4.22.huaidai.ps

是multiuser detection 这本书的习题解答, 很有用的书.

951 y Fj(Huaiyu)40 b(Dai)1561 1269 y Fi(31)e(Mar)g(1999)-186
1730 y Fh(PR)m(OBLEM)e(4.22.)p Fg(Asssume)f Ff(\033)c
Fg(=)d(0)k(and)h Ff(K)i Fg(=)27 b(3.)-186 1936 y(\(a\))i(Find)f(a)h
(set)i(of)d(signature)h(w)m(a)m(v)m(eforms)i(and)e(receiv)m(ed)i
(amplitudes)d(suc)m(h)j(that)e(in)g(a)g(sync)m(hronous)i(CDMA)-186
2056 y(c)m(hannel,)i(the)g(minim)m(um)c(bit-error-rate)i(satis\014es)
1431 2276 y Ff(P)1494 2291 y Fe(k)1536 2276 y Fg(\(0\))d
Ff(>)f Fg(0)p Ff(;)17 b(k)31 b Fg(=)c(1)p Ff(;)17 b Fg(2)p
Ff(;)g Fg(3)p Ff(:)-186 2496 y Fg(\(b\))34 b(With)f(the)i(same)f(set)g
(of)g(signature)g(w)m(a)m(v)m(eforms)h(y)m(ou)g(c)m(hose)g(in)e(\(a\),)
h(\014nd)h(a)f(set)g(of)g(receiv)m(ed)h(amplitudes)-186
2617 y(suc)m(h)f(that)1431 2737 y Ff(P)1494 2752 y Fe(k)1536
2737 y Fg(\(0\))28 b(=)f(0)p Ff(;)17 b(k)31 b Fg(=)c(1)p
Ff(;)17 b Fg(2)p Ff(;)g Fg(3)p Ff(:)-186 2911 y Fg(\(c\))47
b(Find)e(a)h(set)h(of)f(signature)h(w)m(a)m(v)m(eforms)g(and)g(receiv)m
(ed)h(amplitudes)d(suc)m(h)i(that)g(in)e(an)i(async)m(hronous)-186
3032 y(CDMA)33 b(c)m(hannel)g(with)f Ff(\034)773 3047
y Fd(1)841 3032 y Fg(=)27 b(0,)33 b Ff(\034)1095 3047
y Fd(2)1162 3032 y Fg(=)28 b Ff(T)11 b(=)1383 2950 y
Fc(p)p 1465 2950 49 4 v 1465 3032 a Fg(2,)33 b(and)f
Ff(\034)1805 3047 y Fd(3)1873 3032 y Fg(=)27 b Ff(T)11
b(=)2093 2950 y Fc(p)p 2176 2950 V 82 x Fg(3)1406 3252
y Ff(P)1483 3211 y Fe(M)1469 3276 y(k)1561 3252 y Fg(\(0\))28
b Ff(>)f Fg(0)p Ff(;)17 b(k)31 b Fg(=)c(1)p Ff(;)17 b
Fg(2)p Ff(;)g Fg(3)p Ff(:)-186 3472 y Fg(\(d\))32 b(Sho)m(w)g(that)g
(for)f(an)m(y)h(set)h(of)e(receiv)m(ed)i(nonzero)f(amplitudes)f(and)h
(signature)f(w)m(a)m(v)m(eforms,)i(there)g(exists)f(a)-186
3592 y(set)h(of)f(o\013sets)i(suc)m(h)g(that)1406 3712
y Ff(P)1483 3671 y Fe(M)1469 3737 y(k)1561 3712 y Fg(\(0\))28
b(=)f(0)p Ff(;)17 b(k)31 b Fg(=)c(1)p Ff(;)17 b Fg(2)p
Ff(;)g Fg(3)p Ff(:)-186 3973 y Fh(Solution:)-186 4179
y Fg(\(a\))k(The)i(situation)d Ff(P)609 4194 y Fe(k)652
4179 y Fg(\(0\))27 b Ff(>)h Fg(0)21 b(happ)s(ens)i(when)g(some)e
(di\013eren)m(t)h(transmitted)f(information)e(v)m(ectors)k(\()p
Ff(b)3639 4194 y Fd(1)3679 4179 y Ff(;)17 b Fc(\001)g(\001)g(\001)d
Ff(;)j(b)3940 4194 y Fe(K)4009 4179 y Fg(\))-186 4299
