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7/31/2019 Dieu Che FM

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CSVin Thng Phm Vn Tn

Trang V.2

TN S TC THI.

Xem mt sng mang cha b bin iu

sC(t) = A cos(2fCt + ) (5.1)Nu fC b thay i ty theo thng tin m ta mun truyn, sng mang c ni l c biniu tn s. Cn nu b lm thay i, sng mang b bin iu pha. Nhng nu khi fC hay bthay i theo thi gian, th sC(t) khng cn l Sinusoide na. Vy nh ngha v tn s m tadng trc y cn c ci bin cho ph hp.Xem 3 hm thi gian:

s1(t) = A cos 6t (5.2a)s2(t) = A cos (6t +5) (5.2b)s3(t) = A cos (2t e

-t ) (5.2c)

Tn s ca s1(t) v s2(t) r rng l 3Hz. Tn s ca s3(t) hin ti cha xc nh. nh nghatruyn thng ca ta v tn s khng p dng c cho loi sng ny. Vy cn mrng khi nim

v tn s p dng cho nhng trng hp m tn s khng l hng.Ta nh ngha tn stc thi theo cch c th p dng c cho cc sng tng qut. Tn s tcthi c nh ngha nh l nhp thay i ca pha.

t s(t) = A cos (t) dt

d)t(f2 i

= (5.3)

fi : tn s tc thi, Hz. Nhl c 2 v ca phng trnh (5.3) c n v l rad/sec.Nh vy trong th d trn, tn s tc thi ca cc tn hiu cho ln lt l 3Hz; 3Hz v e-t

(1 - t) Hz.

Th d 1:Tm tn s tc thi ca cc sng sau:


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Trang V.3

Hnh 5.2

Th d 2. Tm tn s tc thi ca hm sau y:s(t) = 10 cos2[1000t + sin 10t ]

Gii:Ap dng nh ngha tm:

t10cos101000dt

d

2

1)t(fi +=

=

fic vhnh 5.3.

Hnh 5.3

BIN IU TN S (FREQUENCY MODULATION).

Bin iu FM c pht minh bi Edwin Armstrong nm 1933 [cng l ngi pht minhmy thu kiu i tn (superheterodyne - siu phch)]. Trong bin iu FM, ta bin iu tn stc thi fi (t) bi tn hiu s(t). V cng v c th tch bit cc i vi nhau, ta phi di tn s(t)ln n tn s sng mang fC.

Ta nh ngha bin iu FM nh l mt sng vi tn s tc thi nh sau:fi (t) = fC + Kfs(t) (5.5)

Trong : fC l tn s sng mang (hng s) v Kf l hng s t l, thay i theo bin cas(t). Nu s(t) tnh bng volt, Kfc n v l Hz/v hoc 1/v.sec .

V tn s l o hm ca pha, nn

(t) = 2 fo

t

i ()d = 2 [fCt + Kf ot

s()d] (5.6)Gi siu kin u bng zero, sng bin iu c dng:

fm(t) = A cos (t).

+=

t

0

fcf d)(sKtf2cosA)t(m (5.7)

Nhl, nu t s(t) = 0, phng (5.7) s thnh mt sng mang thun ty.Td . V sng AMSC v FM cho cc tn hiu thng tin nh hnh 5.4.Gii:


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Trang V.4

Hnh 5.4

m1(t)

sm1(t)

s1(t)

t

sm2(t)

s2(t)

t

m2(t)

Hnh 5.4

Tn s ca fm(t) thay i t fC + Kf[min . s(t)] n fC + Kf[max . s(t)].


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Trang V.5

Bng cch lm cho Kf nh mt cch ty , th tn s ca fm(t) c thc gi mt cchty xung quanh fC. iu lm tit gim c kh bng.

Nh l s bin iu th khng tuyn tnh cho s(t). Nu thay s(t) trong phng trnh (5.7)bng mt tng gm nhiu tn hiu th sng FM kt qu khng l tng ca cc sng FM thnhphn. iu ng, v:

Cos (A + B) cosA + cosB.Ta chia bin iu FM lm 2 nhm; ty thuc vo cca Kf. Vi Kf rt nh ta c FM bng

hp; v Kf ln ta c FM bng rng.

BIN IU PHA.

Khng c s khc bit cbn gia bin iu pha v bin iu tn s. Hai ty thngc dng thay i cho nhau. Bin iu mt pha bng mt sng th cng nh bin iu o hmca n (tn s) vi sng y.

Sng bin iu pha cng c dng:pm(t) = A cos (t).

Trong (t) c bin iu bi s(t). Vy:(t) =2 [fCt + Kp s(t)] (5.8)

Hng s t l Kp c n v V-1. Sng PM c dng:

(5.9)pm(t) = A cos 2 [fCt + Kp s(t)]

Khi s(t) = 0, sng PM trthnh sng mang thun ty.Ta c th lin h PM vi FM bng cch dng nh ngha ca tn s tc thi:

fi (t) = fC + Kpds

dt

(5.10)

Trng rt ging vi (5.5), trng hp ca FM.Thc vy, khng c s khc bit gia vic bin iu tn s mt sng mang bng s(t) v

vic bin iu pha ca cng sng mang bng tch phn ca s(t). Ngc li khng c g khcnhau gia vic bin iu pha ca mt sng mang bng s(t) v bin iu tn s cng sng mangy bng o hm ca s(t).

V vy, tt c cc kt qu sau y th chuyn d dng gia 2 loi bin iu.

