3.23.kishore.ps

是multiuser detection 这本书的习题解答, 很有用的书.

2374 46 2 v 1119 2408 a Fi(2)p Fh(\031)q(x)1189 2396
y Fb(3)1616 2375 y Fi(\(12\))725 2485 y(=)846 2457 y(1)p
804 2476 105 2 v 804 2484 a Fg(p)p 839 2484 46 2 v 34
x Fi(2)p Fh(\031)q(x)913 2485 y Fi(\(1)9 b Fg(\000)1017
2457 y Fi(1)p 1006 2476 43 2 v 1006 2514 a Fh(x)1030
2502 y Fb(2)1053 2485 y Fi(\))p Fh(e)1088 2468 y Fe(\000)p
Fd(x)1133 2456 y Fc(2)1150 2468 y Fd(=)p Fb(2)1199 2485
y Fi(,)41 b Fh(x)11 b(>)h Fi(1)264 b(\(13\))963 2628
y(1)p eop
%%Page: 2 2
2 1 bop 320 262 a Fi(\(b\))14 b(Pro)q(of)g(of)f(\(3.38\))g(Start)h
(with)f(the)i(de\014nition)e(of)h(the)g Fh(Q)g Fi(function.)704
376 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))42 b(=)943 348 y(1)p
913 367 81 2 v 913 375 a Fg(p)p 948 375 46 2 v 34 x Fi(2)p
Fh(\031)1006 320 y Ff(Z)1047 330 y Fe(1)1029 414 y Fd(x)1089
376 y Fh(e)1108 359 y Fe(\000)p Fd(t)1147 347 y Fc(2)1163
359 y Fd(=)p Fb(2)1206 376 y Fh(dt)373 b Fi(\(14\))835
490 y Fg(\024)943 462 y Fi(1)p 913 481 81 2 v 913 489
a Fg(p)p 948 489 46 2 v 34 x Fi(2)p Fh(\031)1006 434
y Ff(Z)1047 444 y Fe(1)1029 528 y Fd(x)1089 490 y Fh(e)1108
473 y Fd(xt=)p Fb(2)1183 490 y Fh(dt)396 b Fi(\(15\))835
608 y(=)948 580 y(1)p 918 598 81 2 v 918 607 a Fg(p)p
953 607 46 2 v 34 x Fi(2)p Fh(\031)1009 580 y Fg(\000)p
Fi(2)p 1009 598 54 2 v 1024 636 a Fh(x)1067 608 y(e)1086
591 y Fe(\000)p Fd(xt=)p Fb(2)1179 548 y Ff(\014)1179
573 y(\014)1179 598 y(\014)1179 622 y(\014)1193 558 y
Fe(1)1193 649 y Fd(x)1616 608 y Fi(\(16\))257 738 y(W)m(e)14
b(then)g(see)i(that)d Fh(Q)p Fi(\()p Fh(x)p Fi(\))f Fg(\024)741
698 y Fe(p)p 768 698 17 2 v 24 x Fb(2)p 730 729 67 2
v 730 755 a Fd(x)749 733 y Fe(p)p 776 733 21 2 v 22 x
Fd(\031)801 738 y Fh(e)820 723 y Fe(\000)p Fd(x)865 711
y Fc(2)881 723 y Fd(=)p Fb(2)917 738 y Fi(.)18 b(T)m(aking)12
b(the)j(logarithm)c(of)i(b)q(oth)h(sides:)683 875 y(log)7
b Fh(Q)p Fi(\()p Fh(x)p Fi(\))k Fg(\024)h Fi(log)949
810 y Ff(r)p 990 810 36 2 v 997 847 a Fi(1)p 995 866
26 2 v 995 904 a Fh(\031)1035 875 y Fg(\000)d Fi(log)e
Fh(x)i Fg(\000)1216 847 y Fh(x)1240 832 y Fb(2)p 1216
866 43 2 v 1227 904 a Fi(2)1616 875 y(\(17\))257 983
