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T H0 NG VAHOA GUYEN i LIEU VOH VI/M y VA N il ONQH1 I I A O H J A T VU N G O C PHAN l_Y T H U Y ^ T THONGTINVAMAHOA NHA XUAT BAN B l/U OIEN Ha Noi, th a n g 10 - 0 L d l N O ID A U Ngay nay, cac mang may tinh, mang dien thoai so him tuyen va vo tuyen, mang truyen hinh cap dang co xu the hoi tu thdnh mang chung, da dich vu, co kha nang chuyen tdi thongtin tich horp vai toe len den hang tram Mbit/s Nhin tie gdc khoa hoc, cac he thongthongtin la cac he thong co phirc tap cau true rat lan va chiu tdc dong cua nhieu loqi nhieu khde De cac he thong co the lam viec on dinh vatin cay, dap ung duac yeu cau cua nguai su dung viec lieu giu, chuyen tdi vd chia se thong tin, khong the khong dua vao kien thi'cc ve lythuyetthongtin vd mahoa Co the noi, khong co lythuyetthongtin vd md hoa thi khong the giai quyet duac van de truyen tin chinh xdc tren cac duang truyen ddi hang ngdn km Nham dap ung nhu cau tim hieu ve lythuyetthongtin vd md hoa, Nhd xudt ban Buu dien xudt ban cuon sach Lythuyetthongtin vd m d hoa ” cua tdc gid Vu Ngoc Phan giai thieu den ban doc Lythuyetthongtin vd md hoa Id cong cu huu hieu de giai quyet nhung van de cong nghe thongtin dang dat ret nhu ndng cao tdc truyen dan theo thai gian thuc, nen dir lieu , boo mat dir lieu, Bo cuc cuon sdeh gom chuang Tu chuang den chuang trinh bay cu the cac cac van de ve lythuyetthongtin rai rac nhu: luang tin vd En-tro-py, nguon rai rac vd kenh rai rac, md hoa nguon vd md hoa kenh , cac phuang phdp md hoa vd giai md, mat md Dae biet chuang giai thieu ve lythuyetthongtin cac he lien tuc duac xem nhu la sir m a rong cua lythuyetthongtin cac he rai rac Trong trinh bay nhirng noi dung ly thuyet, cuon such dua nhirng vi du kha true quan, giiip nguai doc de theo ddi Ngodi cudi cuon such co phdn Phil luc giai thieu mot so chuang trinh mo phong cue thudt loan md hoa viet tren Mat Lab giup cho ban doc hieu mot each true quan ve cdc md dd neliien ciru Cuon sach la tai lieu tham khao rat huu ich cho cac chuyen gui ky thuat vien, cung nhu can bo giang day va dac biet hoc vien nganh vien thong muon tim hieu nhung kiin thuc co ban va nghien ciru sdu vcrly thuyetthongtinva md hoa Nha xudt ban xin tran giai thieu den ban doc va rat mong nhan d u a c y kien go p y cua q u i vi M oi y kien gop y xin g u i ve Nha xu d t ban Buu dien - 18 Nguyen Du, Ha Noi Tran cam cm./ Ha Noi, thang 10 nam 2006 NHA XUAT BAN B l/U DIEN Chuang MG DAU 1.1 KHAI QUAT Sau chien tranh the gicri thu hai, co ba lythuyet doi va da cung anh hucmg rat manh me den su phat trien khoa hoc va cong nghe Do la lythuyet he thong (System Theory), lythuyet dieu khien (Control Theory) valythuyetthongtin (Information Theory) Trong cuon sach “ Five More Golden Rules” xuat ban nam 2000 tai My khong phai khong co ly John L Casti xep ba lythuyet tren cung vcri lythuyet day (Knot Theory) va giai tich ham (Functional Analysis) la nam so nhung lythuyet lan cua the ky XX Luc dau lythuyetthongtin duac phat trien chu yeu de phuc vu