6.15.rickard.ps

是multiuser detection 这本书的习题解答, 很有用的书.

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TeXDict begin

1 0 bop 1373 126 a Fn(Multiuser)45 b(Detection)1656 358
y Fm(Scott)33 b(Ric)m(k)-5 b(ard)1645 547 y(April)31
b(16,)h(1999)515 895 y Fl(Solution)45 b(to)h(Problem)f(6.15)515
1060 y Fk(The)33 b(Ca)n(yley-Hamilton)f(theorem)g(states)i(that)f(an)n
(y)f(matrix)g(satis\014es)i(its)f(o)n(wn)g(c)n(harateristic)515
1151 y(function.)1312 1313 y(\()p Fj(A)16 b Fi(\000)h
Fj(\025)1538 1321 y Fh(1)1573 1313 y Fj(I)6 b Fk(\)\()p
Fj(A)15 b Fi(\000)i Fj(\025)1868 1321 y Fh(2)1903 1313
y Fj(I)6 b Fk(\))12 b Fi(\001)h(\001)g(\001)g Fk(\()p
Fj(A)k Fi(\000)f Fj(\025)2313 1321 y Fg(K)2372 1313 y
Fj(I)6 b Fk(\))20 b(=)h(0)515 1475 y(Equiv)l(alen)n(tly)-6
b(,)24 b(w)n(e)j(can)e(write,)1065 1637 y Fj(A)1123 1601
y Fg(K)1198 1637 y Fk(+)17 b Fj(\015)1319 1656 y Fh(1)1353
1637 y Fj(A)1411 1601 y Fg(K)s Ff(\000)p Fh(1)1565 1637
y Fk(+)g Fj(\015)1686 1656 y Fh(2)1720 1637 y Fj(A)1778
1601 y Fg(K)s Ff(\000)p Fh(2)1932 1637 y Fk(+)g Fi(\001)c(\001)g(\001)
18 b Fk(+)e Fj(\015)2236 1656 y Fg(K)s Ff(\000)p Fh(1)2373
1637 y Fj(A)h Fk(+)g Fj(\015)2569 1656 y Fg(K)2628 1637
y Fj(I)26 b Fk(=)21 b(0)p Fj(;)515 1799 y Fk(where)26
b Fj(\015)781 1818 y Fg(i)833 1799 y Fk(is)g(the)f(appropriately)h
(signed)g(sum)f(of)h(all)h(p)r(ossible)g(groups)f(of)g
Fj(i)g Fk(eigen)n(v)l(alues,)1463 1975 y Fj(\015)1507
1994 y Fg(i)1555 1975 y Fk(=)21 b(\()p Fi(\000)p Fk(1\))1794
1939 y Fg(i)1903 1904 y Fe(X)1833 2063 y Fg(l)1853 2073
y Fd(1)1885 2063 y Fg(<)p Ff(\001\001\001)p Fg(<l)2058
2074 y Fc(i)2097 1975 y Fj(\025)2142 1984 y Fg(l)2162
1994 y Fd(1)2211 1975 y Fi(\001)14 b(\001)f(\001)g Fj(\025)2359
1984 y Fg(l)2379 1995 y Fc(i)2409 1975 y Fj(:)515 2225
y Fk(F)-6 b(or)24 b(non-singular)g Fj(A)p Fk(,)g(m)n(ultiplying)f(the)h
(ab)r(o)n(v)n(e)g(equation)g(b)n(y)f Fj(A)2422 2193 y
Ff(\000)p Fh(1)2528 2225 y Fk(and)g(solving)i(yields)f
Fj(A)3204 2193 y Ff(\000)p Fh(1)3310 2225 y Fk(as)515
2316 y(a)i(p)r(olynomial)g(in)f Fj(A)g Fk(of)i(degree)f
Fj(K)c Fi(\000)17 b Fk(1.)935 2506 y Fj(A)993 2470 y
Ff(\000)p Fh(1)1159 2506 y Fk(=)1313 2457 y Fi(\000)p
Fk(1)p 1311 2489 103 4 v 1311 2556 a Fj(\015)1355 2575