y(Note)18 b(that)g(the)g(function)f(on)g(the)h(righ)o(t)f(is)h
(monotically)c(decreasing.)30 b(Therefore,)19 b(w)o(e)257
1046 y(can)14 b(mak)o(e)f(the)h(follo)o(wing)e(observ)n(ation,)h(for)g
Fh(x)f Fg(\025)1063 998 y Ff(q)p 1105 998 31 2 v 1112
1030 a Fb(8)p 1110 1037 21 2 v 1110 1060 a Fd(\031)1135
1046 y Fi(:)628 1185 y(log)7 b Fh(Q)p Fi(\()p Fh(x)p
Fi(\))k Fg(\024)h Fi(log)893 1120 y Ff(r)p 935 1120 36
2 v 942 1157 a Fi(2)p 940 1176 26 2 v 940 1214 a Fh(\031)979
1185 y Fg(\000)e Fi(log)1081 1120 y Ff(r)p 1123 1120
36 2 v 1130 1157 a Fi(8)p 1128 1176 26 2 v 1128 1214
a Fh(\031)1167 1185 y Fg(\000)1209 1120 y Ff(r)p 1250
1120 36 2 v 1257 1157 a Fi(8)p 1255 1176 26 2 v 1255
1214 a Fh(\031)1290 1157 y(x)p 1290 1176 24 2 v 1291
1214 a Fi(2)1616 1185 y(\(18\))257 1293 y(T)m(aking)j(the)h(in)o(v)o
(erse)h(logarithm)c(of)i(b)q(oth)h(sides)h(of)e(the)i(inequalit)o(y)d
(ab)q(o)o(v)o(e,)710 1415 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))g
Fg(\024)859 1387 y Fi(1)p 859 1405 21 2 v 859 1443 a(2)885
1415 y Fh(e)904 1398 y Fe(\000)930 1369 y Fg(p)p 965
1369 28 2 v 972 1387 a Fc(2)p 970 1392 18 2 v 970 1408
a Fa(\031)993 1398 y Fd(x)1028 1415 y Fi(,)41 b Fh(x)11
b Fg(\025)1160 1350 y Ff(r)p 1201 1350 36 2 v 1209 1387
a Fi(8)p 1206 1405 26 2 v 1206 1443 a Fh(\031)1616 1415
y Fi(\(19\))257 1541 y(W)m(e)g(kno)o(w)g Fh(Q)p Fi(\(0\))g(=)h(1)p
Fh(=)p Fi(2.)17 b(Let)11 b Fh(f)t Fi(\()p Fh(x)p Fi(\))i(=)878
1525 y Fb(1)p 878 1532 17 2 v 878 1555 a(2)899 1541 y
Fh(e)918 1525 y Fe(\000)944 1497 y Fg(p)p 979 1497 28
2 v 986 1514 a Fc(2)p 984 1519 18 2 v 984 1535 a Fa(\031)1007
1525 y Fd(x)1028 1541 y Fi(.)k(Note)12 b(again)e(that,)h
Fh(f)t Fi(\(0\))h(=)g(1)p Fh(=)p Fi(2.)17 b(Th)o(us,)576
1603 y Ff(Z)618 1613 y Fd(x)599 1697 y Fb(0)646 1660
y Fh(Q)679 1642 y Fe(0)690 1660 y Fi(\()p Fh(t)p Fi(\))7
b Fh(dt)12 b Fi(=)f Fh(Q)p Fi(\()p Fh(x)p Fi(\))e Fg(\000)h
Fh(Q)p Fi(\(0\))h(=)h Fh(Q)p Fi(\()p Fh(x)p Fi(\))d Fg(\000)1262
1632 y Fi(1)p 1262 1650 21 2 v 1262 1688 a(2)1616 1660
y(\(20\))609 1716 y Ff(Z)650 1727 y Fd(x)632 1811 y Fb(0)678
1773 y Fh(f)702 1756 y Fe(0)714 1773 y Fi(\()p Fh(t)p