cho ky thuat truyen tin Nhung sau nguai ta nhan rang, nhung ung dung cua lythuyetthongtin khong chi dung lai a ky thuat truyen tinma a ca cac linh vuc khac nhu sinh y kinh te, ngon ngu am nhac, nghe thuat hoi hoaLythuyetthongtin co vai tro het sue quan trong viec nghien cuu cac he thong tu to chuc, tu dieu chinh va tu on dinh Lythuyetthongtin cung vai lythuyet he thongvalythuyet dieu khien da dat nen mong cho qua trinh chuyen doi tir cach tiep can dua tren quan he hinh thuc-noi dung (Form-Content-Relation) sang cach tiep can dua tren quan he cau truc-chuc ndng (Structure-FunctionRelation) Cach tiep can sau da gop phan tao nhung tuu vo cung to lan cua nhan loai suot nua the ky qua viec phan tich va thiet ke he thong, dac biet la cac he thong lan (Large-scale Systems) Vao nam 1948 Shannon cong bo lythuyetthongtin cua minh qua cuon sach noi tieng "A Mathematical Theory o f Communication", d u a n g cap lan nhat the giai luc bay gia mai chi cho phep thuc hien dong thai 1800 cuoc thoai D uai tac dong cua lythuyetthongtin 20 nam sau so cuoc thoai dong thai tren d uang truyen da la 230.000 Nam 2001 d u an g cap quang Lythuyetthongtinva md hoa Ngay nguai ta da thira nhan, ben canh qua trinh van dong vat chat la qua trinh van dong thongtin khong kern phan quan N h u da biet, cac nhiem sac the va chat long trang trung tao bo n h a sinh hoc Lythuyetthongtin la mot cong cu huu hieu de hieu ban chat cua nhung bo n h a sinh hoc Duai goc cua lythuyetthongtin ta thay hoat dong cua te bao khong khac gi hoat dong cua mot nha may, nhan te bao la ban giam doc dieu hanh toan bo qua trinh san xuat cac nhiem sac the la ho so ve qui trinh cong nghe va ke hoach san xuat te bao chat la nguyen vat lieu, cac en-zim la doi ngu ky su va cong nhan Neu vi mot ly nao do, cac thongtin ehua nhiem sac the bi sai lech thi hoat dong cua nha m ay te bao se khong dung ke hoach va qui trinh cong nghe, san pham tao khong dam bao tieu chuan ve chat lugng va so lugng, benh tat xuat hien N h u da biet, nhiem sac the la mot chuoi rat dai cac phan tu chua thongtin Trong qua trinh phan chia te bao, cac thongtin phai duac bao toan va chia deu cho ca hai nira (hai te bao mai) Cac thongtin ton tai nhiem sac the d uai dang m a d u ac tao nen tu bon phan tu c a sa (ly thuyetthongtinva m a hoa goi la m a hieu) Do la Adenin (ky hieu la A), Thymin (ky hieu la T), Guanin (ky hieu la G) va Cytosin (ky hieu la C) Theo luat so m u cua sir lira chon (Exponential Law o f Choice), neu mot nhiem sac the gom mot chuoi L phan tu nu-clein thi se co l cau true khac Vai L = 100 ta co 100 « 1,6 x 10 60 cau true nhiem sac the khac Tren thuc te, mot nhiem sac the co the bao gom hang ngan phan tu nu-clein Dieu giai thich day dii tinh da dang phong phu cua the giai sinh vat Trong ITnh \ uc than kinh hoc nguai ta thay rang mot te bao thAn kinh co the d uac mo ta hoan toan bai mot o-to-mat huu han va he than kinh co the xem nhu mot mang cac o-to-mat huu han Mot na-ron bao gom phan than va cac sy-nap Cac sy-nap bat dau tir than cua