y Fg(K)1424 2506 y Fj(A)1482 2470 y Fg(K)s Ff(\000)p
Fh(1)1636 2506 y Fi(\000)1734 2457 y Fj(\015)1778 2476
y Fh(1)p 1722 2489 V 1722 2556 a Fj(\015)1766 2575 y
Fg(K)1835 2506 y Fj(A)1893 2470 y Fg(K)s Ff(\000)p Fh(2)2047
2506 y Fi(\000)2145 2457 y Fj(\015)2189 2476 y Fh(2)p
2133 2489 V 2133 2556 a Fj(\015)2177 2575 y Fg(K)2246
2506 y Fj(A)2304 2470 y Fg(K)s Ff(\000)p Fh(3)2458 2506
y Fi(\000)g(\001)c(\001)g(\001)k(\000)2728 2450 y Fj(\015)2772
2469 y Fg(K)s Ff(\000)p Fh(1)p 2728 2489 182 4 v 2767
2556 a Fj(\015)2811 2575 y Fg(K)2919 2506 y Fj(I)1159
2744 y Fk(=)1301 2651 y Fg(K)s Ff(\000)p Fh(1)1312 2673
y Fe(X)1318 2831 y Fg(i)p Fh(=0)1447 2744 y Fj(\014)1490
2752 y Fg(i)1517 2744 y Fj(A)1575 2708 y Fg(i)1601 2744
y Fj(;)515 2974 y Fk(where)26 b(\(de\014ning)f Fj(\015)1101
2993 y Fh(0)1157 2974 y Fk(=)c(1\),)26 b Fj(\014)1396
2982 y Fg(i)1444 2974 y Fk(=)21 b Fi(\000)p Fj(\015)1629
2993 y Fg(K)s Ff(\000)p Fh(1)p Ff(\000)p Fg(i)1836 2974
y Fj(=\015)1918 2993 y Fg(K)1977 2974 y Fk(.)639 3065
y(No)n(w,)27 b(w)n(e)f(solv)n(e)g(for)g([)p Fj(R)18 b
Fk(+)f Fj(\033)1475 3034 y Fh(2)1509 3065 y Fj(I)6 b
Fk(])1570 3034 y Ff(\000)p Fh(1)1653 3065 y Fk(,)1157
3299 y([)p Fj(R)18 b Fk(+)f Fj(\033)1378 3263 y Fh(2)1412
3299 y Fj(I)6 b Fk(])1473 3263 y Ff(\000)p Fh(1)1638
3299 y Fk(=)1781 3205 y Fg(K)s Ff(\000)p Fh(1)1792 3228
y Fe(X)1798 3386 y Fg(i)p Fh(=0)1927 3299 y Fj(\014)1970
3307 y Fg(i)1997 3299 y Fk([)p Fj(R)18 b Fk(+)e Fj(\033)2217
3263 y Fh(2)2252 3299 y Fj(I)6 b Fk(])2313 3263 y Fg(i)1638
3563 y Fk(=)1781 3469 y Fg(K)s Ff(\000)p Fh(1)1792 3492
y Fe(X)1798 3650 y Fg(i)p Fh(=0)1927 3563 y Fj(\014)1970
3571 y Fg(i)2054 3470 y(i)2009 3492 y Fe(X)2012 3650
y Fg(j)s Fh(=0)2133 3435 y Fe( )2199 3515 y Fj(i)2194
3614 y(j)2230 3435 y Fe(!)2290 3563 y Fk(\()p Fj(\033)2367
3527 y Fh(2)2401 3563 y Fk(\))2431 3527 y Fg(i)p Ff(\000)p
Fg(j)2534 3563 y Fj(R)2593 3527 y Fg(j)1638 3837 y Fk(=)1781
3743 y Fg(K)s Ff(\000)p Fh(1)1792 3766 y Fe(X)1795 3923
y Fg(j)s Fh(=0)1927 3709 y Fe(")1972 3743 y Fg(K)s Ff(\000)p
Fh(1)1983 3766 y Fe(X)1990 3923 y Fg(i)p Fh(=)p Fg(j)2117
3837 y Fj(\014)2160 3845 y Fg(i)2187 3709 y Fe( )2253
3789 y Fj(i)2248 3888 y(j)2284 3709 y Fe(!)2344 3837
y Fk(\()p Fj(\033)2421 3801 y Fh(2)2455 3837 y Fk(\))2485
