www.pudn.com > SolutionsforMultiuserDetection.rar > 4.15.kishore.ps

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b Fg(+)9 b(1])41 b Fc(:)7 b(:)g(:)40 b Fh(b)1269 1673 
y Ff(T)1295 1688 y Fg([)p Fc(M)5 b Fg(])1384 1655 y Fd(\003)1636 
1688 y Fg(\(6\))257 1763 y(where)659 1813 y Fh(b)p Fg([)p 
Fc(i)p Fg(])10 b(=)778 1779 y Fd(\002)816 1813 y Fc(b)834 
1819 y Fb(1)853 1813 y Fg([)p Fc(i)p Fg(])40 b Fc(b)949 
1819 y Fb(2)968 1813 y Fg([)p Fc(i)p Fg(])h Fc(:)7 b(:)g(:)40 
b(b)1155 1819 y Ff(K)1186 1813 y Fg([)p Fc(i)p Fg(])1245 
1779 y Fd(\003)1262 1788 y Ff(T)1636 1813 y Fg(\(7\))257 
1878 y(Let)15 b Fh(c)c Fg(=)h Fh(b)435 1863 y Ff(T)461 
1878 y Fh(A)497 1884 y Ff(m)529 1878 y Fh(RA)601 1884 
y Ff(m)632 1878 y Fg(.)18 b(Th)o(us)c Fh(c)g Fg(is)g(a)g 
Fc(K)s Fg(\(2)p Fc(M)g Fg(+)9 b(1\))14 b(ro)o(w)g(v)o(ector)g(suc)o(h)h 
(that)677 1953 y Fh(c)c Fg(=)753 1919 y Fd(\002)791 1953 
y Fh(c)812 1959 y Fa(\000)p Ff(M)917 1953 y Fh(c)938 
1959 y Fa(\000)p Ff(M)s Fb(+1)1084 1953 y Fc(:)c(:)g(:)40 
b Fh(c)1195 1959 y Ff(M)1253 1919 y Fd(\003)1636 1953 
y Fg(\(8\))257 2027 y(where)480 2102 y Fh(c)501 2108 
y Fa(\000)p Ff(M)606 2102 y Fg(=)i Fh(b)707 2085 y Ff(T)732 
2102 y Fg([)p Fe(\000)p Fc(M)5 b Fg(])p Fh(AR)o Fg([0])p 
Fh(A)j Fg(+)i Fh(b)1062 2085 y Ff(T)1088 2102 y Fg([)p 
Fe(\000)p Fc(M)j Fg(+)d(1])p Fh(AR)o Fg([1])p Fh(A)224 
b Fg(\(9\))438 2165 y Fh(c)459 2171 y Fa(\000)p Ff(M)s 
Fb(+1)606 2165 y Fg(=)42 b Fh(b)707 2147 y Ff(T)732 2165 
y Fg([)p Fe(\000)p Fc(M)5 b Fg(])p Fh(AR)904 2146 y Ff(T)930 
2165 y Fg([1])p Fh(A)k Fg(+)g Fh(b)1088 2147 y Ff(T)1114 
2165 y Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(1])p Fh(AR)o 
Fg([0])p Fh(A)178 b Fg(\(10\))680 2227 y(+)p Fh(b)739 
2210 y Ff(T)765 2227 y Fg([)p Fe(\000)p Fc(M)13 b Fg(+)d(2])p 
Fh(AR)o Fg([1])p Fh(A)527 b Fg(\(11\))438 2289 y Fh(c)459 
2295 y Fa(\000)p Ff(M)s Fb(+2)606 2289 y Fg(=)42 b Fh(b)707 
2272 y Ff(T)732 2289 y Fg([)p Fe(\000)p Fc(M)14 b Fg(+)c(1])p 
Fh(AR)976 2271 y Ff(T)1002 2289 y Fg([1])p Fh(A)e Fg(+)i 
Fh(b)1160 2272 y Ff(T)1185 2289 y Fg([)p Fe(\000)p Fc(M)k 
Fg(+)c(2])p Fh(AR)o Fg([0])p Fh(A)106 b Fg(\(12\))680 
