6.5.pramod.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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%%Page: 1 1
1 0 bop -30 72 1305 2 v 1275 72 a Fj(Problem)11 b(6.5,)f(Autho)o(r:)15
b(Pramo)q(d)d(Visw)o(anath,)-30 184 y Fi(6.5\(a\):)-30
290 y(W)l(e)19 b(shall)h(sho)o(w)g(a)g(general)f(result)g(from)g(whic)o
(h)g(the)g(desired)g(statemen)o(t)f(will)g(follo)o(w.)31
b(Fix)19 b(a)h(linear)-30 350 y(transformation)13 b Fh(c)p
Fi(.)20 b(Let)13 b Fh(\013)469 357 y Fg(c)500 350 y Fi(b)q(e)g(the)g
(scaling)g(that)g(minim)o(ize)o(s)e(the)h(mean-square-error)h(of)g(the)
g(estimate)-30 410 y(using)k(the)f(linear)f(transformation)h
Fh(c)p Fi(:)63 513 y Fh(\013)94 520 y Fg(c)153 513 y
Fi(=)41 b(argmin)381 525 y Fg(\013)p Ff(2)p Fe(R)450
513 y Fd(E)486 465 y Fc(h)506 513 y Fi(\()p Fh(b)p Fb(\000)13
b Fh(<)h(\013c;)8 b(y)16 b(>)p Fi(\))820 489 y Fa(2)840
465 y Fc(i)153 605 y Fi(=)41 b(argmin)381 617 y Fg(\013)406
605 y Fd(E)442 557 y Fc(h)461 605 y Fi(\()p Fh(b)11 b
Fb(\000)g Fh(\013)j(<)g(c;)8 b(bs)13 b(>)h Fb(\000)p
Fh(\013)g(<)g(c;)8 b(m)13 b(>)p Fi(\))1103 581 y Fa(2)1123
557 y Fc(i)153 696 y Fi(=)41 b(argmin)381 708 y Fg(\013)414
648 y Fc(n)442 696 y Fh(\013)473 676 y Fa(2)501 648 y
Fc(\020)526 696 y Fd(E)562 648 y Fc(h)582 696 y Fh(b)603
676 y Fa(2)622 648 y Fc(i)650 696 y Fi(\()p Fh(<)14 b(c;)8
b(s)13 b(>)p Fi(\))857 672 y Fa(2)888 696 y Fi(+)e(\()p
Fh(<)j(c;)8 b(m)13 b(>)p Fi(\))1164 672 y Fa(2)1184 648
y Fc(\021)1220 696 y Fb(\000)d Fi(2)p Fh(\013)p Fd(E)1361
648 y Fc(h)1381 696 y Fh(b)1402 676 y Fa(2)1421 648 y
Fc(i)1455 696 y Fh(<)j(c;)8 b(s)14 b(>)g Fi(+)p Fd(E)1712
648 y Fc(h)1731 696 y Fh(b)1752 676 y Fa(2)1772 648 y
Fc(i)o(o)1858 696 y Fi(\(1\))153 811 y(=)416 777 y Fd(E)f
Fi([)p Fh(b)487 759 y Fa(2)506 777 y Fi(])h Fh(<)g(c;)8
b(s)13 b(>)p 237 799 646 2 v 237 850 a Fd(E)g Fi([)p
Fh(b)308 835 y Fa(2)327 850 y Fi(])8 b(\()p Fh(<)14 b(c;)8
b(s)13 b(>)p Fi(\))556 826 y Fa(2)587 850 y Fi(+)e(\()p
Fh(<)j(c;)8 b(m)13 b(>)p Fi(\))863 826 y Fa(2)1858 811
y Fi(\(2\))-30 944 y(Then,)j(the)g(minim)n(um)c(mean)j(square)h(error)h
(using)f(the)g(linear)g(transformation)g Fh(c)g Fi(is)16
