4.44.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 289 a Fj(Multiuser)45 b(Detection)1656 530
y Fi(Scott)33 b(Ric)m(k)-5 b(ard)1669 725 y(April)31
b(2,)h(1999)515 1079 y Fh(Solution)45 b(to)h(Problem)f(4.44)1737
1303 y Fg(\021)1778 1315 y Ff(1)1839 1303 y Fe(=)1936
1237 y Fg(d)1979 1207 y Ff(2)1979 1259 y(1)p Fd(;)p Ff(min)p
1936 1284 211 4 v 1992 1360 a Fg(A)2054 1331 y Ff(2)2054
1382 y(1)3273 1303 y Fe(\(1\))515 1506 y(W)-7 b(e)25
b(are)f(in)n(terested)g(in)i(an)e(expression)g(for)g
Fg(d)1941 1476 y Ff(2)1941 1528 y(1)p Fd(;)p Ff(min)2133
1506 y Fe(in)h(the)g(t)n(w)n(o-user)e(async)n(hronous)g(case.)515
1606 y(That)37 b(is,)j(w)n(e)e(w)n(an)n(t)e(to)i(\014nd)g
Fg(\017)1518 1618 y Ff(1)1593 1606 y Fe(and)f Fg(\017)1798
1618 y Ff(2)1873 1606 y Fe(that)h(corresp)r(ond)d(to)j(the)g(minim)n
(um)g(energy)515 1705 y(signal,)1175 1778 y Fd(M)1150
1803 y Fc(X)1112 1981 y Fd(i)p Ff(=)p Fb(\000)p Fd(M)1321
1882 y Fg(A)1383 1894 y Ff(1)1421 1882 y Fg(\017)1455
1894 y Ff(1)1492 1882 y Fe([)p Fg(i)p Fe(])p Fg(s)1606
1894 y Ff(1)1643 1882 y Fe(\()p Fg(t)18 b Fa(\000)g Fg(iT)12
b Fe(\))18 b(+)g Fg(A)2091 1894 y Ff(2)2128 1882 y Fg(\017)2162
1894 y Ff(2)2199 1882 y Fe([)p Fg(i)p Fe(])p Fg(s)2313
1894 y Ff(2)2350 1882 y Fe(\()p Fg(t)h Fa(\000)f Fg(iT)29
b Fa(\000)18 b Fg(\034)9 b Fe(\))492 b(\(2\))515 2115
y(Without)28 b(loss)f(of)g(generalit)n(y)-7 b(.)36 b(w)n(e)27
b(will)h(assume)f Fg(\017)2109 2127 y Ff(1)2147 2115
y Fe([0])22 b(=)h(1.)639 2214 y(Consider)i(the)h(case)f(where)g
Fg(\017)1571 2226 y Ff(2)1608 2214 y Fe([)p Fa(\000)p
Fe(1])d(=)h Fg(\017)1904 2226 y Ff(2)1942 2214 y Fe([0])f(=)h(0.)36
b(In)26 b(this)g(case,)f(an)n(y)g(other)g(non-zero)515
2314 y(comp)r(onen)n(t)k(of)g Fg(\017)1071 2326 y Ff(1)1138
2314 y Fe(or)g Fg(\017)1275 2326 y Ff(2)1342 2314 y Fe(\(b)r(esides)h
Fg(\017)1695 2326 y Ff(1)1732 2314 y Fe([0]\),)g(will)g(not)f(o)n(v)n
(erlap)f(in)h(time)i(with)f(the)f(section)515 2414 y(of)c(the)h(signal)
f(corresp)r(onding)e(to)i Fg(\017)1646 2426 y Ff(1)1683
2414 y Fe([0])h(and)f(th)n(us)g(will)h(just)g(add)f(to)h(the)g(total)f
(energy)f(of)515 2513 y(the)30 b(signal.)41 b(F)-7 b(rom)29
b(this,)h(it)g(is)f(clear)f(that)i(the)g(only)f(w)n(a)n(y)f(that)i(a)f
(non-zero)e(comp)r(onen)n(t)515 2613 y(of)e Fg(\017)641
2625 y Ff(1)704 2613 y Fe(or)g Fg(\017)838 2625 y Ff(2)901
2613 y Fe(\(call)h(it)g Fg(\017)1199 2625 y Fd(i)1226
2613 y Fe([)p Fg(j)5 b Fe(]\))26 b(will)g(serv)n(e)f(to)h(reduce)f(the)
h(o)n(v)n(erall)e(signal)h(energy)f(is)i(if)g(all)g(the)515
2712 y(comp)r(onen)n(ts)32 b(b)r(et)n(w)n(een)i Fg(\017)1337
2724 y Fd(i)1365 2712 y Fe([)p Fg(j)5 b Fe(])33 b(and)g
Fg(\017)1684 2724 y Ff(1)1721 2712 y Fe([0])g(\(in)h(b)r(oth)g
Fg(\017)2212 2724 y Ff(1)2283 2712 y Fe(and)f Fg(\017)2484
2724 y Ff(2)2521 2712 y Fe(\))g(are)g(also)f(non-zero)f(and)515
2812 y(some)c(cancellation)f(is)i(o)r(ccuring.)36 b(This)27
b(can)h(b)r(e)g(summerized,)1224 2992 y Fg(\017)1258
3004 y Ff(1)1295 2992 y Fe([)p Fg(i)p Fe(])23 b Fa(6)p
Fe(=)g(0)k(only)g(if)111 b Fg(\017)1925 3004 y Ff(2)1962
2992 y Fe([)p Fg(i)18 b Fa(\000)g Fe(1])23 b Fa(6)p Fe(=)f(0)p
Fg(;)14 b(i)22 b(>)h Fe(0)p Fg(;)1224 3116 y(\017)1258
3128 y Ff(2)1295 3116 y Fe([)p Fg(i)p Fe(])g Fa(6)p Fe(=)g(0)k(only)g
(if)183 b Fg(\017)1996 3128 y Ff(1)2033 3116 y Fe([)p
Fg(i)p Fe(])23 b Fa(6)p Fe(=)g(0)p Fg(;)14 b(i)22 b Fa(\025)h
Fe(0)p Fg(;)1224 3241 y(\017)1258 3253 y Ff(1)1295 3241
y Fe([)p Fg(i)p Fe(])g Fa(6)p Fe(=)g(0)k(only)g(if)183
b Fg(\017)1996 3253 y Ff(2)2033 3241 y Fe([)p Fg(i)p
Fe(])23 b Fa(6)p Fe(=)g(0)p Fg(;)14 b(i)22 b(<)h Fe(0)p
Fg(;)1224 3365 y(\017)1258 3377 y Ff(2)1295 3365 y Fe([)p
Fg(i)p Fe(])g Fa(6)p Fe(=)g(0)k(only)g(if)118 b Fg(\017)1932
3377 y Ff(1)1969 3365 y Fe([)p Fg(i)18 b Fe(+)g(1])23
b Fa(6)p Fe(=)f(0)p Fg(;)14 b(i)22 b(<)h Fe(0)p Fg(:)515
3669 y Fe(Using)31 b(these)g(rules,)h(w)n(e)f(consider)g(some)f
(candidate)h(w)n(a)n(v)n(eforms.)46 b(With)33 b Fg(\017)2942
3681 y Ff(1)2980 3669 y Fe([0])c(=)g(1)i(the)515 3769
y(only)f(non-zero)f(comp)r(onen)n(t,)i(the)g(error)e(signal)h(energy)g
(is)g Fg(A)2500 3739 y Ff(2)2500 3790 y(1)2538 3769 y
Fe(.)46 b(Adding)31 b(one)f(of)h Fg(\017)3190 3781 y
Ff(2)3227 3769 y Fe([)p Fa(\000)p Fe(1])515 3869 y(or)e
Fg(\017)653 3881 y Ff(2)690 3869 y Fe([0])h(\(and)h(selecting)e(the)i
(minim)n(um)g(energy)e(c)n(hoice)h(for)g(eac)n(h\),)g(the)h(signal)e
(energy)515 3968 y(w)n(ould)22 b(b)r(e)i Fg(A)923 3938
y Ff(2)923 3989 y(1)970 3968 y Fe(+)10 b Fg(A)1107 3938
y Ff(2)1107 3989 y(2)1153 3968 y Fa(\000)g Fe(2)p Fg(A)1332
3980 y Ff(1)1368 3968 y Fg(A)1430 3980 y Ff(2)1468 3968
y Fa(j)p Fg(\032)1534 3980 y Ff(21)1604 3968 y Fa(j)23
b Fe(or)g Fg(A)1810 3938 y Ff(2)1810 3989 y(1)1857 3968
y Fe(+)10 b Fg(A)1994 3938 y Ff(2)1994 3989 y(2)2040
3968 y Fa(\000)g Fe(2)p Fg(A)2219 3980 y Ff(1)2255 3968
y Fg(A)2317 3980 y Ff(2)2354 3968 y Fa(j)p Fg(\032)2420
3980 y Ff(12)2491 3968 y Fa(j)p Fe(,)24 b(resp)r(ectiv)n(ely)-7
b(.)34 b(Similarly)-7 b(,)515 4068 y(with)34 b(all)g(three)g(of)g
Fg(\017)1186 4080 y Ff(1)1223 4068 y Fe([0],)i Fg(\017)1404
4080 y Ff(2)1441 4068 y Fe([)p Fa(\000)p Fe(1],)f(and)f
Fg(\017)1854 4080 y Ff(2)1891 4068 y Fe([0])g(non-zero,)g(the)h(signal)
e(energy)g(w)n(ould)h(b)r(e)515 4168 y(\(for)23 b(the)h(appropriate)e
(c)n(hoices)h(of)g Fg(\017)1653 4180 y Ff(1)1690 4168
y Fe([0],)h Fg(\017)1859 4180 y Ff(2)1896 4168 y Fe([)p
Fa(\000)p Fe(1],)g(and)g Fg(\017)2287 4180 y Ff(2)2325
4168 y Fe([0]\),)g Fg(A)2554 4137 y Ff(2)2554 4188 y(1)2602
4168 y Fe(+)11 b(2)p Fg(A)2782 4137 y Ff(2)2782 4188
y(2)2829 4168 y Fa(\000)g Fe(2)p Fg(A)3009 4180 y Ff(1)3045
4168 y Fg(A)3107 4180 y Ff(2)3144 4168 y Fa(j)p Fg(\032)3210
4180 y Ff(12)3281 4168 y Fa(j)g(\000)515 4267 y Fe(2)p
Fg(A)619 4279 y Ff(1)656 4267 y Fg(A)718 4279 y Ff(2)755
4267 y Fa(j)p Fg(\032)821 4279 y Ff(12)892 4267 y Fa(j)21
b Fe(Note)g(that)h(the)g(sections)e(of)h(the)h(signal)e(corresp)r
(onding)g(to)h Fg(\017)2855 4279 y Ff(1)2892 4267 y Fe([0])g(and)g
Fg(\017)3190 4279 y Ff(2)3227 4267 y Fe([)p Fa(\000)p
Fe(1])515 4367 y(are)31 b(non-o)n(v)n(erlapping)e(in)j(time.)51
b(Selecting)32 b(the)h(minim)n(um)f(energy)f(signal)g(from)h(these)515
4466 y(four)27 b(p)r(ossibilities,)g(w)n(e)h(see)f(that,)915
4646 y Fg(d)958 4612 y Ff(2)958 4667 y(1)p Fd(;)p Ff(min)1148
4646 y Fa(\024)22 b Fg(A)1297 4612 y Ff(2)1297 4667 y(1)1353
4646 y Fa(\000)c Fe(\(2)p Fg(A)1572 4658 y Ff(1)1610
4646 y Fg(A)1672 4658 y Ff(2)1709 4646 y Fa(j)p Fg(\032)1775
4658 y Ff(12)1846 4646 y Fa(j)g(\000)g Fg(A)2032 4612
y Ff(2)2032 4667 y(2)2070 4646 y Fe(\))2102 4612 y Ff(+)2176
4646 y Fa(\000)g Fe(\(2)p Fg(A)2395 4658 y Ff(1)2432
4646 y Fg(A)2494 4658 y Ff(2)2531 4646 y Fa(j)p Fg(\032)2597
