3.8.huaidai.ps

是multiuser detection 这本书的习题解答, 很有用的书.

b(Sho)m(w)33 b(that)g(the)g(optim)m(um)d(detector)k(for)e(equiprobable)
g Fh(b)c Fg(2)g(f\000)p Fi(1)p Fh(;)17 b Fi(1)p Fg(g)32
b Fi(is)1692 2862 y(^)1696 2888 y Fh(b)c Fi(=)g Fh(sg)t(n)p
Fi(\()p Fj(y)2123 2847 y Ff(T)2182 2888 y Fj(h)q Fi(\))p
Fh(;)58 3103 y Fi(where)34 b Fj(h)e Fi(is)h(the)g Fe(discr)-5
b(ete-time)34 b(matche)-5 b(d)34 b(\014lter)p Fi(:)1780
3319 y Fj(h)28 b Fi(=)g Fj(\006)2055 3278 y Fd(\000)p
Ff(1)2154 3319 y Fj(s)p Fh(:)-121 3575 y Fi(\(b\))49
b(Sho)m(w)36 b(that)e(among)g(all)f(transmitted)g(v)m(ectors)k
Fj(s)p Fi(,)e(suc)m(h)i(that)d Fg(k)p Fj(s)p Fg(k)e Fi(=)f(1,)k(the)h
(one)f(that)f(minimizes)e(the)58 3696 y(error)g(probabilit)m(y)f(of)h
(the)h(matc)m(hed)g(\014lter)f(detector)h(is)f(a)h(minim)m(um-eigen)m
(v)-5 b(alue)28 b(eigen)m(v)m(ector)34 b(of)e Fj(\006)p
Fi(.)-186 4005 y Fj(Solution:)-186 4211 y Fi(\(a\))-186
4417 y(W)-8 b(e)33 b(will)d(sho)m(w)k(that)e(the)h Fe(discr)-5
b(ete-time)34 b(matche)-5 b(d)34 b(\014lter)f Fi(structure)h(is)e
(optimal)e(in)h(the)i(sense)i(that)d(it)-41 4703 y Fg(\017)49
b Fi(maximizes)31 b(the)i(output)f(Signal-to-Noise)e(Ratio)g(among)i
(all)e(linear)h(\014lters)-41 4905 y Fg(\017)49 b Fi(minimizes)30
b(the)j(error)f(probabilit)m(y)f(among)g(all)g(detecters)j(with)e(the)h
(Gaussian)f(noise)h(assumption)-186 5190 y(b)s(ecause)h
Fj(\006)e Fi(is)g(p)s(ositiv)m(e)g(de\014nite,)h(with)g
Fe(Cholesky)h(De)-5 b(c)g(omp)g(osion)p Fi(,)30 b(it)h(can)i(b)s(e)g
(wriiten)f(as)1666 5406 y Fj(\006)c Fi(=)g Fj(CC)2041
5365 y Fc(T)1857 5655 y Fi(1)p eop
2 1 bop -186 167 a Fi(where)34 b Fj(C)e Fi(is)g(an)h
Fh(L)22 b Fg(\002)h Fh(L)33 b Fi(in)m(v)m(ertible)f(lo)m(w)m(er)g
(triangular)f(matrix.)-186 373 y(W)-8 b(e)33 b(de\014ne)h(new)f(observ)
-5 b(ables)1022 545 y Fj(^)-59 b(y)30 b Fi(=)d Fj(C)1292
504 y Fd(\000)p Fc(1)1386 545 y Fj(y)j Fi(=)d Fh(Ab)p
Fj(C)1773 504 y Fd(\000)p Fc(1)1868 545 y Fj(s)c Fi(+)f
Fj(C)2114 504 y Fd(\000)p Fc(1)2208 545 y Fj(n)28 b Fi(=)f
Fh(Ab)-6 b Fj(^)-50 b(s)23 b Fi(+)i Fj(^)-59 b(n)-186
717 y Fi(After)33 b(transformation)d(the)j(noise)g(b)s(ecomes)g(i.i.d)e
Fh(N)10 b Fi(\(0)p Fh(;)17 b Fi(1\))1273 889 y Fh(E)6