Fi(\))e Fh(dt)12 b Fi(=)f Fh(f)t Fi(\()p Fh(x)p Fi(\))f
Fg(\000)g Fh(f)t Fi(\(0\))i(=)g Fh(f)t Fi(\()p Fh(x)p
Fi(\))e Fg(\000)1262 1745 y Fi(1)p 1262 1763 V 1262 1801
a(2)1616 1773 y(\(21\))257 1887 y(Therefore,)18 b(using)e(fundamen)o
(tal)f(theorem)h(of)f(calculus,)i(w)o(e)g(can)f(sa)o(y)h
Fh(Q)p Fi(\()p Fh(x)p Fi(\))e Fg(\024)h Fh(f)t Fi(\()p
Fh(x)p Fi(\))h(if)257 1937 y Fh(Q)290 1922 y Fe(0)302
1937 y Fi(\()p Fh(t)p Fi(\))12 b Fg(\024)f Fh(f)428 1922
y Fe(0)441 1937 y Fi(\()p Fh(t)p Fi(\))j(for)g(all)e
Fh(t)g Fg(2)f Fi([0)p Fh(;)c(x)p Fi(].)16 b(So)e(w)o(e)g(no)o(w)f
(determine,)h(the)g(v)n(alues)g(of)f Fh(t)h Fi(for)f(whic)o(h:)818
2029 y Fh(Q)851 2011 y Fe(0)862 2029 y Fi(\()p Fh(t)p
Fi(\))42 b Fg(\024)g Fh(f)1049 2011 y Fe(0)1061 2029
y Fi(\()p Fh(t)p Fi(\))508 b(\(22\))728 2082 y Fg(\000)p
Fi(1)p 714 2101 81 2 v 714 2109 a Fg(p)p 749 2109 46
2 v 34 x Fi(2)p Fh(\031)800 2110 y(e)819 2093 y Fe(\000)p
Fd(t)858 2081 y Fc(2)874 2093 y Fd(=)p Fb(2)951 2110
y Fg(\024)1043 2082 y(\000)p Fi(1)p 1030 2101 81 2 v
1030 2109 a Fg(p)p 1064 2109 46 2 v 1064 2143 a Fi(2)p
Fh(\031)1115 2110 y(e)1134 2093 y Fe(\000)1160 2065 y
Fg(p)p 1195 2065 28 2 v 1202 2082 a Fc(2)p 1200 2087
18 2 v 1200 2103 a Fa(\031)1223 2093 y Fd(t)1616 2110
y Fi(\(23\))787 2214 y Fh(e)806 2197 y Fe(\000)832 2169
y Fg(p)p 867 2169 28 2 v 874 2186 a Fc(2)p 872 2191 18
2 v 872 2207 a Fa(\031)895 2197 y Fd(t)951 2214 y Fg(\024)42
b Fh(e)1044 2197 y Fe(\000)p Fd(t)1083 2185 y Fc(2)1099
2197 y Fd(=)p Fb(2)1616 2214 y Fi(\(24\))724 2282 y Fh(t)739
2267 y Fb(2)p 724 2300 34 2 v 730 2338 a Fi(2)774 2310
y Fg(\024)818 2245 y Ff(r)p 859 2245 36 2 v 866 2282
a Fi(2)p 864 2300 26 2 v 864 2338 a Fh(\031)894 2310
y(t)707 b Fi(\(25\))894 2428 y Fh(t)42 b Fg(\024)1025
2362 y Ff(r)p 1066 2362 36 2 v 1073 2400 a Fi(8)p 1071
2418 26 2 v 1071 2456 a Fh(\031)1616 2428 y Fi(\(26\))963
2628 y(2)p eop
%%Page: 3 3
3 2 bop 257 271 a Fi(So,)20 b(for)e Fh(t)i Fg(2)f Fi([0)p
Fh(;)535 223 y Ff(q)p 576 223 31 2 v 582 255 a Fb(8)p
580 262 21 2 v 580 286 a Fd(\031)606 271 y Fi(],)g Fh(Q)682
256 y Fe(0)693 271 y Fi(\()p Fh(t)p Fi(\))h Fg(\024)g
Fh(f)836 256 y Fe(0)849 271 y Fi(\()p Fh(t)p Fi(\).)33