4658 y Ff(21)2668 4646 y Fa(j)g(\000)g Fg(A)2854 4612
y Ff(2)2854 4667 y(2)2892 4646 y Fe(\))2924 4612 y Ff(+)3273
4646 y Fe(\(3\))515 4826 y(No)n(w)40 b(w)n(e)g(sho)n(w)f(this)i(b)r
(ound)g(is)f(tigh)n(t.)75 b(Consider)40 b(adding)g(non-zero)e
Fg(\017)j Fe(comp)r(onen)n(ts)515 4925 y(on)32 b(the)h(righ)n(t)f(of)g
Fg(\017)1122 4937 y Ff(2)1159 4925 y Fe([0].)52 b(W)-7
b(e)32 b(w)n(ould)g(only)h(ha)n(v)n(e)e Fg(\017)2134
4937 y Ff(1)2171 4925 y Fe([1])h(non-zero)f(if)i Fg(\017)2748
4937 y Ff(2)2785 4925 y Fe([0])f(w)n(as)g(non-zero)515
5025 y(\(from)25 b(b)r(efore\),)g(th)n(us)h(the)f(energy)f(of)h(the)g
(new)h(section)e(of)h(the)h(signal)e(w)n(ould)g(con)n(tribute)515
5125 y Fg(A)577 5095 y Ff(2)577 5145 y(1)635 5125 y Fa(\000)c
Fe(2)p Fg(A)824 5137 y Ff(1)861 5125 y Fg(A)923 5137
y Ff(2)960 5125 y Fa(j)p Fg(\032)1026 5137 y Ff(21)1096
5125 y Fa(j)p Fe(.)46 b(W)-7 b(e)31 b(w)n(ould)f(only)g(w)n(an)n(t)g
(to)g(do)g(this)h(if)g Fg(A)2498 5137 y Ff(1)2564 5125
y Fg(<)c Fe(2)p Fg(A)2760 5137 y Ff(2)2797 5125 y Fa(j)p
Fg(\032)2863 5137 y Ff(21)2933 5125 y Fa(j)p Fe(,)32
b(but)f(if)g(this)515 5224 y(is)26 b(the)g(case,)g(w)n(e)g(w)n(ould)g
(not)g(ha)n(v)n(e)f Fg(\017)1668 5236 y Ff(2)1705 5224
y Fe([0])h(non-zero.)35 b(This)26 b(argumen)n(t)f(hold)h(regardless)e
(of)515 5324 y(the)h(n)n(um)n(b)r(er)g(of)h(comp)r(onen)n(ts)e(w)n(e)h
(wish)h(to)f(add,)g(and)g(th)n(us)h(the)f(b)r(ound)h(is)f(tigh)n(t)h
(and)f(the)515 5424 y(optim)n(um)j(asymptotic)f(e\016ciency)g(is,)997
5673 y Fg(\021)1038 5685 y Ff(1)1099 5673 y Fe(=)c(1)18
b Fa(\000)1340 5617 y Fg(A)1402 5629 y Ff(2)p 1340 5654
100 4 v 1340 5730 a Fg(A)1402 5742 y Ff(1)1463 5531 y
Fc(")1511 5556 y(\022)1572 5673 y Fe(2)p Fa(j)p Fg(\032)1680
5685 y Ff(12)1750 5673 y Fa(j)h(\000)1885 5617 y Fg(A)1947
5629 y Ff(2)p 1885 5654 V 1885 5730 a Fg(A)1947 5742
y Ff(1)1994 5556 y Fc(\023)2055 5574 y Ff(+)2129 5673
y Fe(+)2212 5556 y Fc(\022)2273 5673 y Fe(2)p Fa(j)p
Fg(\032)2381 5685 y Ff(21)2451 5673 y Fa(j)f(\000)2585
5617 y Fg(A)2647 5629 y Ff(2)p 2585 5654 V 2585 5730
a Fg(A)2647 5742 y Ff(1)2695 5556 y Fc(\023)2756 5574
y Ff(+)2811 5531 y Fc(#)2873 5673 y Fg(:)377 b Fe(\(4\))1926
6106 y(1)p eop end
userdict /end-hook known{end-hook}if