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📄 7.10.rickard.ps

📁 是multiuser detection 这本书的习题解答, 很有用的书.
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b(B12,)k(the)d(\014rst)h(stage)e(con)n(v)n(en)n(tional)g(detector)h(decides)g(\(+)p Fc(;)14 b Fd(+\).)69 b(Canceling)38b(in)515 3527 y(the)f(second)g(stage,)h(the)g(decision)e(is)h(\(sgn\()pFc(y)1987 3539 y Fb(1)2049 3527 y Fa(\000)24 b Fc(A)22003539 y Fb(2)2238 3527 y Fc(\032)p Fd(\))p Fc(;)14 b Fd(sgn)o(\()pFc(y)2543 3539 y Fb(2)2605 3527 y Fa(\000)25 b Fc(A)27573539 y Fb(1)2794 3527 y Fc(\032)p Fd(\)\))40 b(=)e(\()pFa(\000)p Fc(;)14 b Fd(+\),)39 b(a)515 3627 y(c)n(hange.)79b(Th)n(us)42 b(w)n(e)g(calculate)f(the)i(next)f(decision,)j(canceling)d(the)g(new)g(estimates,)515 3726 y(\(sgn)o(\()p Fc(y)7403738 y Fb(1)803 3726 y Fa(\000)25 b Fc(A)955 3738 y Fb(2)9923726 y Fc(\032)p Fd(\))p Fc(;)14 b Fd(sgn\()p Fc(y)12983738 y Fb(2)1361 3726 y Fd(+)24 b Fc(A)1512 3738 y Fb(1)15503726 y Fc(\032)p Fd(\)\))41 b(=)f(\()p Fa(\000)p Fc(;)14b Fd(+\),)40 b(whic)n(h)e(is)g(the)g(same)g(as)f(the)h(previous)5153826 y(stage)27 b(and)h(th)n(us)g(the)h(same)e(for)h(all)g(remaining)f(stages.)37 b(Similar)28 b(b)r(eha)n(v)n(oir)f(is)h(exhibited)5153926 y(b)n(y)e(regions)f(B32,)h(B34,)g(and)g(B14)g(as)g(detailed)h(in)g(the)g(table.)36 b(In)27 b(con)n(trast)f(to)g(these)h(t)n(w)n(o)5154025 y(b)r(eha)n(v)n(oirs,)f(in)j(region)e(I1)h(the)g(initial)h(decision)e(is)i(\(+)p Fc(;)14 b Fd(+\),)28 b(and)g(the)g(second)g(stage)f(deci-)515 4125 y(sion)i(is)i(\(sgn)o(\()p Fc(y)10004137 y Fb(1)1058 4125 y Fa(\000)19 b Fc(A)1204 4137 yFb(2)1242 4125 y Fc(\032)p Fd(\))p Fc(;)14 b Fd(sgn)o(\()pFc(y)1547 4137 y Fb(2)1605 4125 y Fa(\000)19 b Fc(A)17514137 y Fb(1)1789 4125 y Fc(\032)p Fd(\)\))28 b(=)f(\()pFa(\000)p Fc(;)14 b Fa(\000)p Fd(\).)44 b(Canceling)30b(the)g(new)g(estimates,)515 4224 y(the)c(new)g(3rd)f(stage)f(decision)i(is)f(\(sgn\()p Fc(y)1801 4236 y Fb(1)1853 4224 y Fd(+)14b Fc(A)1994 4236 y Fb(2)2032 4224 y Fc(\032)p Fd(\))pFc(;)g Fd(sgn)o(\()p Fc(y)2337 4236 y Fb(2)2389 4224y Fd(+)g Fc(A)2530 4236 y Fb(1)2568 4224 y Fc(\032)pFd(\)\))23 b(=)g(\(+)p Fc(;)14 b Fd(+\).)36 b(Th)n(us)25b(w)n(e)515 4324 y(are)32 b(stuc)n(k)i(in)g(a)f(lo)r(op)g(deciding)h(\(+)p Fc(;)14 b Fd(+\))33 b(on)h(o)r(dd)f(stages)g(and)g(\()pFa(\000)p Fc(;)14 b Fa(\000)p Fd(\))34 b(on)f(ev)n(en)g(stages.)5154424 y(Region)d(I3)g(b)r(eha)n(v)n(es)f(similarly)h(\(although)g(decisions)g(are)f(\(+)p Fc(;)14 b Fd(+\))31 b(on)f(ev)n(en)g(stages)f(and)515 4523 y(\()p Fa(\000)p Fc(;)14 b Fa(\000)p Fd(\))27b(on)g(o)r(dd)h(stages\).)639 4623 y(Th)n(us)g(the)g(decision)f(regions)f(in)i(Figure)f(7.17)f(are)g(con\014rmed.)19265255 y(1)p eop2 1 bop 515 523 a Fe(\(b\))515 676 y Fd(As)21 b(b)r(efore,)i(w)n(e)e(consider)f(regions)g(in)i(the)g(decision)f(space,)h(although)f(for)g(a)g(decorrelating)515 776 y(\014rst)32 b(stage,)g(w)n(e)g(consider)f(the)i(regions)e(as)g(in)i(Figure)f(2.)50 b(W)-7 b(e)33b(will)f(de\014ne)h(eac)n(h)e(region)515 876 y(with)g(b)r(ounds)f(on)gFc(y)1157 888 y Fb(1)1225 876 y Fd(and)g Fc(y)1430 888y Fb(2)1467 876 y Fd(,)h(but)g(add)f(to)g(the)h(region)e(de\014nition)i(the)g(condition)f(that)515 975 y(a)e(particular)f(decorrelating)f(detector)i(decision)g(is)h(satis\014ed.)39 b(F)-7 b(or)27b(example,)i(Q1)e(is)i(the)515 1075 y(region)d(where,)6161241 y(1.)41 b Fc(y)763 1253 y Fb(1)823 1241 y Fa(2)24b Fd([)p Fc(A)987 1253 y Fb(2)1024 1241 y Fc(\032;)14b Fa(1)p Fd(\),)616 1407 y(2.)41 b Fc(y)763 1419 y Fb(2)8231407 y Fa(2)24 b Fd([)p Fc(A)987 1419 y Fb(1)1024 1407y Fc(\032;)14 b Fa(1)p Fd(\),)28 b(and)616 1573 y(3.)41b(decorrelating)26 b(detector)h(decides)g(\(+)p Fc(;)14b Fd(+\),)515 1739 y(and)27 b(B21)g(is)g(the)h(region)e(where,)6161905 y(1.)41 b Fc(y)763 1917 y Fb(1)823 1905 y Fa(2)24b Fd([)p Fc(A)987 1917 y Fb(2)1024 1905 y Fc(\032;)14b Fa(1)p Fd(\),)616 2071 y(2.)41 b Fc(y)763 2083 y Fb(2)8232071 y Fa(2)24 b Fd([)p Fc(A)987 2083 y Fb(1)1024 2071y Fc(;)14 b Fa(1)p Fd(\),)28 b(and)616 2237 y(3.)41 b(decorrelating)26b(detector)h(decides)g(\()p Fa(\000)p Fc(;)14 b Fd(+\).)5152403 y(First,)26 b(note)f(that)h(b)n(y)f(de\014ning)h(the)g(regions)e(in)i(this)g(w)n(a)n(y)-7 b(,)24 b(regions)g(Q1,)i(Q3,)f(I1,)g(I3,)h(B12,)515 2503 y(B14,)35 b(B32,)g(and)g(B34)e(are)h(all)g(subsets)h(of)f(the)h(corresp)r(onding)e(regions)g(in)i(the)g(m)n(ulti-)5152602 y(stage)f(detector)h(with)g(con)n(v)n(en)n(tional)f(\014rst)h(stage.)59 b(Moreo)n(v)n(er,)34 b(for)h(these)g(regions,)h(the)5152702 y(con)n(v)n(en)n(tional)30 b(and)i(decorrelating)e(detectors)i(ha)n(v)n(e)f(the)h(same)g(decisions,)g(th)n(us)h(result-)5152802 y(ing)g(decisions)g(for)h(the)g(m)n(ultistage)f(detector)g(with)i(decorrelating)c(\014rst)j(stage)f(will)h(b)r(e)515 2901y(the)28 b(same)g(as)f(those)h(for)f(the)i(the)f(m)n(ultistage)g(detector)f(with)i(con)n(v)n(en)n(tional)d(\014rst)i(stage.)5153001 y(Th)n(us,)k(the)g(en)n(tries)e(in)i(T)-7 b(able)31b(2,)h(whic)n(h)g(con)n(tains)e(the)i(sequence)f(of)g(decisions)g(for)g(this)515 3101 