mot na-ron va noi tai mot na-ron khac C u nhu vay nhi^u na-ron k^t n i vai mot mang na-ron Moi aa-ron chi co the co hai trana thai kich hoat (active) hoac khong kich hoat (passive), tucmg d u a n e he nhi phan cac m ay tinh va cac thiet bj dien tu s thong dune Q ua trinh liru trCr thongtin bo nao d uac hinh n h a hoat done cua Chuang 1: M d dau cac sy-nap Den nguai ta biet rang, moi na-ron chira nhieu han bit Theo Schaefer thi moi na-ron co dung lugng khoang 102 bit N hu vay mot bo nao trung binh vai khoang 10 10 na-ron co the luu giu 1000 Gbit thongtinLythuyetthongtin cung da tra lai cau hoi, mot he thong tu thich nghi hay mot he thong tu hoc co the co bac bang bao nhieu moi truang cua no D uai goc nhin cudUy thuyetthong tin, nguai ta co the giai thich van de mot each kha thuyet phuc Truac het chung ta tarn sir dung khai niem du thira va luang tin se duac lam sang to a cac phan sau Ta dat: r = 1— I I ( 1- 1) Trong bieu thuc (1.1-1), r la du thira, I la lugng tin thuc te cua nguon tinva Imax la lugng tin cuc dai co the co Doi vai cac he thong tu thich nghi hoac cac he thong tu hoc, ta luon co: f d\^ dr — > hay - dt dt -I at >o ( - 1- ) iL Tir (1.1-2) suy ra: dl, r*t > i m a x *- v at (1.1-3) Bieu thirc (1.1-3) noi len mot cach tong quat rang, gia tri ciia trang thai thongtinva su bien thien ciia no lien quan chat che vai Tir day ta co the rut ket luan rang, neu lugng tin cuc dai ciia mot he thong tu thich nghi hay he thong tu hoc khong thay doi thi lugng tin thuc te se giant NghTa la: = const dl a —— > dt (M -5) Cac bieu thuc rat don gian vira trinh bay tren cho phep giai quyet mot van de da ton tai rat lau ljch sir nhan loai nhung chua d u a c tra lai mot cach thoa dang, la vi cac cau true sinh hoc co kha nang thich ung vai moi tru ang va tu phat trien Cac cau true sinh hoc khac cac cau true vo c a a cho no co kha nang chong lai su tang en-tro-py Nhieu nghien ciru lythuyetthongtin tren tinh tinh va nguai da di den ket luan, kha nang nhan biet sir vat la ket qua cua mot qua trinh hoc rat da dang Cac thi nghiem da cho thay rang, mot nguai mu bam sinh co dau oc tucmg doi thong minh, sau tru ang d u a c phau thuat mat va nhin duac, nhung ho phai can nhieu thang de co the nhan cac vat het sue dan gian, nguai binh thuang lam d u a c viec tir cai nhin dau tien Doi vai tinh tinh va tre nho, nhung nghien ciru da cho thay rang, quay mot tam giac di mot goc 90 thi tre nho va tinh tinh cung phai nghieng dau di 90 m nhan duac Tom lai, khong co qua trinh hoc thi khong co su giam en-tro-py thongtin Trong dai song hang ngay, nguai da tiep xuc vai rat nhieu hien tugng m a ho khong d u doan truac d uac, hoac chi doan truac d u ac mot cach m a ho, khong cu the Mot can loc xoay ap den bat nga mot tran m u a lut chua tirng thay lich sir hang tram nam, mot tau bong nhien mat tich ngoai bien khai Cac nha ky thuat th u a n g gap nhirng hien tugng kho chiu m a ho goi la tap am (noise) hoac sir thay doi bat thuang (fluctuation) Tat ca nhung hien tugng tren it nhieu lien quan den