1047 y(min)23 1074 y Fg(\013)p Ff(2)p Fe(R)105 1047 y
Fd(E)141 999 y Fc(h)161 1047 y Fi(\()p Fh(b)p Fb(\000)d
Fh(<)h(\013c;)8 b(y)15 b(>)p Fi(\))475 1022 y Fa(2)495
999 y Fc(i)556 1047 y Fi(=)42 b Fd(E)672 999 y Fc(h)691
1047 y Fi(\()p Fh(b)p Fb(\000)14 b Fh(<)f(\013)866 1054
y Fg(c)884 1047 y Fh(c;)8 b(y)15 b(>)p Fi(\))1023 1022
y Fa(2)1043 999 y Fc(i)556 1189 y Fi(=)42 b Fd(E)672
1141 y Fc(h)691 1189 y Fh(b)712 1169 y Fa(2)732 1141
y Fc(i)762 1189 y Fb(\000)915 1146 y Fd(E)13 b Fi([)o
Fh(b)985 1128 y Fa(2)1005 1146 y Fi(])7 b Fd(E)1062 1098
y Fc(h)1082 1146 y Fi(\()p Fh(<)14 b(c;)8 b(bs)13 b(>)p
Fi(\))1310 1122 y Fa(2)1330 1098 y Fc(i)p 817 1178 630
2 v 817 1233 a Fd(E)853 1184 y Fc(h)873 1233 y Fi(\()p
Fh(<)g(c;)8 b(bs)14 b(>)p Fi(\))1101 1208 y Fa(2)1132
1233 y Fi(+)d(\()p Fh(<)i(c;)8 b(m)14 b(>)p Fi(\))1408
1208 y Fa(2)1427 1184 y Fc(i)1523 1189 y Fi(from)i(\(1\))g(and)h(\(2\))
556 1367 y(=)739 1324 y Fd(E)c Fi([)o Fh(b)809 1306 y
Fa(2)829 1324 y Fi(])7 b Fd(E)886 1276 y Fc(h)906 1324
y Fi(\()p Fh(<)14 b(c;)8 b(m)13 b(>)p Fi(\))1133 1300
y Fa(2)1153 1276 y Fc(i)p 641 1355 V 641 1410 a Fd(E)677
1362 y Fc(h)696 1410 y Fi(\()p Fh(<)h(c;)8 b(bs)13 b(>)p
Fi(\))924 1386 y Fa(2)955 1410 y Fi(+)e(\()p Fh(<)j(c;)8
b(m)13 b(>)p Fi(\))1231 1386 y Fa(2)1251 1362 y Fc(i)-30
1517 y Fi(and)k(hence)652 1580 y Fd(E)c Fi([)o Fh(b)722
1562 y Fa(2)742 1580 y Fi(])p 419 1602 569 2 v 419 1657
a(min)501 1664 y Fg(\013)p Ff(2)p Fe(R)578 1657 y Fd(E)614
1609 y Fc(h)634 1657 y Fi(\()p Fh(b)p Fb(\000)g Fh(<)h(\013c;)8
b(y)15 b(>)p Fi(\))948 1633 y Fa(2)968 1609 y Fc(i)1006
1614 y Fi(=)f(1)e(+)1148 1571 y Fd(E)1184 1523 y Fc(h)1203
1571 y Fi(\()p Fh(<)i(c;)8 b(bs)13 b(>)p Fi(\))1431 1546
y Fa(2)1451 1523 y Fc(i)p 1148 1602 323 2 v 1149 1657
a Fd(E)1184 1609 y Fc(h)1204 1657 y Fi(\()p Fh(<)h(c;)8
b(m)13 b(>)p Fi(\))1431 1633 y Fa(2)1451 1609 y Fc(i)1858
1614 y Fi(\(3\))-30 1747 y(Since)i(\(3\))i(is)f(true)g(for)g(ev)o(ery)f
(linear)h(transformation)g Fh(c)p Fi(,)g(the)g(desired)f(result)h
(follo)o(ws)g(immedi)o(ately)l(.)-30 1853 y(6.5\(b\):)43
1960 y(This)h(follo)o(ws)f(imme)o(diatel)o(y)d(from)j(\(3\).)-30