3801 y Fg(i)p Ff(\000)p Fg(j)2588 3709 y Fe(#)2646 3837
y Fj(R)2705 3801 y Fg(j)1638 4111 y Fk(=)1781 4017 y
Fg(K)s Ff(\000)p Fh(1)1792 4040 y Fe(X)1795 4197 y Fg(j)s
Fh(=0)1927 4111 y Fj(\013)1976 4119 y Fg(j)2009 4111
y Fj(R)2068 4075 y Fg(j)2100 4111 y Fj(:)515 4346 y Fk(Th)n(us,)902
4575 y Fj(\013)951 4583 y Fg(j)1067 4575 y Fk(=)1209
4482 y Fg(K)s Ff(\000)p Fh(1)1220 4504 y Fe(X)1227 4662
y Fg(i)p Fh(=)p Fg(j)1355 4575 y Fj(\014)1398 4583 y
Fg(i)1425 4448 y Fe( )1490 4527 y Fj(i)1485 4626 y(j)1521
4448 y Fe(!)1582 4575 y Fk(\()p Fj(\033)1659 4539 y Fh(2)1693
4575 y Fk(\))1723 4539 y Fg(i)p Ff(\000)p Fg(j)1067 4849
y Fk(=)1209 4755 y Fg(K)s Ff(\000)p Fh(1)1220 4778 y
Fe(X)1227 4936 y Fg(i)p Fh(=)p Fg(j)1355 4849 y Fi(\000)p
Fj(\015)1459 4868 y Fg(K)s Ff(\000)p Fh(1)p Ff(\000)p
Fg(i)1666 4849 y Fj(=\015)1748 4868 y Fg(K)1807 4721
y Fe( )1872 4801 y Fj(i)1868 4900 y(j)1903 4721 y Fe(!)1964
4849 y Fk(\()p Fj(\033)2041 4813 y Fh(2)2075 4849 y Fk(\))2105
4813 y Fg(i)p Ff(\000)p Fg(j)1067 5123 y Fk(=)1209 5029
y Fg(K)s Ff(\000)p Fh(1)1220 5052 y Fe(X)1227 5209 y
Fg(i)p Fh(=)p Fg(j)1365 5052 y Fk(\()p Fi(\000)p Fk(1\))1523
5020 y Fg(i)1562 4996 y Fe(P)1643 5075 y Fg(l)1663 5085
y Fd(1)1695 5075 y Fg(<)p Ff(\001\001\001)p Fg(<l)1868
5086 y Fc(K)s Fb(\000)p Fd(1)p Fb(\000)p Fc(i)2080 5052
y Fj(\025)2124 5061 y Fg(l)2144 5071 y Fd(1)2194 5052
y Fi(\001)13 b(\001)g(\001)g Fj(\025)2341 5061 y Fg(l)2361
5072 y Fc(K)s Fb(\000)p Fd(1)p Fb(\000)p Fc(i)p 1365
5106 1195 4 v 1783 5174 a Fk(\()p Fj(\025)1858 5182 y
Fh(1)1906 5174 y Fi(\001)g(\001)g(\001)g Fj(\025)2053
5182 y Fg(K)2111 5174 y Fk(\))2570 4995 y Fe( )2635 5075
y Fj(i)2631 5174 y(j)2666 4995 y Fe(!)2727 5123 y Fk(\()p
Fj(\033)2804 5087 y Fh(2)2838 5123 y Fk(\))2868 5087
y Fg(i)p Ff(\000)p Fg(j)2971 5123 y Fj(;)515 5370 y Fk(where)21
b Fj(\025)777 5378 y Fg(i)824 5370 y Fk(are)g(the)g(eigen)n(v)l(alues)g
(of)h Fj(R)8 b Fi(\000)f Fj(\033)1730 5338 y Fh(2)1764
5370 y Fj(I)f Fk(,)21 b(whic)n(h)g(can)g(b)r(e)g(mapp)r(ed)e(easily)j
(to)f(the)f(eigen)n(v)l(alues)515 5461 y(of)26 b Fj(R)g
Fk(b)n(y)f(adding)h Fj(\033)1092 5430 y Fh(2)1126 5461
y Fk(.)1926 5956 y Fa(1)p eop end
userdict /end-hook known{end-hook}if