y(m)n(ultistage)e(detector,)h(need)h(not)f(b)r(e)g(c)n(hanged)f(in)i(an)n(y)e(w)n(a)n(y)g(\(for)g(these)i(eigh)n(t)e(regions\).)515 3200 y(Th)n(us)h(regions)g(I1)g(and)h(I3)f(still)i(ha)n(v)n(e)d(the)j(\015ip-\015op)e(decisions)h(\(although)f(the)h(shap)r(e)g(of)515 3300 y(I1)c(and)g(I3)g(is)h(di\013eren)n(t)g(b)n(y)f(de\014nition\).)639 3399 y(So)33 b(w)n(e)g(consider)f(the)h(eigth)g(remaining)g(regions.)51 b(In)34 b(region)d(Q4)i(\(for)f(whic)n(h)hFc(y)3254 3411 y Fb(1)3324 3399 y Fa(2)515 3499 y Fd([)pFa(\000)p Fc(A)665 3511 y Fb(2)702 3499 y Fc(\032;)14b Fa(1)p Fd(\))25 b(and)f Fc(y)1121 3511 y Fb(2)11813499 y Fa(2)g Fd(\()p Fa(\0001)p Fc(;)14 b Fd(0]\),)25b(the)g(decorrelating)d(detector)i(decides)h(\(b)n(y)f(de\014nition\))515 3599 y(\(+)p Fc(;)14 b Fa(\000)p Fd(\),)25 b(and)g(the)h(second)f(stage)f(canceler)h(decides,)g(\(sgn\()p Fc(y)2420 3611y Fb(1)2471 3599 y Fd(+)14 b Fc(A)2612 3611 y Fb(2)26493599 y Fc(\032)p Fd(\))p Fc(;)g Fd(sgn\()p Fc(y)29553611 y Fb(2)3006 3599 y Fa(\000)f Fc(A)3146 3611 y Fb(1)31843599 y Fc(\032)p Fd(\)\))23 b(=)515 3698 y(\(+)p Fc(;)14b Fa(\000)p Fd(\),)25 b(and)g(the)h(resulting)f(decision)f(is)i(alw)n(a)n(ys)d(\(+)p Fc(;)14 b Fa(\000)p Fd(\).)36 b(Similar)24b(b)r(eha)n(v)n(oir)g(is)h(found)h(in)515 3798 y(Q1,)j(I2,)h(and)f(I4)h(as)f(detailed)g(in)h(the)g(table)g(\(for)g(these)f(regions,)g(the)h(only)f(thing)h(whic)n(h)515 3898 y(c)n(hanges)c(in)i(the)g(table)f(is)h(the)g(b)r(ounds)g(on)f Fc(y)1941 3910 y Fb(1)2005 3898y Fd(and)h Fc(y)2208 3910 y Fb(2)2245 3898 y Fd(\).)6393997 y(In)35 b(region)e(B21)h(\(for)g(whic)n(h)g Fc(y)16344009 y Fb(1)1706 3997 y Fa(2)h Fd([)p Fc(A)1881 4009y Fb(2)1918 3997 y Fc(\032;)14 b Fa(1)p Fd(\))35 b(and)gFc(y)2358 4009 y Fb(2)2429 3997 y Fa(2)g Fd(\()p Fa(\0001)pFc(;)14 b Fd(+)p Fa(1)p Fd(\)\),)36 b(the)f(decor-)5154097 y(relating)c(detector)g(decides)h(\(b)n(y)g(de\014nition\))g(\()pFa(\000)p Fc(;)14 b Fd(+\),)33 b(and)f(the)g(second)f(stage)g(canceler)515 4196 y(decides,)43 b(\(sgn\()p Fc(y)1066 4208 y Fb(1)11304196 y Fa(\000)26 b Fc(A)1283 4208 y Fb(2)1320 4196 yFc(\032)p Fd(\))p Fc(;)14 b Fd(sgn\()p Fc(y)1626 4208y Fb(2)1690 4196 y Fd(+)26 b Fc(A)1843 4208 y Fb(1)18814196 y Fc(\032)p Fd(\)\))44 b(=)g(\(+)p Fc(;)14 b Fd(+\).)74b(The)40 b(third)g(stage)f(decides,)515 4296 y(\(sgn)o(\()pFc(y)740 4308 y Fb(1)799 4296 y Fa(\000)22 b Fc(A)9484308 y Fb(2)985 4296 y Fc(\032)p Fd(\))p Fc(;)14 b Fd(sgn\()pFc(y)1291 4308 y Fb(2)1350 4296 y Fa(\000)21 b Fc(A)14984308 y Fb(1)1536 4296 y Fc(\032)p