FM BNG HP (NARROW BAND FM).

Nu Kfrt b, ta c th dng php tnh xp xn gin phng trnh ca sng FM.

=

t

0fcf )ds(K+tf2cosA)t(m (5.11)

trnh vic lp li nhiu ln, ta t g(t) l tch phn ca tn hiu cha tin.

= t

0

d)(s)t(g (5.12)

Phng trnh (5.11) trnn:

fm(t) = A cos 2[ ]f t + K g(t)c f (5.13)

Dng lng gic, khai trin hm cosine:


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Trang V.6

fm(t) = Acos2fCt . cos2Kfg(t) - A sin2fCt . sin2Kfg(t) (5.14)

Cosine ca mt gc b 1. Trong khi sin ca n gn bng chnh n.

Vy, nu Kf nh sao cho 2Kf g(t) biu din cho mt gc rt nh, ta c th tnh xp x

phng trnh (5.14):

fm(t) Acos2fCt - 2Ag(t) Kfsin2fCt (5.15)

Php tnh ny tuyn tnh vi g(t) v nh vy tuyn tnh vi s(t). Ta c th tnh bin i F

ca n (vi mt t kh khn) nh sau:

Bin i F ca g(t) lin h vi s(t) bi:

G(f) =S(f)

j2 f

Ly bin i F ca (5.15):

( ) ( )[ ] ( ) ( )

++

+++=

fcf

ffS

fcf

ffS

j4

AK2ffff

2

Afm(f) ccfcc (5.16)

Hnh 5.5: Bin i F ca sng FM.FM bng hp c 3 vn :- Tn s c th tng cao n mc cn thit truyn i c hiu qa, bng cch

iu chnh fCn tr mong mun.- Nu tn s sng mang ca ngun tin ln cn cch n t nht 2fm, th cc tn hiu

cha nhng ngun tin khc nhau c th truyn cng lc trn cng mt knh.- s(t) c th hi phc t sng bin iu. V phn sau ta s thy, cng mt khi

hon iu c th tch sng cho FM trong c 2 trng hp Kfnh v Kfln.Kh bng ca sng FM l 2fm, ng nh trng hp AM hai cnh. Th d dng ting hut

so (ti a 5000Hz) bin iu mt sng mang. Gi s s di tn ti a l 1Hz. Nh vy, tns tc thi thay i t (fC - 1)Hz n (fC + 1)Hz. Bin i F ca sng FM chim mt bng gia

(fC - 5000)Hz v (fC + 5000)Hz.R rng, tn s tc thi v cch thc m n thay i gp phn (c 2) vo kh bng ca

FM.

Gi l Bng hpkhi Kf nh, l v khi Kf tng, kh bng s tng t tr ti thiu 2fm.PM BNG HP.

Bin iu pha bng s(t) th ging nh bin iu tn s bng o hm ca s(t). V o hmca s(t) cha cng khong tn s nh s(t), nn kh bng ca PM bng hp cng chim vng tns t gia fC - fm v fC + fm. Tc l kh bng rng 2fm.

Vi FM bng hp, tr max ca 2kf g(t) l mt gc rt nh (Trong g(t) l tch phn cas(t)).


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Trang V.7

Vi PM bng hp, 2Kp s(t) phi l mt gc rt nh. iu ny cho php tnh xp x cosinev sine (s hng th nht trong chui khai trin).

FM BNG RNG (WIDE BAND FM).

Nu Kfnh khng cho php tnh xp x nhphn trn, ta c FM bng rng. Tn

hiu c truynfm(t) = A cos 2[ ]fct + Kfg(t) (5.17)

Trong g(t) l tch phn ca tn hiu cha tin s(t). Nu g(t) l mt hm bit, bin i Fca sng FM s tnh c. Nhng trong nhng trng hp tng qut, khng th tm bin i Fcho sng FM, v s lin h phi tuyn gia s(t) v sng bin iu. Nhng phn gii thc hintrong phm vi thi gian.

Ta gii hn trong mt trng hp ring, dng tn hiu mang tin l mt Sinusoide thun ty.iu ny cho php dng lng gic trong phn gii.

S(t) = a cos 2fmta: hng s bin .

Tn s tc thi ca sng FM c cho bi:fi (t) = fC + aKfcos 2fmt (5.18)

Sng FM c dng:

fm(t) = A cos 2 fctaKff

sin2 f m tm

+

(5.19)

Ta nh ngha ch s bin iu :

aK

f

fm

, : khng n v (5.20)

fm(t) = A cos (2fCt + sin2fmt)fm(t) = Re {A exp (j2fCt +j sin 2fmt)} (5.21)

Hm expo trong (5.21) phn thnh mt tch, trong tha s th 2 c cha tin. l: expo(j sin 2fmt).

l mt hm tun hon, chu k 1/fm.Khai trin chui F phc, tn s fm.

+

=

=n

tf2jn

n

tf2sinj mm eCe (5.22)

H sF cho bi:

=

m

m

mm

f1

f1

tf2jntf2sinjmn dteefC (5.23)

Tch phn ca (5.23) khng tnh c, n hi t ti mt tr gi thc. Tr gi thc l mthm ca n v . N khng phi l mt hm ca fm. Tch phn c gi l hm Bessel loi mt,k hiu Jn().

HM BESSEL.

Hm Bessel loi 1 l gii p ca phng trnh vi phn:

x2

d y

dxx

dy

dx

2

2

++ ( x

2

- n2

) y( x ) = 0

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