b(Therefore,)21 b(w)o(e)e(see)h(that)f Fh(Q)p Fi(\()p
Fh(x)p Fi(\))g Fg(\024)h Fh(f)t Fi(\()p Fh(x)p Fi(\))g(for)257
352 y(0)12 b Fg(\024)f Fh(x)h Fg(\024)413 304 y Ff(q)p
454 304 31 2 v 461 335 a Fb(8)p 459 342 21 2 v 459 366
a Fd(\031)484 352 y Fi(.)18 b(Com)o(bining)11 b(this)j(with)g(the)g
(result)h(in)f(the)g(range)g Fh(x)d Fg(\025)1400 304
y Ff(q)p 1442 304 31 2 v 1449 335 a Fb(8)p 1447 342 21
2 v 1447 366 a Fd(\031)1472 352 y Fi(,)j(w)o(e)g(get:)738
477 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))e Fg(\024)887 449 y
Fi(1)p 887 468 V 887 506 a(2)913 477 y Fh(e)932 460 y
Fe(\000)958 432 y Fg(p)p 993 432 28 2 v 1000 449 a Fc(2)p
998 454 18 2 v 998 470 a Fa(\031)1021 460 y Fd(x)1056
477 y Fi(,)41 b Fh(x)11 b Fg(\025)h Fi(0)407 b(\(27\))320
583 y(\(c\))18 b(Pro)q(of)f(of)g(\(3.37\))g(via)f(comparisons)h(to)g
(\(3.35\))g(and)g(\(3.38\).)28 b(First,)18 b(note)g(that)257
653 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))12 b Fg(\024)407 637
y Fb(1)p 407 644 17 2 v 407 667 a(2)428 653 y Fh(e)447
631 y Fe(\000)473 601 y Ff(p)p 515 601 37 2 v 526 620
a Fc(2)p 520 625 27 2 v 520 641 a Fa(pi)551 631 y Fd(x)586
653 y Fi(b)o(y)i(\(3.38\).)j(Observ)o(e)e(that:)599 740
y(1)p 599 758 21 2 v 599 796 a(2)625 768 y Fh(e)644 751
y Fe(\000)670 723 y Fg(p)p 705 723 28 2 v 712 740 a Fc(2)p
710 745 18 2 v 710 761 a Fa(\031)733 751 y Fd(x)795 768
y Fg(\024)874 740 y Fi(1)p 874 758 21 2 v 874 796 a(2)900
768 y Fh(e)919 751 y Fe(\000)p Fd(x)964 738 y Fc(2)980
751 y Fd(=)p Fb(2)1071 768 y Fi(whenev)o(er)378 b(\(28\))653
816 y Ff(r)p 695 816 36 2 v 702 854 a Fi(2)p 700 872
26 2 v 700 910 a Fh(\031)730 882 y(x)41 b Fg(\025)874
854 y Fh(x)898 838 y Fb(2)p 874 872 43 2 v 885 910 a
Fi(2)977 882 y(or)13 b(equiv)n(alen)o(tly)g(when)264
b(\(29\))689 999 y(0)11 b Fg(\024)46 b Fh(x)g Fg(\024)913
934 y Ff(r)p 954 934 36 2 v 961 971 a Fi(8)p 959 990
26 2 v 959 1028 a Fh(\031)1616 999 y Fi(\(30\))257 1109
y(Th)o(us,)14 b(from)e(\(\))i(w)o(e)g(see)i(that:)653
1234 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))11 b Fg(\024)802 1206
y Fi(1)p 802 1224 21 2 v 802 1262 a(2)828 1234 y Fh(e)847