Fd(\)\))31 b(=)g(\(+)pFc(;)14 b Fd(+\),)34 b(and)e(th)n(us)h(ev)n(ery)e(stage)h(\(after)g(the)h(\014rst)515 4396 y(decides)28 b(\(+)p Fc(;)14b Fd(+\)\).)40 b(Similar)28 b(baha)n(v)n(oir)f(is)h(found)h(in)g(the)g(remaining)f(regions)f(\(B41,)h(B23,)515 4495 y(and)f(B43\).)6394595 y(Th)n(us)h(the)g(decision)f(regions)f(in)i(Figure)f(7.20)f(are)g(con\014rmed.)1926 5255 y(2)p eop3 2 bop 1197 2074 a @beginspecial 0 @llx 0 @lly 76 @urx76 @ury 1800 @rwi @setspecial/MyAppDict 100 dict dup begin def/$F2psDict 200 dict def$F2psDict begin$F2psDict /mtrx matrix put/col-1 {0 setgray} bind def/col0 {0.000 0.000 0.000 srgb} bind def/col1 {0.000 0.000 1.000 srgb} bind def/col2 {0.000 1.000 0.000 srgb} bind def/col3 {0.000 1.000 1.000 srgb} bind def/col4 {1.000 0.000 0.000 srgb} bind def/col5 {1.000 0.000 1.000 srgb} bind def/col6 {1.000 1.000 0.000 srgb} bind def/col7 {1.000 1.000 1.000 srgb} bind def/col8 {0.000 0.000 0.560 srgb} bind def/col9 {0.000 0.000 0.690 srgb} bind def/col10 {0.000 0.000 0.820 srgb} bind def/col11 {0.530 0.810 1.000 srgb} bind def/col12 {0.000 0.560 0.000 srgb} bind def/col13 {0.000 0.690 0.000 srgb} bind def/col14 {0.000 0.820 0.000 srgb} bind def/col15 {0.000 0.560 0.560 srgb} bind def/col16 {0.000 0.690 0.690 srgb} bind def/col17 {0.000 0.820 0.820 srgb} bind def/col18 {0.560 0.000 0.000 srgb} bind def/col19 {0.690 0.000 0.000 srgb} bind def/col20 {0.820 0.000 0.000 srgb} bind def/col21 {0.560 0.000 0.560 srgb} bind def/col22 {0.690 0.000 0.690 srgb} bind def/col23 {0.820 0.000 0.820 srgb} bind def/col24 {0.500 0.190 0.000 srgb} bind def/col25 {0.630 0.250 0.000 srgb} bind def/col26 {0.750 0.380 0.000 srgb} bind def/col27 {1.000 0.500 0.500 srgb} bind def/col28 {1.000 0.630 0.630 srgb} bind def/col29 {1.000 0.750 0.750 srgb} bind def/col30 {1.000 0.880 0.880 srgb} bind def/col31 {1.000 0.840 0.000 srgb} bind defendsave-8.0 83.0 translate1 -1 scale.9 .9 scale % to make patterns same scale as in xfig% This junk string is used by the show operators/PATsstr 1 string def/PATawidthshow { 	% cx cy cchar rx ry string  % Loop over each character in the string  {  % cx cy cchar rx ry char    % Show the character    dup				% cx cy cchar rx ry char char    PATsstr dup 0 4 -1 roll put	% cx cy cchar rx ry char (char)    false charpath		% cx cy cchar rx ry char    /clip load PATdraw    % Move past the character (charpath modified the    % current point)    currentpoint			% cx cy cchar rx ry char x y    newpath    moveto			% cx cy cchar rx ry char    % Reposition by cx,cy if the character in the string is cchar    3 index eq {			% cx cy cchar rx ry      4 index 4 index rmoveto    } if    % Reposition all characters by rx ry    2 copy rmoveto		% cx cy cchar rx ry  } forall  pop pop pop pop pop		% -  currentpoint  newpath  moveto} bind def/PATcg {  7 dict dup begin    /lw currentlinewidth def    /lc currentlinecap def    /lj currentlinejoin def    /ml currentmiterlimit def    /ds [ currentdash ] def    /cc [ currentrgbcolor ] def    /cm matrix currentmatrix def  end} bind def% PATdraw - calculates the boundaries of the object and% fills it with the current pattern/PATdraw {			% proc  save exch    PATpcalc			% proc nw nh px py    5 -1 roll exec		% nw nh px py    newpath    PATfill			% -  restore} bind def% PATfill - performs the tiling for the shape/PATfill { % nw nh px py PATfill -  PATDict /CurrentPattern get dup begin    setfont    % Set the coordinate system to Pattern Space    PatternGState PATsg    % Set the color for uncolored pattezns    PaintType 2 eq { PATDict /PColor get PATsc } if    % Create the string for showing    3 index string		% nw nh px py str    % Loop for each of the pattern sources    0 1 Multi 1 sub {		% nw nh px py str source	% Move to the starting location	3 index 3 index		% nw nh px py str source px py	moveto			% nw nh px py str source	% For multiple sources, set the appropriate color	Multi 1 ne { dup PC exch get PATsc } if	% Set the appropriate string for the source	0 1 7 index 1 sub { 2 index exch 2 index put } for pop	% Loop over the number of vertical cells	3 index 		% nw nh px py str nh	{			% nw nh px py str	  currentpoint		% nw nh px py str cx cy	  2 index show		% nw nh px py str cx cy	  YStep add moveto	% nw nh px py str	} repeat		% nw nh px py str    } for    5 { pop } repeat  end} bind def% PATkshow - kshow with the current pattezn/PATkshow {			% proc string  exch bind			% string proc  1 index 0 get			% string proc char  % Loop over all but the last character in the string  0 1 4 index length 2 sub {				% string proc char idx    % Find the n+1th character in the string    3 index exch 1 add get	% string proe char char+1    exch 2 copy			% strinq proc char+1 char char+1 char    % Now show the nth character    PATsstr dup 0 4 -1 roll put	% string proc chr+1 chr chr+1 (chr)    false charpath		% string proc char+1 char char+1    /clip load PATdraw    % Move past the character (charpath modified the current point)

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