www.pudn.com > SolutionsforMultiuserDetection.rar > 3.11.rickard.ps

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%%EndProlog
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%%Page: 1 1
1 0 bop 1373 872 a Fn(Multiuser)45 b(Detection)1656 1112
y Fm(Scott)33 b(Ric)m(k)-5 b(ard)1563 1308 y(F)d(ebruary)33
b(22,)f(1999)515 1665 y Fl(Solution)45 b(to)h(Problem)f(3.11)515
1847 y Fk(W)-7 b(e)34 b(will)g(select)g(the)h(h)n(yp)r(othesis)e(that)i
(is)f(most)g(lik)n(ely)f(giv)n(en)g(the)i(observ)-5 b(ation)32
b(v)n(ector)515 1947 y Fj(Z)p Fk(.)54 b(The)33 b(probabilit)n(y)f
(densit)n(y)h(function)g Fi(f)1919 1959 y Fh(i)1947 1947
y Fk(\()p Fj(Z)p Fk(\))h(for)e(the)i(m)n(ultiv)-5 b(ariate)32
b(Gaussian)h(with)515 2047 y(co)n(v)-5 b(ariance)25 b(matrix)i(\004)
1241 2059 y Fh(i)1297 2047 y Fk(is,)1306 2264 y Fi(f)1347
2276 y Fh(i)1375 2264 y Fk(\()p Fj(Z)p Fk(\))d(=)1838
2208 y(1)p 1618 2245 481 4 v 1618 2261 a Fg(p)p 1701
2261 130 4 v 71 x Ff(j)p Fk(\004)1779 2344 y Fh(i)1807
2332 y Ff(j)p Fk(\(2)p Fi(\031)s Fk(\))1986 2308 y Fh(n=)p
Fe(2)2109 2264 y Fi(e)2148 2229 y Fd(\000)2209 2207 y
Fc(1)p 2209 2216 29 4 v 2209 2249 a(2)2248 2229 y Fe(\()p
Fj(Z)2332 2193 y Fb(T)2378 2229 y Fe(\004)2422 2202 y
Fa(\000)p Fc(1)2422 2250 y Fb(i)2499 2229 y Fj(Z)p Fe(\))3273
2264 y Fk(\(1\))515 2506 y(So,)j(the)h(maxim)n(um-lik)n(eliho)r(o)r(d)f
(decision)g(rule)g(is,)1267 2724 y(argmax)1537 2744 y
Fh(i)1565 2724 y Ff(f)1836 2667 y Fk(1)p 1617 2705 481
4 v 1617 2721 a Fg(p)p 1699 2721 130 4 v 1699 2792 a
Ff(j)p Fk(\004)1777 2804 y Fh(i)1805 2792 y Ff(j)p Fk(\(2)p
Fi(\031)s Fk(\))1984 2768 y Fh(n=)p Fe(2)2107 2724 y
Fi(e)2146 2689 y Fd(\000)2207 2667 y Fc(1)p 2207 2676
29 4 v 2207 2709 a(2)2246 2689 y Fe(\()p Fj(Z)2330 2653
y Fb(T)2376 2689 y Fe(\004)2420 2662 y Fa(\000)p Fc(1)2420
2710 y Fb(i)2497 2689 y Fj(Z)p Fe(\))2585 2724 y Ff(g)646
b Fk(\(2\))515 2966 y(The)26 b(argumen)n(t)f(to)g(the)i(argmax)d(is)i
(alw)n(a)n(ys)e(p)r(ositiv)n(e,)i(th)n(us)g(w)n(e)f(can)h(tak)n(e)f
(the)h(log)h(of)f(the)515 3066 y(argumen)n(t,)972 3283
y(argmax)1242 3303 y Fh(i)1270 3283 y Ff(f\000)14 b Fk(log)o(\()1529
3208 y Fg(p)p 1612 3208 130 4 v 75 x Ff(j)p Fk(\004)1690
3295 y Fh(i)1719 3283 y Ff(j)p Fk(\))k Ff(\000)g Fk(log)q(\(\(2)p
Fi(\031)s Fk(\))2171 3249 y Fh(n=)p Fe(2)2284 3283 y
Fk(\))g Ff(\000)2427 3227 y Fk(1)p 2427 3264 42 4 v 2427
3340 a(2)2479 3283 y(\()p Fj(Z)2569 3247 y Fh(T)2622
3283 y Fk(\004)2677 3248 y Fd(\000)p Fe(1)2677 3306 y
Fh(i)2766 3283 y Fj(Z)q Fk(\))p Ff(g)p Fi(:)351 b Fk(\(3\))515
3494 y(Elimating)27 b(the)h(constan)n(t)f(term/factor)f(and)h
(simplifying,)h(w)n(e)f(obtain,)1316 3676 y(argmax)1586
3697 y Fh(i)1613 3676 y Ff(f\000)14 b Fk(log)o(\()p Ff(j)p
Fk(\004)1950 3688 y Fh(i)1979 3676 y Ff(j)p Fk(\))19
b Ff(\000)f Fk(\()p Fj(Z)2226 3640 y Fh(T)2278 3676 y
Fk(\004)2333 3641 y Fd(\000)p Fe(1)2333 3699 y Fh(i)2423
3676 y Fj(Z)p Fk(\))p Ff(g)p Fi(;)695 b Fk(\(4\))515
3859 y(whic)n(h)27 b(is)h(equiv)-5 b(alen)n(t)27 b(to,)1363
4042 y(argmin)1617 4062 y Fh(i)1644 4042 y Ff(f)p Fk(log\()p
Ff(j)p Fk(\004)1903 4054 y Fh(i)1931 4042 y Ff(j)q Fk(\))18
b(+)g(\()p Fj(Z)2179 4005 y Fh(T)2231 4042 y Fk(\004)2286
4006 y Fd(\000)p Fe(1)2286 4065 y Fh(i)2376 4042 y Fj(Z)p
Fk(\))p Ff(g)p Fi(:)742 b Fk(\(5\))1926 5255 y(1)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF