2/5/20151Chng 3CHUI FOURIER V BIN I FOURIERNi Dung1. Mt s dng tn hiu quan trng2. Khi nim hm tun hon3. Chui Fourier Khai trin Fourier4. Tch phn Fourier Bin i Fourier5. Phn tch ph tn hiu2/5/20152MT S DNG TN HIUQUAN TRNGMt S Dng Tn Hiu Quan Trng1. Tn hiu xung vung gc H()2. Hm dc (Ramp function) ()3. Hm bc nhy n v ()4. Hm xung lc n v o()5. Hm du - Tn hiu ()6. Tn hiu xung tam gic ()7. Hm m suy gim ()8. Hm m tng dn 1 ()9. Xung hm m10. Tn hiu xung cosin11. Cp phn b o() chn l12. Phn b lc13. Dy xung vung lng cc14. Dy xung vung n cc15. Tn hiu Sin suy gim theo hm m16. Tn hiu 17. Tn hiu , 18. Tn hiu 2 , 2 2/5/20153Mt S Dng Tn Hiu Quan Trng1. Xung vung gc H() 2. Hm dc ()212110) ( ) ( t t x H == [ =21t , 121,0) ( ) (tt t xc0) ( . ) (bc ta t xH =ba) ( . ) (bc ta t xH ==0 , 00 ,) (tt tt r= a ta t ta t r, 0,) (0ar(t-a)K0r(t)1Mt S Dng Tn Hiu Quan Trng3. Bc nhy n v ()10 t> 10 0 1 0

0 +1 1 = () 0 = 1 0, 0 10 , 0 > 11-1

0 = () = 1 , > 00 , = 01 , < 02/5/20155Mt S Dng Tn Hiu Quan Trng7. Hm m suy gim 8. Hm m tng dn = . . > 0

0 ()

0 () = 1 . > 0Mt S Dng Tn Hiu Quan Trng9. Xung hm m 10. Xung hm Cos = . .

2

> 0

0 ()

0 ()

20 20 = . cos(0) .

/02/5/20156Mt S Dng Tn Hiu Quan Trng11.Cp phn b o() chn l|()|||() 0

12

12 12 0

12 12 12|| = 12 +12+ 12= 12 +12 12

12Mt S Dng Tn Hiu Quan Trng12.Phn b lc Tnh cht phn b lc Hm chn ||| = ||| Tun hon ||| + = ||| Thay i thang |||

= = Ri rc ||| = =() Lp tun hon ||| = = ||| = = .... ....5 4 3 2 1 0 1 2 3 4 5 ||| 2/5/20157Mt S Dng Tn Hiu Quan Trng13.Dy xung vung lngcc14.Dy xung vung ncc... ... 20

2 ... ...2 0 2

Mt S Dng Tn Hiu Quan Trng15.Tn hiu Sin suy gimtheo hm m16.Tn hiu Gausse = . . sin 0 . > 0 = 22/5/20158Mt S Dng Tn Hiu Quan Trng17.Tn hiu () Tn hiu ()()11 32 43 12 4

= sin

, 01 , = 0 = sin

, 01 , = 0Mt S Dng Tn Hiu Quan Trng18.Tn hiu 2 Tn hiu 2

2()1

2 1 2

2 = sin2

2, 01 , = 0

2 = sin2

2, 01 , = 02/5/20159Tnh Cht Hm Xung Lc1. Quan h vi hm ()

0

= = 2. Thay i thang

0= 0 3. Tnh i xng = 4. Nhn i s = 0 0 = 0 05. Nhn tch phn

= 0

0 = 06. Nhn tch chp = = = 0 = ( 0)3.2 KHI NIM HM TUN HON 2/5/201510Khi Nim Hm Tun Hon = + Mt tn hiu tun hon c thuc tnh l khng i khidch thi gian mt on . Chu k c bn: l gi tr nh dng nh nht mhm cn tun hon. Hm M Phc Tun Hon =

Vi l thc v = 0l s thun o. Bin v pha c v (biu din) tch bit.2/5/2015113.3 CHUI FOURIER KHAI TRIN FOURIERChui Fourier Chui lng gic:

02 + =1

cos +

sin

0

; 1

: h s (l cc hng s); = 1 Chui lng gic m rng:

02 + =1

cos

+

sin

0

; 1

: h s (l cc hng s); = 2

=

2/5/201512nh L Dirichlet Nu l hm tun hon, b chn v c mt sim xc nh khng lin tc trong mt chu k can th khi chui FOURIER ca s hi t n ti tt c nhng im m lin tc. Cn tinhng im m khng lin tc, chuiFOURIER ca n s hi t n gi tr trung bnh cagiihn tri v giihn phica tc nu tiim = 0hm s b gin on th:

= 0 =lim0 + lim0+ 2= 0 + 0+2

02 + =1

cos

+

sin

=

1 = = +2

1 < <

2 =

2 <

3 < < +22/5/201513Khai Trin Fourier Cho hm s f(x) tun hon vi chu k T = 2p tha iukin Dirichlet. Khi hm f(x) c th biu din didng chui Fouier theo cng thc sau: = 02 + =1

cos

+

sin

;

0 = 1

2

= 1

2 cos

= 1

2 sin

, 0,

,

Cng Thc Euler M Rnga. 2cos

= 0; 0b. 2sin

= 0c. 2cos

cos

= 0; d.2cos2

= e. 2cos

sin

= 0f.2sin

sin

= 0; g.2sin2

= ; 02/5/201514VD: = 0, < 0,1, 0 < , Tm khai trin Fourier. Bit () = ( +2) = 2 = 2 = ; 0 = 1

2 = 1

0

=1;

= 1

0

cos = 0

= 1

0

sin

= cos()

0

= 1

1 cos = 2

; 0; = 12+ 2

sin 1+sin 33+sin 55+VD: = , < 0,, 0 < , Tm khai trin Fourier. Bit () = ( +2)Nhn xt: f(x) tha mn nh lut Dirichlet (tun hon, b chn vc mt s im xc nh khng lin tc trong mt chu k ca n), nn c th khai trin Fourier. = 2 = 2 = ;

0 = 1

0 +0

= 1

0+

220

= 322/5/201515VD: = , < 0,, 0 < ,

= 1

0 cos +0

cos = 1

+ = 0 cos =

sin 0= 0 = 0

cos = 0

= 0

=sin

0

0

sin

= cos

20

= 1

2 cos 1 = 1 1

2= 2

2; 0;

= 1 1

2VD: = , < 0,, 0 < ,

= 1

0 sin +0

sin = 1

+ = 0 sin =

cos 0= 1 cos

= 0

sin = 0

= 0

=cos

0

0

cos

= cos

+ sin

20

= cos

= 1

2/5/201516VD: = , < 0,, 0 < ,khai trin Fourier

0 = 32 ;

= 1 1

2;

= 1

= 12

0 + =1

cos

+

sin

= 1232 + =11 1

2cos 1

sin = 34 + 2

112cos 132cos 3 152cos 5 + sin 1+sin 22+sin 33+Chui Fourier Chn Lnh l 1 Nu f(t) l hm tun honchn theo t th

= 0 v

0 = 1

2 =2

= 1

2 cos

= 2

cos

nh l 2 Nu f(t) l hm tun honl theo t th

= 0 v

= 1

2 sin

= 2

sin

2/5/201517VD: = 0, < 0,, 0 < , Tm khai trin Fourier. Bit () = () = Hm chn

= 0; v

0 = 2

0

= 2

220

=

= 2

0

cos = 2

cos

20

= 2

1 1

2= 4

2; 0; VD: = 0, < 0,, 0 < , = ; 0 =

= 4

2; 0; ;

= 0Khai trin Fourier = 12

0 + =1

cos = 2 4

cos 12+cos 332+cos 552+2/5/201518VD: = 0, < 0,, 0 < , Tm khai trin Fourier. Bit = = Hm l

= 0; v

= 2

0

sin = 2

cos

= 2

cos = 2

1 ==1

sin =2 sin 1sin 22+sin 33Cc Dng Chuyn i Ca KhaiTrin Fourier =

02 +=1

cos

+

sin

=

02 +

2 +

2

=1

2 +

2cos

+

2 +

2sin

t

0 = 02

=

2 +

2

= tan1

= 2

2/5/201519Cc Dng Chuyn i Ca KhaiTrin Fourier =

02 +

2 +

2

=1

2 +

2cos

+

2 +

2sin

=0+

=1cos

cos

+cos

sin

(1)=0+

=1sin

cos

+sin

sin

(2)1 = 0 +

=1cos

: 2 = 0 +

=1sin

+

: Chui Fourier Dng M Phc = 02 + =1

cos

+

sin

= 02 + =1

+

2+

2= 02 + =1

2

+

+

2

t C0 = a02 ; Cn =

2 Cn =

+

2 =0+=1

+

=0+=1

+ =1

= =

2/5/201520Chui Fourier Dng M Phc =0+=1

+ =1

= =

= 12

2

= 12

2

0 = 12

2 VD: = , < /,, / , Tm chui Fourier dng m phc. -/2 /2 2 -2/5/201521VD: = , < /,, / , = ;

= 12

2

= 12

2

2

= 12

2

2= 1

2

22= sin

2

=12 sin

2

2=12sinc 2

0 = 12

2 =12VD: = 0, < /2,1, /2 , = =

= 0 + =1

+ =1

= 0 + =1

+= 12+ =1sinc 2 cos 2/5/201522WD: Phn Tch PhTn Hiu Tun HonPhng php phn tch ph Tm Cn Tnh F(e) =2tCn Khi : th biu din bin ca F(e) theo tn sgi l ph bin ca tn hiu f(t). Biu din gc pha ca F(e) theo cc tn s cachng gi l ph pha ca tn hiu f(t)V DCho hm s () tun hon vi chu k T nhhnh bn di. V ph ca ()2/5/201523V D

= 12

2

= 1

201 2

+ 1

0

2

2

= 1

12

2

20+12

2

0

2=12 1 +

+ 1= 122 2

+2=

1 cos V D Tm F(e): = 2

= 2

1 cos = 2

1 cos = 0 = 24

= 2 +12/5/201524V D Ph bin Ph pha- - - --e e 2e 3e 4e 5e -2e -3e -4e -5e44/33/4e.. ..|F(e)|- - - --ee 2e 3e 4e 5e-2e -3e -4e -5et/2e.. ..argF(e)-t/2+ ===1 2k nn42k n0) F(< + => + ===0 1 2k nn40 1 2k n22k n0) ) F( arg(t