1216 y Fe(\000)p Fd(x)892 1204 y Fc(2)908 1216 y Fd(=)p
Fb(2)999 1234 y Fi(for)i(0)f Fg(\024)f Fh(x)h Fg(\024)1218
1168 y Ff(r)p 1259 1168 36 2 v 1266 1206 a Fi(8)p 1264
1224 26 2 v 1264 1262 a Fh(\031)1616 1234 y Fi(\(31\))320
1353 y(No)o(w,)h(note)h(that)g(b)o(y)g(\(3.35\),)e Fh(Q)p
Fi(\()p Fh(x)p Fi(\))f Fg(\024)979 1337 y Fb(1)p 946
1344 84 2 v 946 1348 a Fe(p)p 973 1348 38 2 v 24 x Fb(2)p
Fd(\031)q(x)1034 1353 y Fh(e)1053 1338 y Fe(\000)p Fd(x)1098
1325 y Fc(2)1115 1338 y Fd(=)p Fb(2)1150 1353 y Fi(.)18
b(Again,)13 b(observ)o(e)i(that)616 1448 y(1)p 575 1467
105 2 v 575 1475 a Fg(p)p 609 1475 46 2 v 609 1509 a
Fi(2)p Fh(\031)q(x)684 1476 y(e)703 1459 y Fe(\000)p
Fd(x)748 1446 y Fc(2)764 1459 y Fd(=)p Fb(2)841 1476
y Fg(\024)920 1448 y Fi(1)p 920 1467 21 2 v 920 1505
a(2)946 1476 y Fh(e)965 1459 y Fe(\000)p Fd(x)1010 1446
y Fc(2)1026 1459 y Fd(=)p Fb(2)1117 1476 y Fi(whenev)o(er)332
b(\(32\))732 1558 y(1)p 691 1577 105 2 v 691 1585 a Fg(p)p
725 1585 46 2 v 725 1619 a Fi(2)p Fh(\031)q(x)841 1586
y Fg(\024)920 1558 y Fi(1)p 920 1577 21 2 v 920 1615
a(2)1001 1586 y(or)14 b(eqiuv)n(alen)o(tly)f(when)239
b(\(33\))776 1710 y Fh(x)41 b Fg(\025)950 1682 y Fi(2)p
920 1700 81 2 v 920 1709 a Fg(p)p 955 1709 46 2 v 34
x Fi(2)p Fh(\031)1017 1710 y Fi(=)1061 1645 y Ff(r)p
1102 1645 36 2 v 1110 1682 a Fi(2)p 1107 1700 26 2 v
1107 1738 a Fh(\031)1616 1710 y Fi(\(34\))257 1823 y(This)14
b(implies)e(that:)691 1904 y Fh(Q)p Fi(\()p Fh(x)p Fi(\))f
Fg(\024)840 1876 y Fi(1)p 840 1894 21 2 v 840 1932 a(2)866
1904 y Fh(e)885 1887 y Fe(\000)p Fd(x)930 1874 y Fc(2)946
1887 y Fd(=)p Fb(2)1037 1904 y Fi(for)i Fh(x)f Fg(\025)1180
1838 y Ff(r)p 1221 1838 36 2 v 1228 1876 a Fi(2)p 1226
1894 26 2 v 1226 1932 a Fh(\031)1616 1904 y Fi(\(35\))257
1997 y(Putting)i(the)h(results)g(in)e(\(\))h(and)g(\(\))g(together,)g
(w)o(e)h(see)g(that:)719 2108 y Fh(Q)p Fi(\()p Fh(x)p
Fi(\))c Fg(\024)868 2079 y Fi(1)p 868 2098 21 2 v 868
2136 a(2)894 2108 y Fh(e)913 2090 y Fe(\000)p Fd(x)958
2078 y Fc(2)974 2090 y Fd(=)p Fb(2)1065 2108 y Fi(for)i
Fh(x)f Fg(\025)f Fi(0)388 b(\(36\))963 2628 y(3)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF