6.003: Signals and Systems

Transcription

1 6.3: Signals and Sysems Lecure 4 March 3, 6.3: Signals and Sysems Fourier Represenaions Mid-erm Examinaion # Wednesday, April 7, 7:3-9:3pm. No reciaions on he day of he exam. Coverage: Lecures 5 Reciaions 5 Homewors 8 Homewor 8 will no colleced or graded. Soluions will be posed. Closed boo: pages of noes (8 inches; fron and bac). Designed as -hour exam; wo hours o complee. Review sessions during open office hours. March 3, Fourier Represenaions Fourier series represen signals in erms of sinusoids. Represening signals by heir harmonic componens. leads o a new represenaion for sysems as filers harmonic # DC fundamenal second harmonic hird harmonic fourh harmonic fifh harmonic sixh harmonic Musical Insrumens Harmonic conen is naural way o describe some inds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano cello Musical Insrumens Harmonic conen is naural way o describe some inds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano cello oboe horn alosax oboe horn alosax 5 seconds

2 6.3: Signals and Sysems Lecure 4 March 3, Musical Insrumens Harmonic conen is naural way o describe some inds of signals. Ex: musical insrumens (hp://heremin.music.uiowa.edu/mis) piano piano 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 Harmonic Represenaions Wha signals can be represened by sums of harmonic componens? Harmonic Represenaions Is i possible o represen ALL periodic signals wih harmonics? Wha abou disconinuous signals? = = Only periodic signals: all harmonics of are periodic in =/. 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 maes sense. Separaing harmonic componens Separaing harmonic componens Underlying properies.. Muliplying wo harmonics produces a new harmonic wih he same fundamenal frequency: Assume ha x() is periodic in and is composed of a weighed sum of harmonics of =/. x() =x( + ) = a e j j jl e e = e j(+l). hen x()e jl d = a e j e jl d. he inegral of a harmonic over any ime inerval wih lengh = equal o a period is zero unless he harmonic is a DC: = a { e j(l) d + = = j j e d e d =,, = = a δ[ l] =a l = δ[] = = herefore a = x()e j d = x()e j π d

3 6.3: Signals and Sysems Lecure 4 March 3, Deermining harmonic componens of a periodic signal. a = x()e j π d ( analysis equaion) x()= x( + ) = a e j ( synhesis equaion) = Chec Yourself Le a represen he Fourier series coefficiens of he following square wave. How many of he following saemens are rue?. a = if is even. a is real-valued 3. a decreases wih 4. here are an infinie number of non-zero a 5. all of he above Properies If a signal is differeniaed in ime, is Fourier coefficiens are muliplied by j π. Proof: Le x() =x( + ) = a e j π = hen ( ) ẋ() = ẋ( + ) = j a e j π = Chec Yourself Le b represen he Fourier series coefficiens of he following riangle wave. 8 8 How many of he following saemens are rue?. b = if is even. b is real-valued 3. b decreases wih 4. here are an infinie number of non-zero b 5. all of he above One can visualize convergence of he by incremenally adding erms. Example: riangle waveform One can visualize convergence of he by incremenally adding erms. Example: riangle waveform 8 5 = 5 odd π e j 8 39 j π e = 39 odd 8 8 Fourier series represenaions of funcions wih disconinuous slopes converge oward funcions wih disconinuous slopes. 3

4 6.3: Signals and Sysems Lecure 4 March 3, One can visualize convergence of he by incremenally adding erms. Example: square wave One can visualize convergence of he by incremenally adding erms. Example: square wave 5 = 5 odd j jπ e 39 = 39 odd jπ e j : Summary Parial sums of Fourier series of disconinuous funcions ring near disconinuiies: Gibb s phenomenon. 9% 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. his ringing resuls because he magniude of he Fourier coefficiens is only decreasing as (while hey decreased as for he riangle). You can decrease (and even eliminae he ringing) by decreasing he magniudes of he Fourier coefficiens a higher frequencies. Filering he oupu of an LI sysem is a filered version of he inpu. Filering Noion of a filer. Inpu: Fourier series sum of complex exponenials. x() = x( + ) = a e j = LI sysems canno creae new frequencies. can scale magniudes and shif phases of exising componens. Complex exponenials: eigenfuncions of LI sysems. e j H(j )e j Oupu: same eigenfuncions, ampliudes/phases se by sysem. x() = a e j y() = a H(j )e j = = Example: Low-Pass Filering wih an RC circui v i + R C + v o 4

5 6.3: Signals and Sysems Lecure 4 March 3, Lowpass Filer Lowpass Filering Calculae he frequency response of an RC circui. v i + R C H(j) H(j) KVL: v i () = Ri()+ v o () Le he inpu be a square wave. + C: i() = Cv o() Solving: v i () = RCv o()+ v o () v o V i (s) = ( + src)v o (s) (s) x() = V e j ; = o H(s) = Vi(s) = + src odd jπ.... /RC X(j) X(j).... /RC π π.. /RC.. /RC Lowpass Filering Lowpass Filering Low frequency square wave: << /RC. Higher frequency square wave: < /RC. x() = e j ; = x() = e j ; = odd jπ odd jπ H(j) H(j) /RC.. /RC π π.. /RC.. /RC H(j) H(j) Lowpass Filering Lowpass Filering Sill higher frequency square wave: =/RC. High frequency square wave: > /RC. x() = e j ; = x() = e j ; = odd jπ odd jπ H(j) H(j) /RC.. /RC π π.. /RC.. /RC H(j) H(j) 5

6 6.3: Signals and Sysems Lecure 4 March 3, : 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. 6

7 MI OpenCourseWare hp://ocw.mi.edu 6.3 Signals and Sysems Spring For informaion abou ciing hese maerials or our erms of Use, visi: hp://ocw.mi.edu/erms.