b Fi(\()s Fj(^)-59 b(n)20 b(^)-60 b(n)1529 848 y Fb(T)1585
889 y Fi(\))27 b(=)h Fj(C)1835 848 y Fd(\000)p Fc(1)1929
889 y Fj(\006)p Fi(\()p Fj(C)2129 848 y Fd(\000)p Fc(1)2223
889 y Fi(\))2261 848 y Fb(T)2344 889 y Fi(=)g Fj(I)-186
1146 y Fi(No)m(w)38 b(let's)f(consider)g(the)h(new)g(observ)-5
b(ables)40 b Fj(^)-59 b(y)r Fi(.)57 b(Assume)2045 1122
y Fj(^)2042 1146 y(h)37 b Fi(is)g(a)g(linear)e(\014lter)i(and)g(let)
3185 1122 y Fj(~)3182 1146 y(h)3244 1161 y Fb(k)3323
1146 y Fi(=)3437 1122 y Fj(^)3434 1146 y(h)3496 1161
y Fb(L)p Fd(\000)p Fb(k)3642 1146 y Fi(.)57 b(Then)-186
1267 y(the)33 b(output)g(SNR)f(of)g(the)h(\014lter)g(is)971
1440 y Fh(A)1044 1404 y Fc(2)1084 1440 y Fg(j)27 b Fh(<)22
b Fj(^)-50 b(s)p Fh(;)1334 1415 y Fj(~)1331 1440 y(h)27
b Fh(>)h Fg(j)1552 1404 y Fc(2)p 941 1484 680 4 v 941
1588 a Fh(E)6 b Fi(\()p Fg(j)28 b Fh(<)i Fj(^)-59 b(n)p
Fh(;)1325 1563 y Fj(~)1322 1588 y(h)28 b Fh(>)g Fg(j)1544
1560 y Fc(2)1583 1588 y Fi(\))1658 1507 y(=)1772 1440
y Fh(A)1845 1404 y Fc(2)1884 1440 y Fg(j)g Fh(<)21 b
Fj(^)-50 b(s)q Fh(;)2134 1415 y Fj(~)2132 1440 y(h)27
b Fh(>)h Fg(j)2353 1404 y Fc(2)p 1772 1484 620 4 v 1981
1588 a Fg(k)2034 1563 y Fj(~)2031 1588 y(h)p Fg(k)2143
1560 y Fc(2)2429 1507 y Fg(\024)h Fh(A)2608 1466 y Fc(2)2647
1507 y Fg(k)-6 b Fj(^)-50 b(s)p Fg(k)2791 1466 y Fc(2)-186
1742 y Fi(with)32 b(equalit)m(y)g(if)g(and)h(only)f(if)1734
1838 y Fj(~)1731 1863 y(h)c Fi(=)f Fh(\013)-5 b Fj(^)-50
b(s)-186 2016 y Fi(So)32 b(the)h(optimal)d(decision)i(statistic)1118
2188 y Fj(~)-58 b(y)84 b Fi(=)f Fg(h)r Fj(^)-58 b(y)r
Fh(;)1565 2163 y Fj(~)1562 2188 y(h)p Fg(i)27 b Fi(=)h
Fg(h)r Fj(^)-58 b(y)q Fh(;)11 b Fj(^)-50 b(s)p Fg(i)1259
2334 y Fi(=)83 b(\()p Fj(C)1537 2292 y Fd(\000)p Ff(1)1654
2334 y Fj(s)p Fi(\))1736 2292 y Fb(T)1807 2334 y Fj(C)1888
2292 y Fd(\000)p Ff(1)2005 2334 y Fj(y)29 b Fi(=)e(\()p
Fj(\006)2315 2292 y Fd(\000)p Ff(1)2432 2334 y Fj(s)p
Fi(\))2514 2292 y Fb(T)2585 2334 y Fj(y)1259 2479 y Fi(=)83
b Fj(h)1480 2438 y Fb(T)1536 2479 y Fj(y)29 b Fi(=)f
Fj(y)1789 2438 y Fb(T)1843 2479 y Fj(h)-186 2737 y Fi(Since)38
b Fj(^)-59 b(y)34 b Fi(=)e Fj(C)353 2700 y Fd(\000)p
Fc(1)447 2737 y Fj(y)37 b Fi(is)d(a)h(information)d(lossless)j(linear)e
(transformation,)h(w)m(e)i(can)f(see)i(for)d(all)f(linear)h(\014lters)h
Fj(h)-186 2857 y Fi(and)e(decision)f(statistic)f Fg(h)p
Fj(y)r Fh(;)17 b Fj(h)p Fg(i)p Fi(,)32 b Fe(discr)-5
b(ete-time)34 b(matche)-5 b(d)34 b(\014lter)f Fi(giv)m(es)g(the)g
(maximal)c(output)k(SNR.)-186 3063 y(F)-8 b(urther)26
b(more,)i(from)d(Problem)g(3.5.)41 b(w)m(e)28 b(kno)m(w)f(that)i
Fj(^)-58 b(y)29 b Fi(=)f Fj(C)2095 3027 y Fd(\000)p Fc(1)2189
3063 y Fj(y)g Fi(is)e(an)g(eqiv)-5 b(alen)m(t)26 b(observ)-5
b(ation)26 b(to)g Fj(y)j Fi(to)d(mak)m(e)-186 3183 y(optimal)31
b(Maxim)m(um-Lik)m(eliho)s(o)s(d)g(decisions.)48 b(for)34
b(equiprobable)g Fh(b)c Fg(2)h(f\000)p Fi(1)p Fh(;)17
b Fi(1)p Fg(g)p Fi(,)34 b(w)m(e)i(ha)m(v)m(e)f(t)m(w)m(o)g
(equiprobable)-186 3304 y(h)m(yp)s(otheses)g(under)e(whic)m(h)j
Fj(^)-59 b(y)35 b Fi(has)e(m)m(ultiv)-5 b(ariate)29 b(Gaussian)j
(distribution:)1344 3581 y Fh(H)1425 3596 y Fc(1)1492
3581 y Fi(:)e Fj(^)-58 b(y)30 b Fg(\030)e Fh(f)1800 3593
y Ff(^)1789 3609 y(Y)r Fd(j)p Fc(1)1939 3581 y Fi(=)f
Fg(N)15 b Fi(\()p Fh(A)-6 b Fj(^)-50 b(s)p Fh(;)17 b
Fj(I)p Fi(\))1251 3753 y Fh(H)1332 3768 y Fd(\000)p Fc(1)1454
3753 y Fi(:)30 b Fj(^)-59 b(y)30 b Fg(\030)e Fh(f)1761
3765 y Ff(^)1750 3781 y(Y)r Fd(j\000)p Fc(1)1955 3753
y Fi(=)f Fg(N)15 b Fi(\()p Fg(\000)p Fh(A)-6 b Fj(^)-50
b(s)q Fh(;)17 b Fj(I)p Fi(\))-186 3992 y(With)31 b(Maxim)m(um-Lik)m
(eliho)s(o)s(d)d(desion)j(rule,)h(the)g(follo)m(wing)d(decision)i
(regions)g(minimize)d(error)k(probabilit)m(y:)1116 4164
y(\012)1186 4179 y Fc(1)1254 4164 y Fi(=)27 b Fg(f)r
Fj(^)-58 b(y)30 b Fg(2)e Fh(R)1665 4123 y Fb(n)1739 4164
y Fi(:)g Fh(f)1853 4176 y Ff(^)1842 4192 y(Y)r Fd(j)p
Fc(1)1965 4164 y Fi(\()r Fj(^)-58 b(y)q Fi(\))28 b Fg(\025)g
Fh(f)2293 4176 y Ff(^)2282 4192 y(Y)r Fd(j\000)p Fc(1)2459
4164 y Fi(\()r Fj(^)-58 b(y)r Fi(\))p Fg(g)1062 4336
y Fi(\012)1132 4351 y Fd(\000)p Fc(1)1255 4336 y Fi(=)27
b Fg(f)r Fj(^)-58 b(y)29 b Fg(2)f Fh(R)1665 4295 y Fb(n)1740
4336 y Fi(:)g Fh(f)1854 4348 y Ff(^)1843 4364 y(Y)r Fd(j\000)p
Fc(1)2020 4336 y Fi(\()r Fj(^)-58 b(y)r Fi(\))27 b Fh(<)h(f)2347
4348 y Ff(^)2336 4364 y(Y)r Fd(j\000)p Fc(1)2513 4336
y Fi(\()r Fj(^)-58 b(y)r Fi(\))p Fg(g)-186 4604 y Fi(W)-8
b(e)33 b(see)h(that)e(the)h(minim)m(um)c(error)k(probabilit)m(y)d
(detector)k(decides)2405 4577 y(^)2408 4604 y Fh(b)28
b Fi(=)g(1)k(if)355 4747 y(1)p 226 4791 307 4 v 226 4883
a(\(2)p Fh(\031)t Fi(\))410 4854 y Fb(L=)p Fc(2)559 4814
y Fh(exp)p Fg(f\000)845 4747 y Fi(1)p 845 4791 49 4 v
845 4883 a(2)905 4814 y(\()r Fj(^)-58 b(y)23 b Fg(\000)g
Fh(A)-6 b Fj(^)-50 b(s)p Fi(\))1280 4773 y Fb(T)1335
4814 y Fi(\()r Fj(^)-58 b(y)24 b Fg(\000)f Fh(A)-6 b
Fj(^)-50 b(s)p Fi(\))p Fg(g)83 b(\025)2143 4747 y Fi(1)p
2014 4791 307 4 v 2014 4883 a(\(2)p Fh(\031)t Fi(\))2198
4854 y Fb(L=)p Fc(2)2347 4814 y Fh(exp)p Fg(f\000)2633
4747 y Fi(1)p 2633 4791 49 4 v 2633 4883 a(2)2693 4814
y(\()r Fj(^)-58 b(y)24 b Fi(+)e Fh(A)-6 b Fj(^)-50 b(s)p
Fi(\))3067 4773 y Fb(T)3122 4814 y Fi(\()r Fj(^)-58 b(y)24
b Fi(+)e Fh(A)-6 b Fj(^)-50 b(s)p Fi(\))p Fg(g)808 5072
y(\000)895 5005 y Fi(1)p 895 5049 V 895 5141 a(2)954
5072 y(\()r Fj(^)-58 b(y)24 b Fg(\000)f Fh(A)-6 b Fj(^)-50
b(s)p Fi(\))1330 5031 y Fb(T)1385 5072 y Fi(\()r Fj(^)-58
b(y)24 b Fg(\000)f Fh(A)-6 b Fj(^)-50 b(s)p Fi(\))83
b Fg(\025)g(\000)2091 5005 y Fi(1)p 2091 5049 V 2091
5141 a(2)2150 5072 y(\()r Fj(^)-58 b(y)24 b Fi(+)e Fh(A)-6
b Fj(^)-50 b(s)q Fi(\))2525 5031 y Fb(T)2580 5072 y Fi(\()r
Fj(^)-58 b(y)23 b Fi(+)f Fh(A)-6 b Fj(^)-50 b(s)q Fi(\))1603
5260 y Fj(^)-58 b(y)1662 5219 y Fb(T)1711 5260 y Fj(^)-50
b(s)83 b Fg(\025)g Fi(0)1436 5406 y Fj(y)1497 5365 y
Fb(T)1552 5406 y Fi(\006)1622 5365 y Fd(\000)p Fc(1)1717
5406 y Fj(s)g Fg(\025)g Fi(0)1857 5655 y(2)p eop
3 2 bop -186 167 a Fi(Th)m(us)38 b Fh(Y)55 b Fi(=)33
b Fj(y)347 131 y Fb(T)402 167 y Fi(\006)472 131 y Fd(\000)p
Fc(1)567 167 y Fj(s)j Fi(is)g(a)g(su\016cien)m(t)h(statistic)e(and)i
(this)f(detector)h(ac)m(hiev)m(es)g(the)g(lo)m(w)m(est)g(error)f
(probabilit)m(y)-186 287 y(among)31 b(all)g(detectors.)-186
493 y(\(b\))-186 700 y(With)h(the)h(matc)m(hed)g(\014lter)f(detector,)h
(the)g(error)g(probabilit)m(y)d(is)j(giv)m(en)f(b)m(y)1029
961 y Fh(P)41 b Fi(=)1247 893 y(1)p 1247 937 49 4 v 1247
1029 a(2)1322 843 y Fa(Z)1405 870 y Fd(1)1368 1032 y
Fc(0)1497 961 y Fh(f)1545 976 y Fb(Y)15 b Fd(j\000)p
Fc(1)1715 961 y Fi(\()p Fh(v)t Fi(\))p Fh(dv)25 b Fi(+)2073