2351 y(+)p Fh(b)739 2334 y Ff(T)765 2351 y Fg([)p Fe(\000)p 
Fc(M)13 b Fg(+)d(3])p Fh(AR)o Fg([1])p Fh(A)527 b Fg(\(13\))553 
2408 y(.)553 2424 y(.)553 2441 y(.)1051 b(\(14\))506 
2503 y Fh(c)527 2509 y Ff(M)606 2503 y Fg(=)42 b Fh(b)707 
2486 y Ff(T)732 2503 y Fg([)p Fc(M)14 b Fe(\000)9 b Fg(1])p 
Fh(AR)943 2485 y Ff(T)970 2503 y Fg([1])p Fh(A)f Fg(+)h 
Fh(b)1127 2486 y Ff(T)1153 2503 y Fg([)p Fc(M)c Fg(])p 
Fh(AR)o Fg([0])p Fh(A)242 b Fg(\(15\))963 2628 y(1)p 
eop 
%%Page: 2 2 
2 1 bop 257 262 a Fg(W)m(e)14 b(can)g(then)h(write)f 
Fh(cb)d Fg(=)h Fh(b)736 246 y Ff(T)761 262 y Fh(A)797 
268 y Ff(m)829 262 y Fh(RA)901 268 y Ff(m)933 262 y Fh(b)h 
Fg(as)h(the)g(follo)o(wing)e(sum)298 349 y Fh(b)325 332 
y Ff(T)351 349 y Fh(A)387 355 y Ff(m)419 349 y Fh(RA)490 
355 y Ff(m)522 349 y Fh(b)41 b Fg(=)h Fh(b)691 332 y 
Ff(T)716 349 y Fg([)p Fe(\000)p Fc(M)5 b Fg(])p Fh(AR)p 
Fg([0])p Fh(Ab)n Fg([)p Fe(\000)p Fc(M)g Fg(])j(+)i Fh(b)1173 
332 y Ff(T)1199 349 y Fg([)p Fe(\000)p Fc(M)j Fg(+)d(1])p 
Fh(AR)o Fg([1])p Fh(Ab)n Fg([)p Fe(\000)p Fc(M)5 b Fg(])-33 
b(\(16\))664 411 y(+)p Fh(b)723 394 y Ff(T)749 411 y 
Fg([)p Fe(\000)p Fc(M)5 b Fg(])p Fh(AR)921 393 y Ff(T)947 
411 y Fg([1])p Fh(Ab)n Fg([)p Fe(\000)p Fc(M)14 b Fg(+)c(1])390 
b(\(17\))664 474 y(+)p Fh(b)723 456 y Ff(T)749 474 y 
Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(1])p Fh(AR)o Fg([0])p 
Fh(Ab)o Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(1])345 b(\(18\))664 
536 y(+)p Fh(b)723 519 y Ff(T)749 536 y Fg([)p Fe(\000)p 
Fc(M)14 b Fg(+)9 b(2])p Fh(AR)o Fg([1])p Fh(Ab)o Fg([)p 
Fe(\000)p Fc(M)14 b Fg(+)9 b(1])345 b(\(19\))664 598 
y(+)p Fh(b)723 581 y Ff(T)749 598 y Fg([)p Fe(\000)p 
Fc(M)14 b Fg(+)9 b(1])p Fh(AR)992 580 y Ff(T)1018 598 
y Fg([1])p Fh(Ab)o Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 
b(2])319 b(\(20\))664 660 y(+)p Fh(b)723 643 y Ff(T)749 
660 y Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(2])p Fh(AR)o 
Fg([0])p Fh(Ab)o Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(2])345 
b(\(21\))664 723 y(+)p Fh(b)723 706 y Ff(T)749 723 y 
Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(3])p Fh(AR)o Fg([1])p 