y(result)38 b(in)g(the)g(same)h(receiv)m(ed)g(signals)e(without)h(bac)m
(kground)i(noise.)60 b(When)39 b(w)m(e)h(receiv)m(e)f(that)f(w)m(a)m(v)
m(eform,)-186 4420 y(w)m(e)c(can)e(just)h(mak)m(e)g(a)f(guess)i(and)f
(error)f(is)g(una)m(v)m(oidable.)44 b(Let)32 b(us)i(assume)1311
4640 y Ff(s)1357 4655 y Fd(1)1396 4640 y Fg(\()p Ff(t)p
Fg(\))28 b(=)g Ff(s)1685 4655 y Fd(2)1724 4640 y Fg(\()p
Ff(t)p Fg(\))g(=)f Ff(s)2012 4655 y Fd(3)2052 4640 y
Fg(\()p Ff(t)p Fg(\))g(=)h Ff(s)p Fg(\()p Ff(t)p Fg(\))1372
b(\(1\))-186 4860 y(and)1479 4980 y Ff(A)1552 4995 y
Fd(1)1619 4980 y Fg(=)27 b Ff(A)1795 4995 y Fd(2)1863
4980 y Fg(=)g Ff(A)2039 4995 y Fd(3)2106 4980 y Fg(=)h
Ff(A)1540 b Fg(\(2\))-186 5154 y(then)33 b(when)h(w)m(e)g(receiv)m(e)
1619 5275 y Ff(y)t Fg(\()p Ff(t)p Fg(\))27 b(=)h Ff(As)p
Fg(\()p Ff(t)p Fg(\))1680 b(\(3\))1857 5655 y(1)p eop
2 1 bop -186 167 a Fg(the)39 b(transmitted)f(information)d(v)m(ector)40
b(could)e(b)s(e)h(\(1)p Ff(;)17 b Fc(\000)p Fg(1)p Ff(;)g
Fg(1\))p Ff(;)g Fg(\(1)p Ff(;)g Fg(1)p Ff(;)g Fc(\000)p
Fg(1\),)38 b(or)g(\()p Fc(\000)p Fg(1)p Ff(;)17 b Fg(1)p
Ff(;)g Fg(1\).)61 b(No)39 b(matter)f(what)-186 287 y(decision)32
b(is)g(made,)g(w)m(e)i(will)c(ha)m(v)m(e)1431 408 y Ff(P)1494
423 y Fe(k)1536 408 y Fg(\(0\))e Ff(>)f Fg(0)p Ff(;)17
b(k)31 b Fg(=)c(1)p Ff(;)17 b Fg(2)p Ff(;)g Fg(3)p Ff(:)-186
668 y Fg(\(b\))33 b(If)f(w)m(e)i(mo)s(di\014y)d(\(2\))h(suc)m(h)i(that)
1406 788 y Ff(A)1479 803 y Fd(1)1546 788 y Fg(=)27 b(2)p
Ff(A)1771 803 y Fd(2)1838 788 y Fg(=)h(4)p Ff(A)2064
803 y Fd(3)2131 788 y Fg(=)f(4)p Ff(A)1467 b Fg(\(4\))-186
962 y(then)33 b(it)f(is)g(easy)h(to)g(c)m(hec)m(k)i(that)d(di\013eren)m
(t)h(information)c(v)m(ectors)34 b(lead)e(to)g(di\013eren)m(t)h(receiv)
m(ed)h(signals:)506 1182 y(7)p Ff(As)p Fg(\()p Ff(t)p
Fg(\))p Ff(;)17 b Fg(5)p Ff(As)p Fg(\()p Ff(t)p Fg(\))p
Ff(;)g Fg(3)p Ff(As)p Fg(\()p Ff(t)p Fg(\))p Ff(;)g(As)p
Fg(\()p Ff(t)p Fg(\))p Ff(;)g Fc(\000)p Ff(As)p Fg(\()p
Ff(t)p Fg(\))p Ff(;)g Fc(\000)p Fg(3)p Ff(As)p Fg(\()p
Ff(t)p Fg(\))p Ff(;)g Fc(\000)p Fg(5)p Ff(As)p Fg(\()p
Ff(t)p Fg(\))p Ff(;)g Fc(\000)p Fg(7)p Ff(As)p Fg(\()p
Ff(t)p Fg(\))-186 1402 y(so)1431 1523 y Ff(P)1494 1538
y Fe(k)1536 1523 y Fg(\(0\))28 b(=)f(0)p Ff(;)17 b(k)31
b Fg(=)c(1)p Ff(;)17 b Fg(2)p Ff(;)g Fg(3)p Ff(:)-186
1783 y Fg(\(c\)As)44 b(in)e(\(a\))h(w)m(e)h(will)d(design)i(suc)m(h)h
(a)f(set)h(of)f(signature)f(w)m(a)m(v)m(eforms)j(that)e(cause)h
(di\013eren)m(t)f(information)-186 1903 y(v)m(ectors)32
b(to)e(coincide)f(in)h(the)h(receiv)m(ed)g(signal)e(space.)44
b(F)-8 b(or)29 b(simplicit)m(y)f(w)m(e)j(assume)g(the)g(receiv)m(ed)g
(amplitudes)-186 2024 y(are)i(the)g(same.)-186 2230 y(In)g(async)m
(hronous)i(c)m(hannel)e(w)m(e)g(consider)g(the)g(su\016cien)m(t)h
(statistics)e(of)g(the)h(matc)m(hed)g(\014lter)f(output:)792
2450 y Fh(y)r Fg([)p Ff(i)p Fg(])c(=)g Fh(R)1156 2409
y Fe(T)1210 2450 y Fg([1])p Fh(Ab)p Fg([)p Ff(i)23 b
Fg(+)f(1])g(+)g Fh(R)p Fg([0])p Fh(Ab)p Fg([)p Ff(i)p
Fg(])g(+)g Fh(R)p Fg([1])p Fh(Ab)p Fg([)p Ff(i)g Fc(\000)h
Fg(1])853 b(\(5\))-186 2670 y(where)1355 2884 y Fh(R)p
Fg([0])27 b(=)1672 2688 y Fb(2)1672 2834 y(6)1672 2887
y(4)1807 2762 y Fg(1)121 b Ff(\032)2027 2777 y Fd(12)2185
2762 y Ff(\032)2235 2777 y Fd(13)1769 2883 y Ff(\032)1819
2898 y Fd(12)2015 2883 y Fg(1)g Ff(\032)2235 2898 y Fd(23)1769
3003 y Ff(\032)1819 3018 y Fd(13)1977 3003 y Ff(\032)2027
3018 y Fd(23)2223 3003 y Fg(1)2352 2688 y Fb(3)2352 2834
y(7)2352 2887 y(5)3823 2884 y Fg(\(6\))1393 3401 y Fh(R)p
Fg([1])27 b(=)1711 3205 y Fb(2)1711 3351 y(6)1711 3405
y(4)1807 3280 y Fg(0)83 b Ff(\032)1989 3295 y Fd(21)2147
3280 y Ff(\032)2197 3295 y Fd(31)1807 3400 y Fg(0)121
b(0)g Ff(\032)2197 3415 y Fd(31)1807 3521 y Fg(0)g(0)159
b(0)2314 3205 y Fb(3)2314 3351 y(7)2314 3405 y(5)3823
3401 y Fg(\(7\))-186 3688 y(If)33 b(w)m(e)g(c)m(ho)s(ose)h(suc)m(h)g(a)
e(set)h(of)f(signature)h(w)m(a)m(v)m(eforms)g(that)1245
4021 y Fh(R)p Fg([0])27 b(=)1563 3825 y Fb(2)1563 3971
y(6)1563 4024 y(4)1709 3900 y Fg(1)170 b(1)p Ff(=)p Fg(2)160
b(1)p Ff(=)p Fg(2)1660 4020 y(1)p Ff(=)p Fg(2)170 b(1)g
Fc(\000)p Fg(1)p Ff(=)p Fg(2)1660 4141 y(1)p Ff(=)p Fg(2)82
b Fc(\000)p Fg(1)p Ff(=)p Fg(2)171 b(1)2461 3825 y Fb(3)2461
3971 y(7)2461 4024 y(5)3823 4021 y Fg(\(8\))1469 4584
y Fh(R)p Fg([1])27 b(=)1787 4388 y Fb(2)1787 4534 y(6)1787
4588 y(4)1884 4463 y Fg(0)82 b(0)h(0)1884 4583 y(0)f(0)h(0)1884
4704 y(0)f(0)h(0)2238 4388 y Fb(3)2238 4534 y(7)2238
4588 y(5)3823 4584 y Fg(\(9\))-186 4963 y(Then)50 b(with)e
Fh(b)p Fg([)p Ff(i)p Fg(])h(b)s(eing)f(\(1)p Ff(;)17
b Fc(\000)p Fg(1)p Ff(;)g Fc(\000)p Fg(1\))48 b(or)g(\()p
Fc(\000)p Fg(1)p Ff(;)17 b Fg(1)p Ff(;)g Fg(1\))48 b(while)g(k)m
(eeping)h(the)g(information)c(v)m(ectors)51 b(of)d(other)-186
5083 y(in)m(terv)-5 b(als)32 b(unc)m(hanged,)i(w)m(e)f(will)e(get)h
(the)h(same)g(receiv)m(ed)h(signal)d([)p Fh(y)r Fg([)p
Fc(\000)p Ff(M)10 b Fg(])p Ff(;)17 b Fc(\001)g(\001)g(\001)e
Ff(;)i Fh(y)r Fg([)p Ff(M)10 b Fg(]])33 b(with)g Fh(y)r
Fg([)p Ff(i)p Fg(])27 b(=)h Fh(0)p Fg(.)44 b(So)1406
5303 y Ff(P)1483 5262 y Fe(M)1469 5328 y(k)1561 5303
y Fg(\(0\))28 b Ff(>)f Fg(0)p Ff(;)17 b(k)31 b Fg(=)c(1)p
Ff(;)17 b Fg(2)p Ff(;)g Fg(3)p Ff(:)1857 5655 y Fg(2)p
eop
3 2 bop -186 167 a Fg(\(d\))41 b(W)-8 b(e)41 b(can)g(view)g(the)g
(3-user,)i Ff(M)10 b Fg(-frame,)42 b(async)m(hronous)h(c)m(hannel)e(as)
g(a)f(3\(2)p Ff(M)e Fg(+)28 b(1\)-user)41 b(sync)m(hronous)-186
287 y(c)m(hannel)35 b(as)g(in)f(section)h(4.2)g(and)g(consider)g(the)g
(su\013cien)m(t)h(statistics)e(of)g(3\(2)p Ff(M)g Fg(+)24
b(1\)-dimensional)31 b(v)m(ector)36 b(of)-186 408 y(matc)m
(hed-\014lter)c(output)1629 528 y Fh(y)e Fg(=)d Fh(RA)1990
543 y Fa(M)2070 528 y Fh(b)1642 b Fg(\(10\))-186 702
y(where)890 1036 y Fh(R)27 b Fg(=)1104 716 y Fb(2)1104
862 y(6)1104 912 y(6)1104 962 y(6)1104 1012 y(6)1104
1061 y(6)1104 1111 y(6)1104 1164 y(4)1201 795 y Fh(R)p
Fg([0])83 b Fh(R)1555 759 y Fe(T)1609 795 y Fg([1])180
b(0)f Fc(\001)17 b(\001)g(\001)150 b Fg(0)221 b(0)1201
915 y Fh(R)p Fg([1])110 b Fh(R)p Fg([0])g Fh(R)1879 879
y Fe(T)1934 915 y Fg([1])83 b Fc(\001)17 b(\001)g(\001)150
b Fg(0)221 b(0)1270 1036 y(0)179 b Fh(R)p Fg([1])138
b Fh(R)p Fg([0])110 b Fc(\001)17 b(\001)g(\001)150 b
Fg(0)221 b(0)1236 1156 y Fc(\001)17 b(\001)g(\001)179
b(\001)17 b(\001)g(\001)207 b(\001)17 b(\001)g(\001)144
b(\001)17 b(\001)g(\001)116 b(\001)17 b(\001)g(\001)152
b(\001)17 b(\001)g(\001)1270 1276 y Fg(0)248 b(0)276
b(0)179 b Fc(\001)17 b(\001)g(\001)81 b Fh(R)p Fg([1])i
Fh(R)p Fg([0])2817 716 y Fb(3)2817 862 y(7)2817 912 y(7)2817
962 y(7)2817 1012 y(7)2817 1061 y(7)2817 1111 y(7)2817
1164 y(5)3774 1036 y Fg(\(11\))-186 1535 y Fh(R)p Fg([0])32
b(and)h Fh(R)p Fg([1])f(are)g(de\014ned)i(as)f(\(6\))f(and)h(\(7\).)