khai niem en-tro-py ta vira nhac den va se dugc lam ro dan nhirng phan sau cua cuon sach Xet a mot goc nao do, muc dich cuoi cung cua lythuyetthongtin chinh la giup ngu viec lam giam en-tro-py thongtin Khai niem en-tro-py dung lythuyetthongtinma chunt; ta vira goi la en-tro-py thong tin, co nguon goc tir khai niem en-tro-p\ troniz nhiet dong hoc Dinh luat thir cua nhiet dong hoc chi rang, nhiet chi co the truyen tir nai co nhiet cao han den nai co nhiet thap hon \ a khong the ngugc lai Trong tu nhien nguai ta nhan thay co hai loai qua Chuang 1: Ma dau 11 trinh: qua trinh dao nguac duac (Reversible Process) va qua trinh khong dao nguac dupe (Non-Reversible Process) Qua trinh truyen nhiet la mot qua trinh khong dao nguac duac Chung ta se phac hoa rat so luge khai niem en-tro-py cua nhiet dong hoc Goi S la trang thai nhiet cua mot he thongva Q la nhiet lugng, ta co: Lay tich phan tir trang thai Si den trang thai S 2, ta co: f ^ Q = s - S |= A S s T (1.1-7) AS dugc goi la en-tro-py Nam 1829, Cac-no, mot nha khoa hoc nguai Phap, da chirng minh rang, mot he thong kin, AS > Dinh luat ve sau dugc m a rong thanh: mot he thong kin, en-tro-py khong tu giam theo nghTa chung nhat N h u vay nghTa la, mot he thong khong tiep xuc vai mot he nao khac (khong co quan he trao doi vai moi truang cua no), luon luon co xu hucrng tra ve trang thai xac suat dong deu, trang thai co en-tro-py cuc dai Trang thai xac suat dong deu la trang thai hoan toan hon loan Theo dinh luat nay, cac he thong kin cuoi cung se rai vao trang thai hon loan va huy diet Trong ky thuat la su hao mon, sinh hoc la su gia coi, hoa hoc la sir phan huy, xa hoi la sir phan hoa, lich sir la sir suy tan Dieu cung dung vai cac he thongthongtinma a sir tang en-tro-py thongtin se dan tai sir bat dinh hoan toan 1.2 NHUNG DINH HITONG CHINH CUA LYTHUYETTHONGTINVAMAHOALythuyetthongtin de cap den tat ca cac hinh thai van dong cua thongtin nhu: qua trinh hinh thongtin ciia mot nguon tin qua trinh thu nhan thong tin, qua trinh bien doi thong tin, qua trinh truyen dan thong tin, qua trinh xir lythongtinva qua trinh luu trir thongtin Nhimg qua trinh co the dien mot cach tuang minh nhu cac qua trinh thongtin ky thuat vien thong, nhung cung co khong tu ang minh nhir Chuang 5: M d hoa kenh rai rac 135 gay loi Day la mot nguyen tac rat quan nhung dang dec la hay bj bo qua xay dung cac mo hinh kenh vat ly Hinh 5.5-5: Cac giai han xdc suat loi giai md 5.4 GIAI MA TRONG TRUOTVG HOP NHIEU MA TlT KENH 5.4.1 Gioi han tren cua xac suat loi giai ma Trong muc 5.3 chung ta da gia thiet rang chi co hai ma tu kenh duoc truyen qua kenh Trong muc ta se xet giai han tren cua xac suat loi giai ma kenh Pe m co nhieu ma tu dugc truyen qua kenh Truac het chung ta nhac lai mot ket qua cua lythuyet xac suat qua dinh ly sau day D inh ly 5.4-1: Kv hieu P(A/), P(Aj) P(A m) la xdc suat ciia mot tap cac bien cd A, vd cho truac mot so duang < p < J Xdc suat cua bien cd hap M A= Am thoa man bat dang thuc: m=I Lythuyetthongtin vd md hoa 136 (5.4-1) /,( - ) = / { l