Fh(Ab)o Fg([)p Fe(\000)p Fc(M)14 b Fg(+)9 b(2])345 b(\(22\))664 
785 y(+)7 b Fc(:)g(:)g(:)h Fg(+)h Fh(b)829 768 y Ff(T)855 
785 y Fg([)p Fc(M)k Fe(\000)d Fg(1])p Fh(AR)1066 767 
y Ff(T)1092 785 y Fg([1])p Fh(Ab)o Fg([)p Fc(M)5 b Fg(])348 
b(\(23\))664 847 y(+)p Fh(b)723 830 y Ff(T)749 847 y 
Fg([)p Fc(M)5 b Fg(])p Fh(AR)o Fg([0])p Fh(Ab)n Fg([)p 
Fc(M)g Fg(])552 b(\(24\))257 935 y(Note)12 b(that)f Fh(b)469 
920 y Ff(T)494 935 y Fg([)p Fc(i)p Fg(])p Fh(AR)604 917 
y Ff(T)630 935 y Fg([1])p Fh(Ab)n Fg([)p Fc(i)s Fg(+)s(1])s(+)s 
Fh(b)898 920 y Ff(T)924 935 y Fg([)p Fc(i)s Fg(+)s(1])p 
Fh(AR)p Fg([1])p Fh(Ab)n Fg([)p Fc(i)p Fg(])g(=)h(2)p 
Fh(b)1340 920 y Ff(T)1365 935 y Fg([)p Fc(i)s Fg(+)s(1])p 
Fh(AR)p Fg([1])p Fh(Ab)o Fg([)p Fc(i)p Fg(].)257 985 
y(This)17 b(is)f(b)q(ecause)i Fh(b)582 970 y Ff(T)608 
985 y Fg([)p Fc(i)p Fg(])p Fh(AR)717 966 y Ff(T)744 985 
y Fg([1])p Fh(Ab)n Fg([)p Fc(i)11 b Fg(+)g(1])k(=)i Fc(w)g 
Fg(where)h Fc(w)f Fg(is)f(some)g(scalar.)26 b(W)m(e)16 
b(note)257 1034 y(the)f(ab)q(o)o(v)o(e)e(since)i Fc(w)579 
1019 y Ff(T)617 1034 y Fg(=)d Fc(w)q Fg(.)17 b(Therefore,)e(w)o(e)f 
(rewrite)h Fh(b)1150 1019 y Ff(T)1176 1034 y Fh(A)1212 
1040 y Ff(m)1243 1034 y Fh(RA)1315 1040 y Ff(m)1347 1034 
y Fh(b)e Fg(as)h(the)h(follo)o(wing)477 1122 y Fh(b)504 
1105 y Ff(T)530 1122 y Fh(A)566 1128 y Ff(m)597 1122 
y Fh(RA)669 1128 y Ff(m)701 1122 y Fh(b)41 b Fg(=)h Fh(b)870 
1105 y Ff(T)895 1122 y Fg([)p Fe(\000)p Fc(M)5 b Fg(])p 
Fh(AR)o Fg([0])p Fh(Ab)o Fg([)p Fe(\000)p Fc(M)g Fg(])341 
b(\(25\))843 1184 y(+2)p Fh(b)923 1167 y Ff(T)948 1184 
y Fg([)p Fe(\000)p Fc(M)14 b Fg(+)c(1])p Fh(AR)o Fg([1])p 
Fh(Ab)n Fg([)p Fe(\000)p Fc(M)5 b Fg(])217 b(\(26\))843 
1247 y(+)p Fh(b)902 1229 y Ff(T)928 1247 y Fg([)p Fe(\000)p 
Fc(M)13 b Fg(+)d(1])p Fh(AR)o Fg([0])p Fh(Ab)n Fg([)p 
Fe(\000)p Fc(M)k Fg(+)c(1])166 b(\(27\))843 1309 y(+2)p 
Fh(b)923 1292 y Ff(T)948 1309 y Fg([)p Fe(\000)p Fc(M)14 
b Fg(+)c(2])p Fh(AR)o Fg([1])p Fh(Ab)n Fg([)p Fe(\000)p 
Fc(M)k Fg(+)c(1])145 b(\(28\))843 1371 y(+)p Fh(b)902 
1354 y Ff(T)928 1371 y Fg([)p Fe(\000)p Fc(M)13 b Fg(+)d(2])p 
Fh(AR)o Fg([0])p Fh(Ab)n Fg([)p Fe(\000)p Fc(M)k Fg(+)c(2])166 