2066 y(6.5\(c\):)43 2172 y(W)l(e)g(shall)f(\014rst)h(c)o(haracterize)f
(the)g(linear)g(transformations)h(that)h(satisfy)e(the)h(un)o
(biasedness)g(crite-)-30 2232 y(rion:)456 2335 y Fd(E)d
Fi([)p Fh(<)h(c;)8 b(y)15 b(>)p Fb(j)f Fh(b)p Fi(])40
b(=)i Fd(E)13 b Fi([)o Fh(<)h(c;)8 b(s)14 b(>)f(b)p Fi(+)h
Fh(<)g(c;)8 b(m)13 b(>)p Fb(j)h Fh(b)p Fi(])781 2408
y(=)42 b Fh(<)14 b(c;)8 b(s)13 b(>)h(b)d Fi(+)g Fd(E)i
Fi([)o Fh(<)h(c;)8 b(m)13 b(>)p Fb(j)h Fh(b)p Fi(])781
2480 y(=)42 b Fh(<)14 b(c;)8 b(s)13 b(>)h(b)-30 2583
y Fi(where)i(in)f(the)h(last)g(step)g(w)o(e)g(used)g(the)f(fact)h(that)
h Fh(m)e Fi(is)h(uncorrelated)g(with)f Fh(b)h Fi(and)h
Fh(m)e Fi(has)i(zero)f(mean.)-30 2643 y(Hence)714 2704
y Fh(c)e Fb(2)g(C)j(\()-8 b(\))p Fh(<)13 b(c;)8 b(s)14
b(>)p Fi(=)f(1)933 2828 y(1)p eop
%%Page: 2 2
2 1 bop -30 98 a Fi(Then)16 b Fb(8)p Fh(c)d Fb(2)h(C)s
Fi(,)i(the)g(mean)f(square)h(error)h(in)e(the)h(estimate)f(is)600
208 y Fd(E)636 160 y Fc(h)655 208 y Fi(\()p Fh(b)p Fb(\000)f
Fh(<)f(c;)8 b(y)16 b(>)p Fi(\))939 184 y Fa(2)958 160
y Fc(i)992 208 y Fi(=)e(\()o Fh(<)g(c;)8 b(m)14 b(>)p
Fi(\))1270 184 y Fa(2)-30 318 y Fi(and)j(hence)e Fb(8)p
Fh(c)e Fb(2)h(C)s Fi(,)495 441 y Fd(E)f Fi([)p Fh(b)566
423 y Fa(2)585 441 y Fi(])p 358 463 379 2 v 358 518 a
Fd(E)394 470 y Fc(h)414 518 y Fi(\()p Fh(b)p Fb(\000)g
Fh(<)h(c;)8 b(y)15 b(>)p Fi(\))697 494 y Fa(2)716 470
y Fc(i)755 475 y Fi(=)921 441 y Fd(E)e Fi([)o Fh(b)991
423 y Fa(2)1011 441 y Fi(])p 812 463 322 2 v 812 518
a Fd(E)848 470 y Fc(h)867 518 y Fi(\()p Fh(<)h(c;)8 b(m)13
b(>)p Fi(\))1094 494 y Fa(2)1114 470 y Fc(i)1152 475
y Fi(=)1209 432 y Fd(E)1245 384 y Fc(h)1265 432 y Fi(\()p
Fh(<)g(c;)8 b(bs)14 b(>)p Fi(\))1493 408 y Fa(2)1512
384 y Fc(i)p 1209 463 323 2 v 1210 518 a Fd(E)1246 470
y Fc(h)1265 518 y Fi(\()p Fh(<)g(c;)8 b(m)13 b(>)p Fi(\))1492
494 y Fa(2)1512 470 y Fc(i)-30 632 y Fi(Since)e(the)g(signal-to-in)o
(terference)g(ratio)g(is)h(indep)q(enden)o(t)f(of)h(the)f(scaling)h(of)
g(the)f(linear)g(transformation)-30 692 y Fh(c)j Fb(2)g(C)s
Fi(,)i(the)g(desired)f(results)h(no)o(w)h(follo)o(ws)f(immedi)o(ately)l
(.)933 2828 y(2)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF