5.16.huaidai.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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1 0 bop 1146 569 a Fl(ELE533)52 b(Homew)l(ork)f(7)1586
951 y Fk(Huaiyu)40 b(Dai)1596 1269 y Fj(7)f(Apr)f(1999)-186
1712 y Fi(PR)m(OBLEM)i(5.16.)54 b Fh(Let)36 b Fg(f)p
Fi(e)986 1727 y Ff(1)1031 1712 y Fe(;)17 b(:)g(:)g(:)f(;)h
Fi(e)1301 1727 y Ff(L)1354 1712 y Fg(g)35 b Fh(and)h
Fg(f)p Fe(\025)1739 1727 y Fd(1)1778 1712 y Fe(;)17 b(:)g(:)g(:)f(;)h
(\025)2054 1727 y Fc(L)2106 1712 y Fg(g)36 b Fh(denote)g(the)h
(orthonormal)c(eigen)m(v)m(ectors)38 b(and)-186 1832
y(corresp)s(onding)32 b(eigen)m(v)-5 b(alues)33 b(of)f(the)h(matrix)
1256 2106 y Fe(E)6 b Fh([)p Fi(rr)1453 2064 y Fc(T)1507
2106 y Fh(])28 b(=)g([)p Fe(\033)1752 2064 y Fd(2)1791
2106 y Fi(I)22 b Fh(+)1985 1998 y Fc(K)1958 2023 y Fb(X)1953
2207 y Fc(k)r Fd(=1)2099 2106 y Fe(A)2172 2064 y Fd(2)2172
2130 y Fc(k)2214 2106 y Fi(s)2258 2121 y Ff(k)2306 2106
y Fi(s)2350 2121 y Ff(k)2397 2064 y Fc(T)2452 2106 y
Fh(])p Fe(:)-186 2377 y Fh(Supp)s(ose)34 b(that)e(the)h(eigen)m(v)-5
b(alues)33 b(are)f(n)m(um)m(b)s(ered)i(in)d(decreasing)i(order.)44
b(De\014ne)33 b(the)g Fe(L)23 b Fg(\002)f Fe(K)40 b Fh(matrix)1520
2581 y Fi(P)27 b Fh(=)h([)p Fi(e)1806 2596 y Ff(1)1851
2581 y Fe(;)17 b(:)g(:)g(:)e(;)i Fi(e)2120 2596 y Ff(K)2188
2581 y Fh(])p Fe(;)-186 2786 y Fh(whic)m(h)33 b(satis\014es)g
Fi(P)529 2750 y Fc(T)584 2786 y Fi(P)27 b Fh(=)h Fi(I)p
Fh(.)43 b(According)32 b(to)h(problem)e(2.47,)1068 3047
y Fe(\025)1125 3062 y Fc(i)1181 3047 y Fh(=)1285 2901
y Fb(\()1393 2986 y Fe(\015)1449 2950 y Fd(2)1444 3010
y Fc(i)1511 2986 y Fh(+)22 b Fe(\033)1668 2950 y Fd(2)1707
2986 y Fe(;)83 b Fh(if)32 b Fe(i)c Fh(=)f(1)p Fe(;)17
b(:)g(:)g(:)f(;)h(K)r(;)1393 3106 y(\033)1452 3070 y
Fd(2)1817 3106 y Fh(if)32 b Fe(i)c Fh(=)f Fe(K)i Fh(+)22
b(1)p Fe(;)17 b(:)g(:)g(:)f(;)h(L;)-186 3308 y Fh(and)1189
3428 y Fi(SA)1336 3387 y Fd(2)1376 3428 y Fi(S)1438 3387
y Fc(T)1521 3428 y Fh(=)27 b Fi(P)p Fh(diag)p Fg(f)p
Fe(\015)1987 3387 y Fd(2)1982 3453 y(1)2026 3428 y Fe(;)17
b(:)g(:)g(:)f(;)h(\015)2301 3387 y Fd(2)2296 3453 y Fc(K)2364
3428 y Fg(g)p Fi(P)2491 3387 y Fc(T)2545 3428 y Fe(:)-186
3596 y Fh(Henceforth,)42 b(w)m(e)e(will)d(assume)i(that)g(the)g(set)h
(of)f(signature)f(v)m(ectors)j(is)e(linearly)e(indep)s(enden)m(t.)64
b(Therefore)-186 3716 y Fe(\015)-130 3680 y Fd(2)-135
3741 y Fc(k)-63 3716 y Fe(>)28 b Fh(0)k(for)g Fe(k)f
Fh(=)c(1)p Fe(;)17 b(:)g(:)g(:)f(;)h(K)7 b Fh(.)-186
3923 y(\(a\))32 b(V)-8 b(erify)32 b(that)1115 4043 y
Fi(S)1177 4002 y Fc(T)1232 4043 y Fi(P)p Fh(diag)1489
4000 y Fa(\000)p Fd(1)1583 4043 y Fg(f)p Fe(\015)1689
4002 y Fd(2)1684 4068 y(1)1729 4043 y Fe(;)17 b(:)g(:)g(:)e(;)i(\015)
2003 4002 y Fd(2)1998 4068 y Fc(K)2066 4043 y Fg(g)p
Fi(P)2193 4002 y Fc(T)2247 4043 y Fi(S)28 b Fh(=)g Fi(A)2526
4002 y Fa(\000)p Fd(2)2620 4043 y Fe(:)-186 4297 y Fh(\(b\)Conclude)33
b(that)g(the)g(decorrelator)f(for)g(the)h Fe(k)s Fh(th)f(user)i(admits)
d(the)i(expression)972 4501 y Fi(d)1034 4516 y Ff(k)1109
4501 y Fh(=)28 b Fi(P)p Fh(diag)1470 4458 y Fa(\000)p
Fd(1)1564 4501 y Fg(f)p Fe(\025)1671 4516 y Fd(1)1732
4501 y Fg(\000)23 b Fe(\033)1891 4460 y Fd(2)1930 4501
y Fe(;)17 b(:)g(:)g(:)f(;)h(\025)2206 4516 y Fc(K)2296
4501 y Fg(\000)23 b Fe(\033)2455 4460 y Fd(2)2494 4501
y Fg(g)p Fi(P)2621 4460 y Fc(T)2676 4501 y Fi(s)2720
4516 y Fc(k)2763 4501 y Fe(:)-186 4791 y Fi(Solution:)-186
4997 y Fh(\(a\)F)-8 b(rom)1189 5118 y Fi(SA)1336 5077
y Fd(2)1376 5118 y Fi(S)1438 5077 y Fc(T)1521 5118 y
Fh(=)27 b Fi(P)p Fh(diag)p Fg(f)p Fe(\015)1987 5077 y
Fd(2)1982 5142 y(1)2026 5118 y Fe(;)17 b(:)g(:)g(:)f(;)h(\015)2301
5077 y Fd(2)2296 5142 y Fc(K)2364 5118 y Fg(g)p Fi(P)2491
5077 y Fc(T)2545 5118 y Fe(:)1251 b Fh(\(1\))-186 5285
y(and)1690 5406 y Fi(P)1767 5365 y Fc(T)1822 5406 y Fi(P)27
b Fh(=)h Fi(I)1751 b Fh(\(2\))1857 5655 y(1)p eop
2 1 bop -186 167 a Fh(W)-8 b(e)33 b(ha)m(v)m(e)905 287
y(diag)p Fg(f)p Fe(\015)1191 246 y Fd(2)1186 312 y(1)1231
287 y Fe(;)17 b(:)g(:)g(:)e(;)i(\015)1505 246 y Fd(2)1500
312 y Fc(K)1568 287 y Fg(g)28 b Fh(=)f(\()p Fi(P)1864
246 y Fc(T)1919 287 y Fi(SA)p Fh(\)\()p Fi(AS)2289 246
y Fc(T)2343 287 y Fi(P)p Fh(\))h(=)f Fi(M)2695 246 y
Fc(T)2750 287 y Fi(M)967 b Fh(\(3\))-186 459 y(where)24
b Fe(M)38 b Fh(=)28 b Fi(AS)469 423 y Fc(T)524 459 y
Fi(P)22 b Fh(is)h(in)m(v)m(ertible)f(b)s(ecause)i(diag)p
Fg(f)p Fe(\015)1770 423 y Fd(2)1765 484 y(1)1809 459
y Fe(;)17 b(:)g(:)g(:)f(;)h(\015)2084 423 y Fd(2)2079
484 y Fc(K)2147 459 y Fg(g)22 b Fh(is)g(p)s(ositiv)m(e)h(de\014nite,)i
(and)e(so)g(is)f Fi(S)3465 423 y Fc(T)3520 459 y Fi(P)g
Fh(b)s(ecause)-186 580 y Fi(A)32 b Fh(has)h(full)e(rank.)44
b(Th)m(us)532 795 y(diag)712 752 y Fa(\000)p Fd(1)807
795 y Fg(f)p Fe(\015)913 754 y Fd(2)908 819 y(1)952 795
y Fe(;)17 b(:)g(:)g(:)f(;)h(\015)1227 754 y Fd(2)1222
819 y Fc(K)1289 795 y Fg(g)28 b Fh(=)g Fi(M)1577 754
y Fa(\000)p Fd(1)1671 795 y Fi(M)1777 754 y Fa(\000)p
Fc(T)1915 795 y Fh(=)g([\()p Fi(S)2146 754 y Fc(T)2201
795 y Fi(P)p Fh(\))2316 754 y Fa(\000)p Fd(1)2410 795
y Fi(A)2495 754 y Fa(\000)p Fd(1)2589 795 y Fh(])17 b([)p
Fi(A)2745 754 y Fa(\000)p Fd(1)2838 795 y Fh(\()p Fi(P)2953
754 y Fc(T)3008 795 y Fi(S)p Fh(\))3108 754 y Fa(\000)p
Fd(1)3203 795 y Fh(])593 b(\(4\))-186 1010 y(So)1115
1130 y Fi(S)1177 1089 y Fc(T)1232 1130 y Fi(P)p Fh(diag)1489
1087 y Fa(\000)p Fd(1)1583 1130 y Fg(f)p Fe(\015)1689
1089 y Fd(2)1684 1155 y(1)1729 1130 y Fe(;)17 b(:)g(:)g(:)e(;)i(\015)
2003 1089 y Fd(2)1998 1155 y Fc(K)2066 1130 y Fg(g)p
Fi(P)2193 1089 y Fc(T)2247 1130 y Fi(S)28 b Fh(=)g Fi(A)2526
1089 y Fa(\000)p Fd(2)2620 1130 y Fe(:)1176 b Fh(\(5\))-186
1388 y(\(b\)A)24 b(linear)d(m)m(ultiuser)i(detector)h(for)f(demo)s
(dulating)e(the)j Fe(k)s Fh(th)f(user's)i(data)e(bit)g(is)g(in)g(the)h
(form)e(of)h(a)g(correlator)-186 1508 y(follo)m(w)m(ed)32
b(b)m(y)h(a)f(hard)h(limiter)1556 1602 y(^)1560 1629
y Fe(b)1601 1644 y Fc(k)1671 1629 y Fh(=)28 b(sgn\()p
Fi(d)2016 1644 y Ff(k)2064 1586 y Fc(T)2119 1629 y Fi(r)p
Fh(\))1620 b(\(6\))-186 1801 y(The)33 b(decorrelator)f(is)g(designed)g
(to)g(completely)g(eliminate)d(the)k(Multiple)d(Acccess)35
b(In)m(terference)f(caused)g(b)m(y)-186 1921 y(other)f(users)h(and)e
(has)h(the)g(form)f(of)g(\(6\))g(with)1517 2211 y Fi(d)1579
2226 y Fc(k)1650 2211 y Fh(=)1781 2103 y Fc(K)1753 2128
y Fb(X)1756 2310 y Fc(i)p Fd(=1)1873 2211 y Fh([)p Fi(R)1984
2169 y Fa(\000)p Fd(1)2078 2211 y Fh(])2105 2226 y Fc(k)r(i)2172
2211 y Fi(s)2216 2226 y Fc(i)3823 2211 y Fh(\(7\))-186
2502 y(where)43 b Fi(R)h Fh(=)f Fi(S)414 2466 y Fc(T)469
2502 y Fi(S)g Fh(is)e(the)h(correlation)f(matrix)f(of)i(the)g(linearly)
e(indep)s(enden)m(t)j(signature)f(w)m(a)m(v)m(eforms.)73
b(It)-186 2623 y(can)32 b(b)s(e)h(sho)m(wn)g(that)f(the)h(decorrelator)
f(in)f(\(7\))h(is)f(the)i(unique)g(decorrelating)e(linear)f
(transformation)g(for)i Fi(s)3905 2638 y Fc(k)-186 2743
y Fh(against)g Fg(f)p Fi(s)245 2758 y Fc(j)281 2743 y
Fg(g)331 2758 y Fc(j)t Fa(6)p Fd(=)p Fc(k)493 2743 y
Fh(in)g(the)h(sense)i(that)1368 3023 y Fi(d)1430 2982
y Fc(T)1430 3048 y(k)1486 3023 y Fi(s)1530 3038 y Fc(j)1594
3023 y Fh(=)1697 2877 y Fb(\()1806 2962 y Fe(c)28 b Fg(6)p
Fh(=)f(0)83 b Fe(j)34 b Fh(=)27 b Fe(k)1806 3083 y Fh(0)256
b Fe(j)34 b Fg(6)p Fh(=)27 b Fe(k)3823 3023 y Fh(\(8\))-186
3301 y(Actually)32 b(from)f(\(7\))1342 3590 y Fi(d)1404
3549 y Fc(T)1404 3615 y(k)1459 3590 y Fi(s)1503 3605
y Fc(j)1623 3590 y Fh(=)1809 3482 y Fc(K)1782 3507 y
Fb(X)1784 3689 y Fc(i)p Fd(=1)1901 3590 y Fh([)p Fi(R)2012
3549 y Fa(\000)p Fd(1)2106 3590 y Fh(])2133 3605 y Fc(k)r(i)2200
3590 y Fi(s)2244 3549 y Fc(T)2244 3615 y(i)2300 3590
y Fi(s)2344 3605 y Fc(j)1623 3892 y Fh(=)1809 3784 y
Fc(K)1782 3809 y Fb(X)1784 3992 y Fc(i)p Fd(=1)1901 3892
y Fh([)p Fi(R)2012 3851 y Fa(\000)p Fd(1)2106 3892 y
Fh(])2133 3907 y Fc(k)r(i)2200 3892 y Fh([)p Fi(R)p Fh(])2338
3907 y Fc(ij)1623 4117 y Fh(=)83 b([)p Fi(R)1893 4076
y Fa(\000)p Fd(1)1987 4117 y Fi(R)p Fh(])2098 4132 y
Fc(k)r(j)2200 4117 y Fh(=)28 b Fi(I)2346 4132 y Fc(k)r(j)3823
4117 y Fh(\(9\))-186 4332 y(so)36 b(\(8\))f(is)g(obtained)h(straitforw)
m(ardly)-8 b(.)51 b(F)-8 b(urthermore,)36 b(since)g Fi(S)g
Fh(has)g(full)e(column)h(rank)h(the)g(solution)e(of)h(\(8\))-186
4452 y(is)d(unique.)-186 4658 y(W)-8 b(e)33 b(will)d(adopt)j(this)f
(criterion)f(to)h(c)m(hec)m(k)j(the)e(giv)m(en)g(expression.)45
b(It)32 b(can)h(b)s(e)g(sho)m(wn)h(that)913 4873 y Fi(d)975
4832 y Fc(T)975 4898 y(k)1031 4873 y Fi(S)28 b Fh(=)f
Fi(s)1268 4832 y Fc(T)1268 4898 y(k)1323 4873 y Fi(P)p
Fh(diag)1580 4831 y Fa(\000)p Fd(1)1674 4873 y Fg(f)p
Fe(\025)1781 4888 y Fd(1)1843 4873 y Fg(\000)22 b Fe(\033)2001
4832 y Fd(2)2041 4873 y Fe(;)17 b(:)g(:)g(:)f(;)h(\025)2317
4888 y Fc(K)2407 4873 y Fg(\000)22 b Fe(\033)2565 4832
y Fd(2)2605 4873 y Fg(g)p Fi(P)2732 4832 y Fc(T)2786
4873 y Fi(S)-186 5088 y Fh(is)32 b(the)h Fe(k)s Fh(th)g(ro)m(w)g(of)
1115 5209 y Fi(S)1177 5168 y Fc(T)1232 5209 y Fi(P)p
Fh(diag)1489 5166 y Fa(\000)p Fd(1)1583 5209 y Fg(f)p
Fe(\015)1689 5168 y Fd(2)1684 5233 y(1)1729 5209 y Fe(;)17
b(:)g(:)g(:)e(;)i(\015)2003 5168 y Fd(2)1998 5233 y Fc(K)2066
5209 y Fg(g)p Fi(P)2193 5168 y Fc(T)2247 5209 y Fi(S)28
b Fh(=)g Fi(A)2526 5168 y Fa(\000)p Fd(2)2620 5209 y
Fe(:)-186 5381 y Fh(th)m(us)34 b(\(8\))e(is)g(ful\014lled)e(and)j(the)g
(giv)m(en)g(expression)h(is)e(indeed)h(the)g(decorrelator)f(for)g(the)h
Fe(k)s Fh(th)f(user.)1857 5655 y(2)p eop end
userdict /end-hook known{end-hook}if