b(\(29\))843 1433 y(+2)p Fh(b)923 1416 y Ff(T)948 1433 
y Fg([)p Fe(\000)p Fc(M)14 b Fg(+)c(3])p Fh(AR)o Fg([1])p 
Fh(Ab)n Fg([)p Fe(\000)p Fc(M)k Fg(+)c(2])145 b(\(30\))843 
1496 y(+)7 b Fc(:)g(:)g(:)g Fg(+)j Fh(b)1008 1478 y Ff(T)1034 
1496 y Fg([)p Fc(M)5 b Fg(])p Fh(AR)n Fg([0])p Fh(Ab)o 
Fg([)p Fc(M)g Fg(])267 b(\(31\))257 1583 y(W)m(e)14 b(can)g(rewrite)h 
(this)f(as)364 1701 y Fh(b)391 1684 y Ff(T)417 1701 y 
Fh(A)453 1707 y Ff(m)485 1701 y Fh(RA)556 1707 y Ff(m)588 
1701 y Fh(b)41 b Fg(=)761 1649 y Ff(M)749 1662 y Fd(X)730 
1751 y Ff(i)p Fb(=)p Fa(\000)p Ff(M)835 1701 y Fh(b)862 
1684 y Ff(T)887 1701 y Fg([)p Fc(i)p Fg(])p Fh(AR)p Fg([0])p 
Fh(Ab)n Fg([)p Fc(i)p Fg(])9 b(+)g(2)p Fh(b)1239 1684 
y Ff(T)1265 1701 y Fg([)p Fc(i)p Fg(])p Fh(AR)o Fg([)p 
Fc(i)p Fg(])p Fh(Ab)o Fg([)p Fc(i)g Fe(\000)g Fg(1])33 
b(\(32\))656 1851 y(=)761 1799 y Ff(M)749 1811 y Fd(X)730 
1900 y Ff(i)p Fb(=)p Fa(\000)p Ff(M)835 1851 y Fh(b)862 
1834 y Ff(T)887 1851 y Fg([)p Fc(i)p Fg(])p Fh(AR)p Fg([0])p 
Fh(Ab)n Fg([)p Fc(i)p Fg(])9 b(+)g(2)p Fh(b)1239 1834 
y Ff(T)1265 1851 y Fg([)p Fc(i)p Fg(])p Fh(AR)o Fg([)p 
Fc(i)p Fg(])p Fh(Ax)o Fg([)p Fc(i)p Fg(])106 b(\(33\))257 
1979 y(Similiarly)l(,)11 b(observ)o(e)j(that)609 2066 
y Fh(b)636 2049 y Ff(T)662 2066 y Fh(A)698 2072 y Ff(m)741 
2066 y Fg(=)785 2033 y Fd(\002)823 2066 y Fh(b)850 2051 
y Ff(T)875 2066 y Fg([)p Fe(\000)p Fc(M)5 b Fg(])p Fh(A)41 
b Fc(:)7 b(:)g(:)40 b Fh(b)1170 2051 y Ff(T)1196 2066 
y Fg([)p Fc(M)5 b Fg(])p Fh(A)1321 2033 y Fd(\003)1616 
2066 y Fg(\(34\))257 2154 y(Then,)515 2204 y Fh(b)542 
2187 y Ff(T)568 2204 y Fh(A)604 2210 y Ff(m)635 2204 
y Fh(y)13 b Fg(=)f Fh(b)744 2187 y Ff(T)769 2204 y Fg([)p 
Fe(\000)p Fc(M)5 b Fg(])p Fh(Ay)q Fg([)p Fe(\000)p Fc(M)g 
Fg(])j(+)h Fc(:)e(:)g(:)h Fg(+)h Fh(b)1208 2187 y Ff(T)1234 
2204 y Fg([)p Fc(M)c Fg(])p Fh(Ay)q Fg([)p Fc(M)g Fg(])182 
b(\(35\))257 2276 y(Since)15 b Fh(b)393 2261 y Ff(T)418 
2276 y Fh(A)454 2282 y Ff(m)486 2276 y Fh(y)g Fg(is)e(a)h(scalar,)g(w)o 
(e)g(can)g(sa)o(y)g(that)f Fh(b)1057 2261 y Ff(T)1083 
2276 y Fh(A)1119 2282 y Ff(m)1151 2276 y Fh(y)f Fg(=)g 
Fh(y)1258 2261 y Ff(T)1284 2276 y Fh(A)1320 2282 y Ff(m)1352 