K ] £ £m=pI ( ) Vrn=l J Ta se ap dung djnh ly de chung minh djnh ly mS hoa kenh sS dugc phat bieu sau day Goi Pn(Y /X ) la phan bo xac suat chuyen tai nhung chuoi co dai N > tren mot kenh rcri rac Ky hieu Q n(X ) la xac suat xuat hien mot chuoi dau vao Tap co M > m a tu kenh co dai bang N D jnh ly 5.3-2 Gia thiet mot thong diep m bat ky (1 i P N < Y / X m> /v 0 (5.4-6) s Vi X m, tong (5.4-6) la mot bien cam nen ta co the bo qua Vi co tit ca M -1 kha nang chon m V m , thay the (5.4-6) vao (5.4-5) ta thu duoc: P ( e / m , X m, Y ) < ( m - i ) ^ ] q n (X) p* ( Y / X ) r N pn ( y / x j s (5.4-7) Tiep tuc thay the (5.4-7) vao (5.4-3) ta thu dugc l Q N( X J P N( Y / X J ' - - X Q » ( X ) P b( Y / X ) - (5.4-8) Luu y rSng, n lu PN( Y / X n,) = thi s6 hang tuong ung se khong xuat hien bieu thuc (5.4-8) Do vay, ta co the dat PN(Y / X m)1_sp = n lu nhu PN(Y / X m) = Cuoi cung chung ta thay s = 1/(1 + p) bieu thuc (5.4-8) va ghi nhan rang Xm cung la mot bien cam tong nen ta suy (5.4-2) Djnh ly tren day kha tong quat va co pham vi ap dung kha rong rai No co th i vua ap dung cho cac kenh khong nho va cac kenh co nho Doi voi trucrng hgp kenh roi rac khong nho ta co PN( Y / X ) = n p( y / x „) (5.4-9) n Gpi Q(k) (k = 1, 2, , K - l) la phan bo xac suat dau vao Vi cac ma tu dugc chon mot cach doc lap nen: Lythuyetthongtinvamahoa 138 (5.4-10) < w x )= riQ < * > 11=1 Khi ta co: Up P , „ S ( M - I ) PX x ( x - l f l Q ( x n) P ( y n/ x „ ) >1 yN I '| I (Np) ' n n=l Up = ( M - i y n ^ l Q ( x n) P ( y „ / x „ ) w * »=' y» L '» J-l = (M - l ) p p) Up K-l X Q t k t P f j / k ) 1" * 1" j=o (5.4-11) k=0 Truoc tiep tuc nghien ciru xac suat loi giai m a kenh lien quan den toe ma khoi, chung ta luu y rang toe ma khoi duoc dinh nghTa boi R = (In M/N) Dieu co nghTa la M = e NR, R khong doi thi M se tang theo so mu N va M khong phai la mot so nguyen Ky hieu int(v) la so nguyen nho nhat bang hoac Ion hon v Voi mot so nguyen N va mot so duong R y, ma khoi (N, R) la mot ma co dai khoi la N va bao gom int(eNR) ma tu Bieu thirc (5.4-11) co the viet lai nhu sau: J-l Pe m < e NRp K-l Up ^ Q ( k ) P ( j / k ) ' ll+pl j=0 (5.4-12) k=0 Bieu thirc (5.4-12) co the phat bieu qua dinh ly sau: Dinh ly 5.4-3 Cho mot kenh roi rac khong nho voi xdc suat chuyen tai P(j k) Voi cac so nguyen duong A va R bat ky■xay dung dupe tap cac md tu cua md khoi ( \ , R) Cac md tu cua tap dupe lay mot cach doc lap voi xdc sudt Q(k) Mot thong diep m, < m < int(eSR) dupe chuyen tic nguon den bo md hoa kenh C o che giai md o dau cua kenh Id co che hop ly nhat Khi voi moi < p < xdc sudt loi giai md trung binh thoa man bat dang thicc: K ,, Z e x p { - X [ E , ( p , Q) - p R ] } (5.4-13) 139 Chuang 5: Md hoa kenh rai rac Trong do: -ii+p £ 0(p > = - l n X Y Q ( k ) P ( j I k ) " " 1" /=0 k=Q (5.4-14) Vi bieu thue (5.4-13) thoa man cho moi thong diep di den bo mahoa kenh, chung ta thay rang xac suat loi trung binh tren tap cac thong diep voi xac suat xuat hien Pr(m) thoa man: m _ P« = £ P r (r n ) P ,m < e x p j - N [ E „ ( p , Q ) - p R ] ( (5.4-15) m=l Vi cac bieu thuc tren p va Q la y, gioi han xac suat loi se duoc xac dinh thong qua cuc dai hoa [E 0( p , Q ) - p R ] , Chung ta dinh nghTa so mu md hoa ngdu nhien (random coding exponent) boi bieu thuc: E r(R ) = max m a x { E ( p , Q ) - p R } Ospsl Q (5.4-16) Trong (5.4-16) cuc dai hoa theo Q duoc thuc hien theo nghTa tim mot phan bo xac suat [Q(0), Q(l), , Q (K -l)] cho (5.4-16) nhan gia trj cuc dai Voi cach lam cho ta ket qua: Pe m < e x p { - N E , ( R ) } ; < m < M (5.4-17) Pc < e x p { - N E r(R)} (5.4-18) Hinh 5.4-1 phac hoa thi cua so mu mahoa ngau nhien cho hai truong hop Ta co the thay rang Er(R) > voi moi R Trong hinh 5.4-1 C la thong lugng kenh tinh bang don vi nat Tren hinh ve ta thay, neu chon ma mot cach thich hgp ta co the lam cho xac suat loi giai ma tiem can theo ham mu bang cach tang dai khoi giu toe ma khoi nho hon thong lugng kenh Vi xac suat loi trung binh tren toan tap ma tu kenh thoa man bat dang thuc (5.4-18) nen ta thay rang it nhat ton tai mot ma cung cap xac suat loi giai ma nho Tuy nhien chung ta chua biet lam cach nao de tim dugc mS Viec chon ma ngau nhien co the dan toi xac suat loi giai ma Ion hon xac suat loi giai ma trung binh Su dung bat dang thuc Chebyshev ta co: Lythuyetthongtinva md hoa 140 EJ1.Q) maxE0(p, Q) Q = (1/2, 1/2, 1/2) Hinh 5.4-1: So mu md hoa ngdu nhien Pr |P, > fiP} < — [X (5.4-19) Ket luan tren day rat quan doi voi viec ung dung thuc te Ket luan tren toat len rang, van de kho khan khong phai o cho tim duoc mot ma tot ma cho tim duoc ky thuat mahoava giai m a cho mot ma mong muon N ket luan tren cung cho phep ta di sau vao ban chat cua quan he: P , = > ( m ) P cm (5.4-20) m doi voi ma chon ngau nhien 5.4.2 Gioi han diroi ciia xac suat loi giai m§ Trong cac phan tren chung ta da nghien cuu gioi han tren cua xac suat loi giai ma Sau day chung ta se nghien cuu gioi han ducri cua xac suat loi giai ma doi voi cac kenh roi rac khong nho, n hu la mot tham so phu thuoc vao dai khoi N va toe ma khoi R C ta se tap trung vao viec xac dinh gioi han ducri cua xac suat loi giai ma doi vcri ma khoi (N, R) Trong toan bo phan ta gia thiet rang, sir chon lua cac ma tu la dong kha nang Giai thiet lam cho gioi han duoi ciia xac suat loi giai 141 Chuang 5: Md hoa kenh rai rac ma khong tam thuang, boi vi khong co gia thiet thi gioi han duai se bang Ta co the thay thirc te mot each true quari rat don gian Ch&ng han neu xac suat lay mot ma tu nao bang thi voi co che giai ma hop ly nhat bo giai ma se luon luon cho ket qua dung va gioi han duoi ciia xac suat loi giai ma se bang Doi voi cac thong diep gan giong nhau, xac suat loi giai ma ung voi mot ban m a chua M thong diep ta co: i m pe = — x/f y 1p e ’ m (5.4-21) Trong (5.4-21) Pe m la xac suat loi giai ma biet thong diep m duoc giri den bo mahoa kenh Trong cac phan truoc ta da biet, voi mot khoi co dai N va toe R > ton tai mot ma (N, R) voi gioi han tren cua xac suat loi giai ma tuan theo bat dang thuc: Pe < e x p { - N E r(R)} (5.4-22) Gioi han duoi cua Pe kho xac dinh hon so voi gioi han tren Ta se neu day mot dinh ly Shannon phat bieu D inh ty 5.4-4 Ky hieu -il+p E „(p,Q ) = - m £ 1=0 £ Q ( k ) P ( j / k ) l'|,‘ p) ( R ) = SUP \ max E0(p, Q) - pR P>° { (5.4-23) k=0 Q xr, In K \ n N oA N ) = -+ ■ Ar) = (5.4-24) (5.4-25) (5.4-26) Pmin Id xdc sudt chuyen tai nho nhat (Ian hon 0) cua kenh vd K Id lan cua an-pha-be dau vao Khi giai han duai cua xdc sudt loi giai md (N, R) tren kenh rai rac khong nha duac xdc dinh bai: Lythuyetthongtinva md hoa 142 Pt > e x p { - N [ E ^ ( R - o , ( N ) ) ] + o ; (N)} (5.4-27) Can luu y rang, Esp(R) dugc goi la so mu bao cau (sphere-packing exponent) Ve co ban Esp(R) dugc djnh nghTa giong nhu so mu mahoa ngau nhien (random-coding exponent) Er(R), chi khac o khoang bien doi cua p ma tren toan tir max dugc thuc hien Ngoai ta thay rang Esp(R) la mot ham loi va nghjch bien voi R khoang < R < C Hinh 5.4-2: Esp(R) vd Er(R) cuu cac md hinh kenh rai rac Hinh 5.4-2 phac hoa sir so sanh giua Esp(R) va Er(R) cho mot so kenh tieu bieu Tren hinh ve ta thay, Esp(R) d4n toi vo cung t c ma khoi dan toi mot gia tri Rx nao De tim R x ta dien dat: m a x E 0( p , Q ) - p R (5.4-28) nhu la mot tap cac ham phu thuoc R vcri p > la mot tham s True R cua ham tren ung voi mot p cho truoc se la: max E ,(p,Q) / p (5.4-29) 143 Chuang 5: Md hoa kenh roi rac Khi p -> oo, nhung d uang thang dan toi vo cung Vi Esp(R) la mot bao loi cua cac ham neu tren, R*, se duoc xac dinh boi true R bi gioi han p —►oo R„ = lim max E ° -^ p-*« o p )- (5.4-30) Su dung luat L ’Hospital de lay gioi han (5.4-30) ta thu duoc: oo max ^ Q(k) Rcr Dieu dan toi ket luan rang, ton tai khoang gia tri toe ma khoi Rcr < R < C cho xac suat loi giai ma phu thuoc theo ham mu vao N Ta dinh nghTa ham tin cay (reliability function) cua kenh nhu sau: - l n ( P e(N, R ) E (R ) = lim s u p - ^ N— ^ ( 4- 33) Trong PC(N, R) la gia tri nho nhat cua Pc tren moi ma (N, R) ung voi mot cap (N, R) cho trucrc Ham tin cay E(R) co gioi han duoi va gioi han tren tuong ung la Er(R) va Esp(R) Co the xay tinh trang R ^ = C Dieu ham nghTa la, voi nhung gia tri y cua R gan vcri C thi: Lythuyetthongtin vd md hoa 144 [max E 0( p , Q ) - p R ] (5.4-34) khong phai dirge cue dai hoa theo p khoang < p < Dieu cung co nghTa duong chan tren true R se Ion hon h o ic bang C voi nhung gia trj p > 1, hay noi cach khac: max E (p, Q) / p tiem can toi C p -> oo (5.4-35) Trong cac truong hgp khac, vi E 0(p, Q) loi p Q co djnh nen ta se phai co E 0(p, Q) = pC ung vcri mot vai su lira chon Q nao Tir cac lap luan tren day ta co djnh ly sau day D inh ly 5.4-5: Ky hieu , xrx In In + ATIn TV o3( N) = - j = + Jn n (5.4-36 J K m ax[ - In Y P ( j / i ) P( j I k ) ] t k oa( N) =,/log, ~Jn ,+ l n + l n + ^ < > A fln ' L -I n N (5_4 _37) N Cho mot kenh roi rac khong nho voi Eex(0) < oo Goi Esi(R) Id ham tuyen tinh nho nhat phu thuoc R gap duong cong Esp(R) vd thoa man dieu kien E J 0) = Eex(0) Ky hieu R/ la gia tri R thoa man Es/(R) = E$P(R) Khi voi so nguyen duong N bdt ky vd s