879 4125 y Fg(i)p
1817 4154 91 4 v 1820 4245 a Fi(\025)1877 4260 y Fg(i)1934
4177 y Fi(e)1979 4136 y Ff(\000)p Fg(u=\025)2151 4146
y Fc(i)2182 4177 y Fi(;)17 b(u)27 b(>)g Fj(0)1313 b(\(10\))-186
4527 y(So)806 4763 y Fi(P)14 b Fj(\()p Fe(k)p Fk(X)p
Fe(k)26 b(\024)i Fi(x)p Fj(\))g(=)g Fi(P)14 b Fj(\()p
Fi(U)38 b Fe(\024)28 b Fi(x)1841 4722 y Fh(2)1881 4763
y Fj(\))83 b(=)2160 4646 y Fd(Z)2244 4672 y Fg(x)2284
4648 y Fa(2)2207 4834 y Fh(0)2339 4763 y Fi(f)2387 4778
y Fg(U)2446 4763 y Fj(\()p Fi(u)p Fj(\))p Fi(du)2002
5029 y Fj(=)2160 4912 y Fd(Z)2244 4938 y Fg(x)2284 4915
y Fa(2)2207 5101 y Fh(0)2375 4921 y Fg(L)2339 4946 y
Fd(X)2341 5128 y Fg(i)p Fh(=1)2485 4962 y Fi(\013)2547
4977 y Fg(i)p 2485 5006 V 2488 5097 a Fi(\025)2545 5112
y Fg(i)2602 5029 y Fi(e)2647 4988 y Ff(\000)p Fg(u=\025)2819
4998 y Fc(i)2850 5029 y Fi(du)2002 5331 y Fj(=)2196 5223
y Fg(L)2160 5248 y Fd(X)2163 5431 y Fg(i)p Fh(=1)2297
5331 y Fi(\013)2359 5346 y Fg(i)2387 5331 y Fj(\(1)22
b Fe(\000)h Fi(e)2641 5290 y Ff(\000)p Fg(x)2736 5267
y Fa(2)2770 5290 y Fg(=\025)2846 5300 y Fc(i)2877 5331
y Fj(\))859 b(\(11\))1857 5655 y(2)p eop
3 2 bop -186 167 a Fj(Finally)761 426 y Fi(E)6 b Fj([)p
Fi(Q)p Fj(\()p Fe(k)p Fk(X)p Fe(k)p Fj(\)])83 b(=)1554
358 y(1)p 1483 402 191 4 v 1483 420 a Fe(p)p 1566 420
108 4 v 82 x Fj(2)p Fi(\031)1700 309 y Fd(Z)1783 335
y Ff(1)1746 497 y Fh(0)1874 426 y Fi(P)14 b Fj(\()p Fe(k)p
Fk(X)p Fe(k)27 b(\024)h Fi(x)p Fj(\))17 b Fi(e)2461 385
y Ff(\000)p Fg(x)2556 361 y Fa(2)2590 385 y Fg(=)p Fh(2)2665
426 y Fi(dx)1314 711 y Fj(=)1554 643 y(1)p 1483 688 191
4 v 1483 706 a Fe(p)p 1566 706 108 4 v 82 x Fj(2)p Fi(\031)1700
594 y Fd(Z)1783 620 y Ff(1)1746 782 y Fh(0)1910 603 y
Fg(L)1874 628 y Fd(X)1877 810 y Fg(i)p Fh(=1)2011 711
y Fi(\013)2073 726 y Fg(i)2101 711 y Fj(\(1)22 b Fe(\000)h
Fi(e)2355 670 y Ff(\000)p Fg(x)2450 646 y Fa(2)2484 670
y Fg(=\025)2560 680 y Fc(i)2591 711 y Fj(\))17 b Fi(e)2691
670 y Ff(\000)p Fg(x)2786 646 y Fa(2)2820 670 y Fg(=)p
Fh(2)2895 711 y Fi(dx)1314 1013 y Fj(=)1509 905 y Fg(L)1473
930 y Fd(X)1476 1113 y Fg(i)p Fh(=1)1609 1013 y Fi(\013)1671
1028 y Fg(i)1797 946 y Fj(1)p 1726 990 191 4 v 1726 1008
a Fe(p)p 1809 1008 108 4 v 82 x Fj(2)p Fi(\031)1943 896
y Fd(Z)2026 922 y Ff(1)1989 1085 y Fh(0)2101 1013 y Fj(\(1)22
b Fe(\000)h Fi(e)2355 972 y Ff(\000)p Fg(x)2450 949 y
Fa(2)2484 972 y Fg(=\025)2560 982 y Fc(i)2591 1013 y
Fj(\))17 b Fi(e)2691 972 y Ff(\000)p Fg(x)2786 949 y
Fa(2)2820 972 y Fg(=)p Fh(2)2895 1013 y Fi(dx)1314 1317
y Fj(=)1509 1209 y Fg(L)1473 1234 y Fd(X)1476 1416 y
Fg(i)p Fh(=1)1609 1317 y Fi(\013)1671 1332 y Fg(i)1716
1317 y Fj(\()1764 1250 y(1)p 1764 1294 49 4 v 1764 1386
a(2)1845 1317 y Fe(\000)1954 1250 y Fj(1)p 1954 1294
V 1954 1386 a(2)2013 1161 y Fd(s)p 2096 1161 274 4 v
2191 1250 a Fi(\025)2248 1265 y Fg(i)p 2106 1294 255
4 v 2106 1386 a Fi(\025)2163 1401 y Fg(i)2213 1386 y
Fj(+)22 b(2)2370 1317 y(\))1314 1620 y(=)1509 1512 y
Fg(L)1473 1537 y Fd(X)1476 1719 y Fg(i)p Fh(=1)1619 1552
y Fi(\013)1681 1567 y Fg(i)p 1619 1596 91 4 v 1640 1688
a Fj(2)1719 1620 y(\(1)g Fe(\000)2083 1552 y Fj(1)p 1938
1596 340 4 v 1938 1613 a Fd(q)p 2021 1613 257 4 v 94
x Fj(1)g(+)2216 1668 y Fh(2)p 2200 1684 68 4 v 2200 1741
a Fg(\025)2241 1751 y Fc(i)2287 1620 y Fj(\))1449 b(\(12\))1857
5655 y(3)p eop end
userdict /end-hook known{end-hook}if