2276 y Fh(b)p Fg(.)17 b(Th)o(us,)485 2364 y Fh(b)512 
2347 y Ff(T)538 2364 y Fh(A)574 2370 y Ff(m)605 2364 
y Fh(y)43 b Fg(=)f Fh(y)773 2347 y Ff(T)799 2364 y Fg([)p 
Fe(\000)p Fc(M)5 b Fg(])p Fh(Ab)p Fg([)p Fe(\000)p Fc(M)g 
Fg(])i(+)i Fc(:)e(:)g(:)h Fg(+)h Fh(y)1237 2347 y Ff(T)1263 
2364 y Fg([)p Fc(M)c Fg(])p Fh(Ab)p Fg([)p Fc(M)g Fg(])152 
b(\(36\))673 2467 y(=)778 2415 y Ff(M)766 2427 y Fd(X)747 
2516 y Ff(i)p Fb(=)p Fa(\000)p Ff(M)851 2467 y Fh(y)877 
2450 y Ff(T)903 2467 y Fg([)p Fc(i)p Fg(])p Fh(Ab)o Fg([)p 
Fc(i)p Fg(])575 b(\(37\))963 2628 y(2)p eop 
%%Page: 3 3 
3 2 bop 257 262 a Fg(Substituting)16 b(the)h(results)g(in)f(\(33\))g 
(and)g(\(37\))g(in)o(to)f(the)i(de\014nition)e(in)h(\(2\),)g(w)o(e)g 
(get)h(the)257 311 y(follo)o(wing)12 b(decomp)q(osition)753 
441 y(\012\()p Fh(b)p Fg(\))f(=)929 389 y Ff(M)916 402 
y Fd(X)897 491 y Ff(i)p Fb(=)p Fa(\000)p Ff(M)1002 441 
y Fc(\025)1026 447 y Ff(i)1040 441 y Fg(\()p Fh(x)1081 
447 y Ff(i)1095 441 y Fc(;)c Fh(b)p Fg([)p Fc(i)p Fg(]\))421 
b(\(38\))257 573 y(where)293 694 y Fc(\025)317 700 y 
Ff(i)331 694 y Fg(\()p Fh(x)372 700 y Ff(i)386 694 y 
Fc(;)7 b Fh(b)p Fg([)p Fc(i)p Fg(]\))j(=)571 642 y Ff(M)559 
655 y Fd(X)540 744 y Ff(i)p Fb(=)p Fa(\000)p Ff(M)644 
694 y Fg(2)p Fh(y)691 677 y Ff(T)717 694 y Fg([)p Fc(i)p 
Fg(])p Fh(Ab)o Fg([)p Fc(i)p Fg(])f Fe(\000)g Fh(b)932 
677 y Ff(T)958 694 y Fg([)p Fc(i)p Fg(])p Fh(AR)o Fg([0])p 
Fh(Ab)o Fg([)p Fc(i)p Fg(])f Fe(\000)i Fg(2)p Fh(b)1310 
677 y Ff(T)1335 694 y Fg([)p Fc(i)p Fg(])p Fh(AR)o Fg([)p 
Fc(i)p Fg(])p Fh(Ax)p Fg([)p Fc(i)p Fg(])35 b(\(39\))257 
828 y(\(b\))17 b(No)o(w)g(observ)o(e)h(that)e(this)h(decomp)q(osition)e 
(can)i(b)q(e)h(used)f(to)g(create)h(a)f(trellis)f(as)h(in)257 
878 y(Figure)c(\(4.19\).)k(The)d(w)o(eigh)o(t)e(along)g(eac)o(h)i(link) 
e(at)g(a)h(particular)g(stage)g Fc(i)h Fg(in)e(the)i(trellis)f(is)257 
928 y(describ)q(ed)j(b)o(y)e Fc(\025)523 934 y Ff(i)550 
928 y Fg(whic)o(h)g(in)f(turn)i(dep)q(ends)g(on)f(the)g(v)n(alue)f(of)h 
Fh(b)p Fg([)p Fc(i)p Fg(])e(and)i Fh(x)1437 934 y Ff(i)1451 
928 y Fg(.)k(Eac)o(h)c Fh(b)p Fg([)p Fc(i)p Fg(])f(is)257 