-186 1741 y Ff(A)-113 1756 y Fe(M)1 1741 y Fg(is)h(the)h(3\(2)p
Ff(M)e Fg(+)24 b(1\))f Fc(\002)h Fg(3\(2)p Ff(M)34 b
Fg(+)23 b(1\))34 b(diagonal)f(matrix)g(whose)i Ff(k)27
b Fg(+)c(3)p Ff(i)35 b Fg(diagoal)d(elemen)m(t)i(is)g(equal)h(to)f
Ff(A)3878 1756 y Fe(k)3921 1741 y Fg(,)-186 1862 y(receiv)m(ed)i
(amplitude)d(for)h(the)h Ff(k)s Fg(th)g(user.)51 b(As)35
b(long)f(as)h(the)g(receiv)m(ed)h(amplitudes)d(are)i(nonzero,)h(the)f
(matrix)-186 1982 y Ff(A)-113 1997 y Fe(M)-1 1982 y Fg(has)e(full)d
(rank.)-186 2188 y(W)-8 b(e)35 b(can)f(see)h(that)f(if)f
Fh(R)h Fg(is)g(in)m(v)m(ertible,)g(it)f(will)f(uniquely)i(map)f(the)i
(information)c(v)m(ectors)k(in)m(to)f(the)g(receiv)m(ed)-186
2309 y(signal)d(space)j(without)e(am)m(biguit)m(y)-8
b(.)41 b(F)-8 b(urther)33 b(insp)s(ection)f(rev)m(eal)h(that)f
Fh(R)p Fg([0])g(is)g(p)s(ositiv)m(e)g(semide\014nite)g(since)1244
2529 y Fh(R)p Fg([0])27 b(=)p Ff(<)h Fh(s)p Fg(\()p Fh(t)22
b Fc(\000)g Ff(\034)11 b Fg(\))q Ff(;)17 b Fh(s)2065
2488 y Fe(T)2120 2529 y Fg(\()p Fh(t)k Fc(\000)i Ff(\034)11
b Fg(\))28 b Ff(>)1256 b Fg(\(12\))-186 2749 y(where)937
2869 y Fh(s)p Fg(\()p Fh(t)21 b Fc(\000)i Ff(\034)11
b Fg(\))28 b(=)g(\()p Ff(s)1491 2884 y Fd(1)1530 2869
y Fg(\()p Ff(t)22 b Fc(\000)h Ff(\034)1767 2884 y Fd(1)1807
2869 y Fg(\))p Ff(;)17 b(s)1935 2884 y Fd(2)1974 2869
y Fg(\()p Ff(t)22 b Fc(\000)h Ff(\034)2211 2884 y Fd(2)2251
2869 y Fg(\))p Ff(;)17 b(s)2379 2884 y Fd(3)2418 2869
y Fg(\()p Ff(t)22 b Fc(\000)h Ff(\034)2655 2884 y Fd(3)2694
2869 y Fg(\)\))2770 2828 y Fe(T)-186 3044 y Fg(So)g(b)m(y)i(c)m(ho)s
(osing)e(the)g(appropriate)g(o\013sets)h Ff(\034)39 b
Fg(=)28 b(\()p Ff(\034)1681 3059 y Fd(1)1721 3044 y Ff(;)17
b(\034)1807 3059 y Fd(2)1846 3044 y Ff(;)g(\034)1932
3059 y Fd(3)1972 3044 y Fg(\))2010 3007 y Fe(T)2088 3044
y Fg(b)s(et)m(w)m(een)25 b(the)f(signature)f(w)m(a)m(v)m(eforms)i(of)e
(di\013eren)m(t)-186 3164 y(users,)k(w)m(e)e(can)e(mak)m(e)h
Fh(R)p Fg([0])f(p)s(ositiv)m(e)g(de\014nite,)j(i.e.,)f(ha)m(ving)e
(full)f(rank.)41 b(Then)25 b(b)m(y)f(the)g(structure)h(of)e(the)h
(matrix)-186 3284 y Fh(R)p Fg(,)32 b(it)g(also)f(has)i(full)e(rank.)44
b(Th)m(us)34 b(w)m(e)g(will)c(ha)m(v)m(e)1406 3504 y
Ff(P)1483 3463 y Fe(M)1469 3529 y(k)1561 3504 y Fg(\(0\))e(=)f(0)p
Ff(;)17 b(k)31 b Fg(=)c(1)p Ff(;)17 b Fg(2)p Ff(;)g Fg(3)p
Ff(:)1857 5655 y Fg(3)p eop end
userdict /end-hook known{end-hook}if