Signals and Systems Fourier Series Representation of Periodic Signals

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1 Signals and Systms Fourir Sris Rprsntation of Priodic Signals Chang-Su Kim

2 Introduction

3 Why do W Nd Fourir Analysis? Th ssnc of Fourir analysis is to rprsnt a signal in trms of complx xponntials x t a a a a a a 0t 2 0t 0t 0t 2 0t Many rasons: Almost any signal can b rprsntd as a sris or sum or intgral of complx xponntials Signal is priodic Fourir Sris DT, CT Signal is non-priodic Fourir Transform DT, CT W will larn ths CTFS, DTFS, CTFT, DTFT on by on Rspons of an LTI systm to a complx xponntial is also a complx xponntial with a scald magnitud.

4 Rspons of LTI systms to Complx Exponntials CT wt H w c H w h d c wt Proprty wt w2t x t a a a 2 3 w t 3 wt w2t y t a H w a H w a H w w t 3 This is usful bcaus almost vry signal can b rprsntd as a sum of complx xponntial functions So w can asily comput th rspons of th systm to almost vry input signal using th abov quation

5 Rspons of LTI systms to Complx Exponntials DT wn H c wn c H w h[ ] w Proprty wn w2n x[ n] a a a 2 3 y t a H a H a H w n 3 w w n w2 w2n w w n This is usful bcaus almost vry signal can b rprsntd as a sum of complx xponntial functions So w can asily comput th rspons of th systm to almost vry input signal using th abov quation

6 Continuous Tim Fourir Sris How to rprsnt a priodic function xt as x t a a a a a a 0t 2 0t 0t 0t 2 0t continuous tim discrt tim priodic sris CTFS DTFS apriodic transform CTFT DTFT

7 Harmonically Rlatd Complx Exponntials Basic priodic signal 0t Fundamntal frquncy: Fundamntal priod: 0 T 2 / 0 Th st of harmonically rlatd complx xponntials 0 t 2 / T t { 0,, 2, 3,...} t t Each of th signals is priodic with T Thus, a linar combination of thm is also priodic with priod T: 0t a

8 Continuous Tim Fourir Sris CTFS Most all in nginring sns functions with priod T=2/w 0 can b rprsntd as a CTFS x t 0t a, T 0t a x t dt T

9 Exampl T = /4T 0 0 T T 0 T T a x 0t 0t t dt dt T0 T 0 0 sin sin 0T 2. T a /5 0 -/3 0 / /2 / 0 -/3 0 /5 x t cos0t cos 30t 2 3 3w0t w0t w0t 3w0t

10 Exampl continud Finit Approximation x N t and Gibbs phnomnon N 0 xn t a N t Ovrshoot is always about 9% of th hight of th discontinuity

11 Exampl 2

12 Exampl 3

13 Continuous Tim Fourir Sris CTFT Drivation of th formula x t 0t a, T 0t a x t dt T

14 Proprtis of CTFS Thr ar 5 proprtis in Tabl 3. in pag 206 of th txtboo Do w hav to rmmbr thm all? No Instad, familiariz yourslf with thm and b abl to driv thm whnvr ncssary Thy all com from th singl formula x t 0t a, T 0t a x t dt T

15 Slctd Proprtis Givn two priodic signals with sam priod T and fundamntal frquncy 0 =2/T: x t y t a b. Linarity: z t Ax t By t Aa Bb 2. Tim-Shifting: z t x t t a 0 t Tim-Rvrsal Flip: z t x t a 4. Conugat Symmtry: z t x t a * *

16 Slctd Proprtis 5. xt is ral and vn a is ral and vn 6. xt is ral and odd a is purly imaginary and odd 7. Multiplication: z t x t y t alb l l 8. Parsval s Rlation: x t dt T T 2 a 2

17 Othr Forms of CTFS ral

18 Discrt Tim Fourir Sris How to rprsnt a priodic function x[n] with priod N as N x[ n] a a a a... a 0 2 / N n 2 / N n 22 / N n N 2 / N n 0 2 N continuous tim discrt tim priodic sris CTFS DTFS apriodic transform CTFT DTFT

19 Discrt Tim Priod Functions with Priod N x[n]=x[n+n] Fundamntal frquncy 0 =2/N { [n] = 0n = 2/Nn : =0, ±, ±2,...} is a st of signals, consisting of all discrt-tim complx xponntials that ar priodic with priod N [n] = [n+n] [n] = +N [n] = +2N [n] =... Only N distinct signals in th st Thrfor, whil an infinit numbr of complx xponntials ar rquird in CTFS, only N complx xponntials ar usd in DTFS.

20 Discrt Tim Fourir Sris DTFS All functions with priod N can b rprsntd as a DTFS x[ n] a N N n N a 2 N x[ n] n 2 N n n N : dnots th sum ovr any intrval of N succssiv valus of n. Driv it!

21 Proprtis

22 Exampl x[n]

23 Fourir Sris and LTI Systms

24 Frquncy Rspons CT DT wt H c wt wn H c wn w c H w h d, x t a t y t a H w 0 0 t 0 w w c H h[ ], x[ n] N 2 N 2 2 N N y[ n] a H a N n n w H w or H ar calld frquncy rsponss.

25 Exampl Exampl 3.6 in pp. 228 in txtboo

26 Filtring Filtring is a procss that changs th amplitud and phas of frquncy componnts of an input signal All LTI systms can b thought as filtrs Ex Diffrntiator H dx t y t H dt High frquncy componnt is magnifid, whil low frquncy componnt is supprssd It is a ind of highpass filtr /2 -/2 H

27 Lowpass, Bandpass and Highpass Filtrs CT stop band H c pass band stop band H c H Idal low-pass filtr LPF H = in pass band 0 in stop band Idal high-pass filtr HPF Idal band-pass filtr BPF c c2

28 Lowpass, Bandpass and Highpass Filtrs DT LPF H HPF H - c 2-2 BPF H - 2 H is priodic with priod 2 low frquncis: at around =0, 2,... high frquncis: at around =, 3,...

29 Lowpass Filtring

30 Highpass Filtring

31 Exampl of CT Filtr RC lowpass filtr V s + V r - V c + - : output : input t V t V t V t V dt t dv RC c s s c c tan 2 RC t t t t t t RC RC H H H RC H H dt d RC

32 Exampl of CT Filtr continud It is a lowpass filtr H RC RC 2 tan RC H H /2 /RC -/4 -/2

33 Exampl of DT Filtr y[n] - ay[n-] = x[n] W now if x[n]= n thn y[n]=h n n n n a H ah H

34 Exampl of DT Filtr Continud H H a H H H /2 -/2 a a=0.6 LPF b a=-0.6 HPF

35 Exampl of DT Filtr 2 M N M N n x n y ] [ ] [,, 0 ] [ othrwis M n N for n h M N M N M N h H ] [

36 Exampl of DT Filtr 2 Continud H a N=M=6 H b N=M=32 It is a lowpass filtr

Introduction to Medical Imaging. Signal Processing Basics. Strange Effects. Ever tried to reduce the size of an image and you got this?

Introduction to Medical Imaging. Signal Processing Basics. Strange Effects. Ever tried to reduce the size of an image and you got this? Strang Effcts Introduction to Mdical Imaging Evr trid to rduc th siz of an imag and you got this? Signal Procssing Basics Klaus Mullr Computr Scinc Dpartmnt Stony Brook Univrsity W call this ffct aliasing