6.003: Signals and Systems. Fourier Representations

Transcription

1 6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20

2 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers.

3 Fourier Series Represening signals by heir harmonic componens. ω 0 2ω0 3ω 0 4ω 0 5ω 0 6ω 0 ω harmonic # DC fundamenal second harmonic hird harmonic fourh harmonic fifh harmonic sixh harmonic

4 Musical Insrumens Harmonic conen is naural way o describe some kinds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano cello bassoon oboe horn alosax violin bassoon 252 seconds

5 Musical Insrumens Harmonic conen is naural way o describe some kinds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano cello bassoon k k k oboe horn alosax k k k violin k

6 Musical Insrumens Harmonic conen is naural way o describe some kinds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano piano violin violin k bassoon bassoon k k

7 Harmonics Harmonic srucure deermines consonance and dissonance. ocave (D+D ) fifh (D+A) D+E ime(periods of "D") D' A E D D harmonics D

8 Harmonic Represenaions Wha signals can be represened by sums of harmonic componens? ω 0 2ω0 3ω 0 4ω 0 5ω 0 6ω 0 ω T = 2π ω 0 T = 2π ω 0 Only periodic signals: all harmonics of ω 0 are periodic in T = 2π/ω 0.

9 Harmonic Represenaions Is i possible o represen ALL periodic signals wih harmonics? Wha abou disconinuous signals? 2π ω 0 Fourier claimed YES even hough all harmonics are coninuous! Lagrange ridiculed he idea ha a disconinuous signal could be wrien as a sum of coninuous signals. We will assume he answer is YES and see if he answer makes sense. 2π ω 0

10 Separaing harmonic componens Underlying properies.. Muliplying wo harmonics produces a new harmonic wih he same fundamenal frequency: e jkω 0 e jlω 0 = e j(k+l)ω The inegral of a harmonic over any ime inerval wih lengh equal o a period T is zero unless he harmonic is a DC: 0 +T { 0, k 0 e jkω0 d e jkω0 d = 0 T T, k = 0 = T δ[k]

11 Separaing harmonic componens Assume ha x() is periodic in T and is composed of a weighed sum of harmonics of ω 0 = 2π/T. x() = x( + T ) = a k e jω 0k k= Then x()e jlω0 d = a k e jω0k e jω0l d T T k= = a k e jω0(k l) d k= T = a k T δ[k l] = T a l k= Therefore a k = x()e jω0k d T T = T T j 2π x()e T k d

12 Fourier Series Deermining harmonic componens of a periodic signal. a k = T T x()= x( + T ) = j 2π x()e T k d k= a k e j 2π T k ( analysis equaion) ( synhesis equaion)

13 Check Yourself Le a k represen he Fourier series coefficiens of he following square wave How many of he following saemens are rue?. a k = 0 if k is even 2. a k is real-valued 3. a k decreases wih k 2 4. here are an infinie number of non-zero a k 5. all of he above

14 Check Yourself Le a k represen he Fourier series coefficiens of he following square wave. 2 0 a k = T = j4πk = 2 j 2π x()e T k d = 0 e j2πk d + 2 e j2πk d (2 e jπk e jπk) jπk ; if k is odd 0 ; oherwise

15 Check Yourself Le a k represen he Fourier series coefficiens of he following square wave. { a k = jπk ; if k is odd 0 ; oherwise How many of he following saemens are rue?. a k = 0 if k is even 2. a k is real-valued X 3. a k decreases wih k 2 X 4. here are an infinie number of non-zero a k 5. all of he above X

16 Check Yourself Le a k represen he Fourier series coefficiens of he following square wave How many of he following saemens are rue? 2. a k = 0 if k is even 2. a k is real-valued X 3. a k decreases wih k 2 X 4. here are an infinie number of non-zero a k 5. all of he above X

17 Fourier Series Properies If a signal is differeniaed in ime, is Fourier coefficiens are muliplied by j 2π T k. Proof: Le hen x() = x( + T ) = ẋ() = ẋ( + T ) = k= k= a k e j 2π T k ( j 2π ) T k a k e j 2π T k

18 Check Yourself Le b k represen he Fourier series coefficiens of he following riangle wave How many of he following saemens are rue?. b k = 0 if k is even 2. b k is real-valued 3. b k decreases wih k 2 4. here are an infinie number of non-zero b k 5. all of he above

19 Check Yourself The riangle waveform is he inegral of he square wave Therefore he Fourier coefficiens of he riangle waveform are imes hose of he square wave. b k = jkπ j2πk = 2k 2 π 2 ; k odd j2πk

20 Check Yourself Le b k represen he Fourier series coefficiens of he following riangle wave. b k = 2k 2 π 2 ; k odd How many of he following saemens are rue?. b k = 0 if k is even 2. b k is real-valued 3. b k decreases wih k 2 4. here are an infinie number of non-zero b k 5. all of he above

21 Check Yourself Le b k represen he Fourier series coefficiens of he following riangle wave How many of he following saemens are rue? 5. b k = 0 if k is even 2. b k is real-valued 3. b k decreases wih k 2 4. here are an infinie number of non-zero b k 5. all of he above

22 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 0 k = 0 k odd 2k 2 π 2 e j2πk 0 8

23 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 k = k odd 2k 2 π 2 e j2πk 0 8

24 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 3 k = 3 k odd 2k 2 π 2 e j2πk 0 8

25 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 5 k = 5 k odd 2k 2 π 2 e j2πk 0 8

26 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 7 k = 7 k odd 2k 2 π 2 e j2πk 0 8

27 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 9 k = 9 k odd 2k 2 π 2 e j2πk 0 8

28 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 9 k = 9 k odd 2k 2 π 2 e j2πk 0 8

29 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 29 k = 29 k odd 2k 2 π 2 e j2πk 0 8

30 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: riangle waveform 8 39 k = 39 k odd 2k 2 π 2 e j2πk 0 8 Fourier series represenaions of funcions wih disconinuous slopes converge oward funcions wih disconinuous slopes.

31 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 0 k = 0 k odd jkπ e j2πk 0 2

32 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 k = k odd jkπ e j2πk 0 2

33 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 3 k = 3 k odd jkπ e j2πk 0 2

34 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 5 k = 5 k odd jkπ e j2πk 0 2

35 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 7 k = 7 k odd jkπ e j2πk 0 2

36 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 9 k = 9 k odd jkπ e j2πk 0 2

37 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 9 k = 9 k odd jkπ e j2πk 0 2

38 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 29 k = 29 k odd jkπ e j2πk 0 2

39 Fourier Series One can visualize convergence of he Fourier Series by incremenally adding erms. Example: square wave 2 39 k = 39 k odd jkπ e j2πk 0 2

40 Fourier Series Parial sums of Fourier series of disconinuous funcions ring near disconinuiies: Gibb s phenomenon. 2 9% 0 2 This ringing resuls because he magniude of he Fourier coefficiens is only decreasing as k (while hey decreased as k2 for he riangle). You can decrease (and even eliminae he ringing) by decreasing he magniudes of he Fourier coefficiens a higher frequencies.

41 Fourier Series: Summary Fourier series represen periodic signals as sums of sinusoids. valid for an exremely large class of periodic signals valid even for disconinuous signals such as square wave However, convergence as # harmonics increases can be complicaed.

42 Filering The oupu of an LTI sysem is a filered version of he inpu. Inpu: Fourier series sum of complex exponenials. x() = x( + T ) = a k e j 2π T k k= Complex exponenials: eigenfuncions of LTI sysems. e j 2π T k H(j 2π j 2π k)e T k T Oupu: same eigenfuncions, ampliudes/phases se by sysem. x() = a k e j 2π T k y() = a k H(j 2π j 2π k)e T k T k= k=

43 Filering Noion of a filer. LTI sysems canno creae new frequencies. can scale magniudes and shif phases of exising componens. Example: Low-Pass Filering wih an RC circui R + v + i v o C

44 Lowpass Filer Calculae he frequency response of an RC circui. R + v + i v o C KVL: v i () = Ri() + v o () C: i() = C v o () Solving: v i () = RC v o () + v o () V i (s) H(s) = ( + src)v o (s) = V o(s) V i (s) = + src H(jω) H(jω) π ω /RC ω /RC

45 Lowpass Filering Le he inpu be a square wave. 2 x() = k odd 0 T 2 jπk e jω 0k ; ω 0 = 2π T X(jω) X(jω) π ω /RC ω /RC

46 Lowpass Filering Low frequency square wave: ω 0 << /RC. 2 x() = k odd 0 T 2 jπk e jω 0k ; ω 0 = 2π T H(jω) H(jω) π ω /RC ω /RC

47 Lowpass Filering Higher frequency square wave: ω 0 < /RC. 2 x() = k odd 0 T 2 jπk e jω 0k ; ω 0 = 2π T H(jω) H(jω) π ω /RC ω /RC

48 Lowpass Filering Sill higher frequency square wave: ω 0 = /RC. 2 x() = k odd 0 T 2 jπk e jω 0k ; ω 0 = 2π T H(jω) H(jω) π ω /RC ω /RC

49 Lowpass Filering High frequency square wave: ω 0 > /RC. 2 x() = k odd 0 T 2 jπk e jω 0k ; ω 0 = 2π T H(jω) H(jω) π ω /RC ω /RC

50 Fourier Series: Summary Fourier series represen signals by heir frequency conen. Represening a signal by is frequency conen is useful for many signals, e.g., music. Fourier series moivae a new represenaion of a sysem as a filer.