893 y(1)p 2073 937 V 2073 1029 a(2)2149 843 y Fa(Z)2232
870 y Fc(0)2195 1032 y Fd(\0001)2341 961 y Fh(f)2389
976 y Fb(Y)16 b Fd(j)p Fc(1)2505 961 y Fi(\()p Fh(v)t
Fi(\))p Fh(dv)-186 1314 y Fi(where)34 b Fh(f)144 1330
y Fb(Y)15 b Fd(j\000)p Fc(1)314 1314 y Fi(\()p Fh(v)t
Fi(\))28 b(=)f Fg(N)15 b Fi(\()p Fg(\000)p Fh(A)p Fj(s)901
1278 y Fb(T)956 1314 y Fj(\006)1037 1278 y Fd(\000)p
Fc(1)1132 1314 y Fj(s)p Fh(;)i Fj(s)1264 1278 y Fb(T)1319
1314 y Fj(\006)1400 1278 y Fd(\000)p Fc(1)1494 1314 y
Fj(s)p Fi(\),)33 b(and)g Fh(f)1874 1330 y Fb(Y)15 b Fd(j)p
Fc(1)1989 1314 y Fi(\()p Fh(v)t Fi(\))28 b(=)f Fg(N)15
b Fi(\()p Fh(A)p Fj(s)2499 1278 y Fb(T)2554 1314 y Fj(\006)2635
1278 y Fd(\000)p Fc(1)2729 1314 y Fj(s)p Fh(;)i Fj(s)2861
1278 y Fb(T)2916 1314 y Fj(\006)2997 1278 y Fd(\000)p
Fc(1)3092 1314 y Fj(s)p Fi(\).)43 b(Then)1286 1603 y
Fh(P)e Fi(=)27 b Fh(Q)p Fi(\()1623 1536 y Fh(A)p Fj(s)1740
1500 y Fb(T)1796 1536 y Fj(\006)1877 1500 y Fd(\000)p
Fc(1)1971 1536 y Fj(s)p 1618 1580 402 4 v 1618 1598 a
Fg(p)p 1701 1598 319 4 v 87 x Fj(s)1745 1656 y Fb(T)1801
1685 y Fj(\006)1882 1656 y Fd(\000)p Fc(1)1976 1685 y
Fj(s)2030 1603 y Fi(\))h(=)f Fh(Q)p Fi(\()p Fh(Ad)p Fi(\))-186
1883 y(where)34 b Fh(d)147 1847 y Fc(2)214 1883 y Fi(=)27
b Fj(s)361 1847 y Fb(T)416 1883 y Fj(\006)497 1847 y
Fd(\000)p Fc(1)592 1883 y Fj(s)p Fi(.)-186 2089 y(The)h(minim)m(um)23
b(error)k(probabilit)m(y)d(is)i(ac)m(hiev)m(ed)i(b)m(y)g(maximizing)23
b Fh(d)2341 2053 y Fc(2)2380 2089 y Fi(.)42 b(According)26
b(to)g Fe(R)-5 b(ayleigh-R)n(itz)26 b Fi(inequal-)-186
2209 y(it)m(y)-8 b(,)1442 2369 y Fj(s)1486 2328 y Fb(T)1541
2369 y Fj(\006)1622 2328 y Fd(\000)p Fc(1)1716 2369 y
Fj(s)28 b Fg(\024)1991 2301 y Fi(1)p 1903 2346 224 4
v 1903 2437 a Fh(\025)1960 2469 y Fi(min)2137 2369 y
Fg(k)p Fj(s)p Fg(k)2281 2328 y Fc(2)-186 2610 y Fi(with)k(equalit)m(y)g
(if)g(and)g(only)g(if)g Fj(s)g Fi(is)g(a)h(eigen)m(v)m(ector)g(corresp)
s(onding)g(to)f Fh(\025)2555 2642 y Fi(min)2721 2610
y(,)h(i.e.)43 b Fj(\006s)28 b Fi(=)f Fh(\025)3262 2642
y Fi(min)3429 2610 y Fj(s)33 b Fi(or)f Fj(\006)3706 2574
y Fd(\000)p Fc(1)3800 2610 y Fj(s)c Fi(=)-90 2691 y Fc(1)p
-176 2707 208 4 v -176 2765 a Fb(\025)-135 2808 y Fi(min)42
2731 y Fj(s