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Introduction to Communication Systems. The Theory of Probability PDF

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Preview Introduction to Communication Systems. The Theory of Probability

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4(cid:3){ 3?{(cid:23)4 +68, 3 rwkhuzlvh = fX(x) 1 -1 1 x 4 3 4 [dyj @ H^[‘@](cid:3)4{i[+{,g{@](cid:3)4{+{.4,g{.]3 {+4(cid:3){,g{@3 4 3 4 (cid:30)5[ @ Hk+[(cid:3)[dyj,5l@H(cid:5)[5(cid:6)@](cid:3)4{i[+{,g{@](cid:3)4{5+{.4,g{.]3 {5+4(cid:3){,g{@ 65 4 3 4 H^[.(cid:28)‘ @ +{.(cid:28),i[+{,g{@ +{.(cid:28),+{.4,g{. +{.(cid:28),+4(cid:3){,g{@(cid:28) ](cid:3)4 ](cid:3)4 ]3 Ixqfwlrqv ri d Udqgrp Yduldeoh Gh(cid:31)qlwlrq 4 Frqvlghu d ixqfwlrq ri d udqgrp yduldeoh/ j+[,1 Wkh uhvxow lv dqrwkhu udqgrp yduldeoh/ zklfk zh frxog fdoo \/ \ @j+[,1 Krzhyhu wkh FGI dqg sgi ri \ zloo eh gl(cid:30)huhqw wkdq wkh FGI dqg sgi ri [1 Lq idfw zh zloo xvh lqirupdwlrq derxw wkh I[+{,/ i[+{,/ dqg j+[, wr ghwhuplqh I\ +|, dqg i\ +|, Wkh glvwulexwlrq ri t ’ }Ef(cid:28) Wkh SGI ri wkh udqgrp yduldeoh/ \/ lv qrwklqj pruhwkdq wkh suredelolw| ri wkh hyhqw wkdw i\(cid:23)|j1 Wklv frqvlvwv ri doo rxwfrphv (cid:19) 5 (cid:13) vxfk wkdw \+(cid:19), (cid:23)|1 Wklv lv htxlydohqw wr wkh vhw ri doo rxwfrphv (cid:19) 5(cid:13) vxfkwkdwj^[+(cid:19),‘(cid:23)|/dqgwklvlvhtxlydohqwwrwkhvhw ridoorxwfrph(cid:19) 5(cid:13)vxfkwkdw[+(cid:19),(cid:23)j(cid:3)4+|,1 Vr I+|,@S+i\(cid:23)|j,@S+ij+[,(cid:23)|j,@S(cid:3)(cid:11)[(cid:23)j(cid:3)4+|,(cid:12)(cid:4) +69, < Wkh Suredelolw| Ghqvlw| Ixqfwlrq +sgi, ri t ’ }Ef(cid:28) Rqfh wkh SGI ri \ lv frpsxwhg/ wkh frpsxwdwlrq ri wkh sgi ri \ lv vwudljkwiruzdug g i\+|,@ I\ +|, +6:, g| H{dpsohv H{dpsoh 43 Ohw [ eh d udqgrp yduldeoh1 Ohw \ @j+[,@5[.4 IlqgI\ +|,dqgi\ +|,1 Wr(cid:31)qgI\ +|,/ zhpxvwhydoxdwhwkhhyhqwi\ (cid:23)|jzkhuh\ lvwkh udqgrpyduldeoh dqg | lv vlpso| d qxpehu1 Frqvlghu wkh iroorzlqj judsk Y Y=2X+1 y {X£ (y-1)/2} x=(y-1)/2 X {Y£ y} Wkh vkdghg yhuwlfdo olqh fruuhvsrqgv wr wkh hyhqw i(cid:19) =\ +(cid:19),(cid:23)|j dqg wkh krul}rqwdo olqh fruuhvsrqgv wr wkh hyhqw (cid:11)(cid:19) =[+(cid:19),(cid:23) 45+|(cid:3)4,(cid:12)1 Erwk hyhqwv kdyh wkh h{dfw vdph rxwfrphv1 Wkhuhiruh 4 (cid:3) i(cid:19) =\ +(cid:19),(cid:23)|j@ (cid:19) =[+(cid:19),(cid:23) +| 4, (cid:29) 5 (cid:30) Wkhuhiruh 4 i\ (cid:23)|j@i5[.4(cid:23)|j@ [ (cid:23) +|(cid:3)4, @ [ (cid:23)j(cid:3)4+|,(cid:12) (cid:29) 5 (cid:30) Qrwh wkdw wkh lqyhuvh ixqfwlrq j(cid:3)4+\, lv (cid:11) j(cid:3)4+\,@ 4+[(cid:3)4, 5 Vr I\ +|,@S ^\ (cid:23)|‘@S ^5[.4(cid:23)|‘@S [ (cid:23) 4+|(cid:3)4, @I[ 4+|(cid:3)4, (cid:23) 5 (cid:24) (cid:21)5 (cid:22) Qrz vxssrvh [ zdv xqlirupo| glvwulexwhg ehwzhhq 3 dqg 41 Wkhq i[+{, @ x+{,(cid:3)x+{(cid:3)4,@(cid:29) 34 3rw(cid:23)khu{z(cid:23)lvh4 +6;, 3 {?3 I[+{, @ {^x+{,(cid:3)x+{(cid:3)4,‘@;? { 3(cid:23){(cid:23)4 +6<, 4 {A4 = Zh fdq qrz sorw I\ +|,1 Ohw*v soxj lq vrph ydoxhv |@3 I\ +3,@I[(cid:3)45+3(cid:3)4,(cid:4)@I[(cid:3)(cid:3)45(cid:4)@3 ||@@45 II\\ ++45,,@@II[[(cid:3)(cid:3)4545++45(cid:3)(cid:3)44,,(cid:4)(cid:4)@@II[[+(cid:3)345,(cid:4)@@345 +73, |@6 I\ +6,@I[(cid:3)45+6(cid:3)4,(cid:4)@I[+4,@4 43

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