978 y(a)h(an)g(elemen)o(t)f(of)g Fe(f\000)p Fg(1)p Fc(;)7 
b Fg(1)p Fe(g)685 963 y Ff(K)730 978 y Fg(and)14 b(th)o(us)g(can)g(tak) 
o(e)g(on)g(2)1147 963 y Ff(K)1192 978 y Fg(p)q(ossible)g(v)n(alues)g 
(at)g(a)f(stage)i Fc(i)p Fg(.)257 1027 y(Since)h Fh(x)392 
1033 y Ff(i)421 1027 y Fg(=)f Fh(b)p Fg([)p Fc(i)9 b 
Fe(\000)i Fg(1],)k(it)g(to)q(o)g(can)h(tak)o(e)g(on)f(2)1000 
1012 y Ff(K)1047 1027 y Fg(p)q(ossible)h(v)n(alues.)23 
b(Th)o(us)16 b(to)g(determine)257 1077 y(the)h(optimal)c 
Fc(\025)509 1083 y Ff(i)538 1077 y Fg(at)j(a)f(stage)h 
Fc(i)p Fg(,)g(w)o(e)g(need)h(to)e(compute)g(\(2)1198 
1062 y Ff(K)1230 1077 y Fg(\))1246 1062 y Fb(2)1281 1077 
y Fg(p)q(ossible)g(outcomes)h(and)257 1127 y(select)h(the)f(maxim)n(um) 
11 b(to)j(determine)i(the)f(next)h(path)f(in)g(the)g(trellis.)22 
b(In)15 b(other)h(w)o(ords,)257 1177 y(w)o(e)e(note)f(that)h(eac)o(h)f 
(stage)h(of)f(the)g(trellis)g(has)h(2)1024 1162 y Ff(K)1069 
1177 y Fg(states)g(e\013ectiv)o(ely)g(describing)g(all)e(the)257 
1227 y(p)q(ossible)i(com)o(binations)d(of)h(bits)h(that)g(could)g(b)q 
(e)h(transmitted)f(b)o(y)g(the)g Fc(K)k Fg(system)12 
b(users.)257 1277 y(Since)20 b(eac)o(h)g(state)h(is)e(connected)i(to)e 
(eac)o(h)h(of)f(the)h(previous)g(states)h(and)e(since)h(there)257 
1326 y(are)e(2)351 1311 y Ff(K)401 1326 y Fg(states)h(in)e(eac)o(h)h 
(stage,)h(w)o(e)f(need)g(to)g(p)q(erform)f(\(2)1213 1311 
y Ff(K)1245 1326 y Fg(\))1261 1311 y Ff(K)1310 1326 y 
Fg(calculations)g(to)h(mak)o(e)257 1376 y(an)e(optimal)e(decision)i(ab) 
q(out)g(whic)o(h)g(set)h(of)f(bits)g(w)o(ere)h(transmitted)e(b)o(y)h 
(the)h Fc(K)i Fg(users.)257 1426 y(Th)o(us)f(the)h(time)d(complexit)o 
(y)g(to)h(determine)h(the)g(v)n(alues)g(of)f(the)h Fc(K)j 
Fg(bits)d(transmitted)257 1476 y(b)o(y)e(the)g(system)f(users)i(is)e 
Fc(O)q Fg(\(\(2)764 1461 y Ff(K)796 1476 y Fg(\))812 
1461 y Fb(2)831 1476 y Fg(\))f(=)g Fc(O)q Fg(\(4)977 
1461 y Ff(K)1009 1476 y Fg(\).)23 b(This)15 b(in)g(turn)h(tells)f(us)h 
(that)g(the)g(time)257 1526 y(complexit)o(y)c(p)q(er)j(bit)f(is)f 
Fc(O)q Fg(\(4)717 1511 y Ff(K)749 1526 y Fc(=K)s Fg(\).)963 
2628 y(3)p eop 
%%Trailer 
end 
userdict /end-hook known{end-hook}if 
%%EOF