6.003: Signals and Systems

Transcription

1 6.3: Signals and Sysems Lecure 7 April 8, 6.3: Signals and Sysems C Fourier ransform C Fourier ransform Represening signals by heir frequency conen. X(j)= x()e j d ( analysis equaion) x()= π X(j)e j d ( synhesis equaion) April 8, generalizes Fourier series o represen aperiodic signals. equals Laplace ransform X(s) s=j if ROC includes j axis. inheris properies of Laplace ransform. complex-valued funcion of real domain. simple inverse relaion more general han able-lookup mehod for inverse Laplace. dualiy. filering. applicaions in physics. Filering Lowpass Filering Noion of a filer. LI sysems canno creae new frequencies. can only scale magniudes and shif phases of exising componens. Example: Low-Pass Filering wih an RC circui Higher frequency square wave: < /RC. j k π x() = e ; = k odd jπk R v i C v o.. /RC H(j) H(j) π.. /RC Source-Filer Model of Speech Producion Vibraions of he vocal cords are filered by he mouh and nasal caviies o generae speech. Filering LI sysems filer signals based on heir frequency conen. Fourier ransforms represen signals as sums of complex exponenials. x() = π X(j)e j d Complex exponenials are eigenfuncions of LI sysems. e j H(j)e j LI sysems filer signals by adjusing he ampliudes and phases of each frequency componen. x() = X(j)e j d y() = H(j)X(j)e j d π π buzz from vocal cords hroa and nasal caviies speech

2 6.3: Signals and Sysems Lecure 7 April 8, Filering Sysems can be designed o selecively pass cerain frequency bands. Examples: low-pass filer (LPF) and high-pass filer (HPF). Filering Example: Elecrocardiogram An elecrocardiogram is a record of elecrical poenials ha are generaed by he hear and measured on he surface of he ches. LPF HPF LPF x() [mv] [s] HPF ECG and analysis by. F. Weiss Filering Example: Elecrocardiogram In addiion o picking up elecrical responses of he hear, elecrodes on he skin also pick up a variey of oher elecrical signals ha we regard as. Filering Example: Elecrocardiogram We can idenify he by breaking he elecrocardiogram ino frequency componens using he Fourier ransform. We wish o design a filer o eliminae he. 6 Hz x() x() [mv] [s] filer y() y() [mv] [s] X(j) [μv]... low-freq. cardiac signal high-freq.... f = [Hz] π Filering Example: Elecrocardiogram Elecrocardiogram: Check Yourself Filer design: low-pass fler + high-pass filer + noch. Which poles and zeros are associaed wih he high-pass filer? he low-pass filer? H(j).. he noch filer? s-plane... f = [Hz] π () ()()

3 6.3: Signals and Sysems Lecure 7 April 8, Filering Example: Elecrocardiogram By placing he poles of he noch filer very close o he zeros, he widh of he noch can be made quie small. H(j) f = [Hz] π Filering Example: Elecrocardiogram Comparision of filered and unfilered elecrocardiograms. X(j) [μv] x() [mv] [s] low-freq. cardiac signal 6 Hz high-freq.... f = π [Hz] Y (j) [μv] y() [mv] [s] f = π [Hz] Filering Example: Elecrocardiogram Reducing he frequency componens ha are no generaed by he hear simplifies he oupu, making i easier o diagnose cardiac problems. Unfilered ECG x() [mv ] [s] Coninuous-ime Fourier ransform: Summary Fourier ransforms represen signals by heir frequency conen. useful for many signals, e.g., elecrocardiogram. moivaes represening a sysem as a filer. useful for many sysems. Filered ECG y() [mv ] [s] Visualizing he Fourier ransform Fourier ransforms provide alernae views of signals. Fourier ransforms in Physics: iffracion A diffracion graing breaks a laser beam inpu ino muliple beams. π Pulses conain all frequencies excep harmonics of π/widh. π Wider pulses conain more low frequencies han narrow pulses. 4 π emonsraion. Consans (in ime) conain only frequencies a =. 3

4 6.3: Signals and Sysems Lecure 7 April 8, Fourier ransforms in Physics: iffracion he graing has a periodic srucure (period = ). Check Yourself C demonsraion. 3 fee λ sin = λ laser poiner λ = 5 nm C screen fee he far field image is formed by inerference of scaered ligh. Viewed from angle, he scaerers are separaed by sin. If his disance is an ineger number of wavelenghs λ consrucive inerference. Wha is he spacing of he racks on he C?. 6 nm. 6 nm 3. 6μm 4. 6μm Check Yourself Fourier ransforms in Physics: iffracion V demonsraion. fee Macroscopic informaion in he far field provides microscopic (invisible) informaion abou he graing. laser poiner λ = 5 nm fee λ V screen Wha is rack spacing on V divided by ha for C? sin = λ Fourier ransforms in Physics: Crysallography Wha if he arge is more complicaed han a graing? Fourier ransforms in Physics: Crysallography Par of image a angle has conribuions for all pars of he arge. arge arge image? image? 4

5 6.3: Signals and Sysems Lecure 7 April 8, Fourier ransforms in Physics: Crysallography he phase of ligh scaered from differen pars of he arge undergo differen amouns of phase delay. Fourier ransforms in Physics: Crysallography oal ligh F () a angle is he inegral of amoun scaered from each par of he arge (f(x)) appropriaely shifed in phase. F () = jπ x sin f(x)e λ dx x x sin Phase a a poin x is delayed (i.e., negaive) relaive o ha a : x sin φ = π λ Assume small angles so sin. Le = π λ. hen he paern of ligh a he deecor is F () = f(x)e jx dx which is he Fourier ransform of f(x)! Fourier ransforms in Physics: iffracion here is a Fourier ransform relaion beween his srucure and he far-field inensiy paern. Impulse rain he Fourier ransform of an impulse rain is an impulse rain. x() = k= δ( k ) graing impulse rain wih pich a k = k k far-field inensiy impulse rain wih reciprocal pich λ π X(j)= k= π π δ( k ) π π wo imensions emonsraion: graing. An Hisoric Fourier ransform aken by Rosalind Franklin, his image sparked Wason and Crick s insigh ino he double helix. Source unknown. All righs reserved. his conen is excluded from our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/fairuse. 5

6 6.3: Signals and Sysems Lecure 7 April 8, An Hisoric Fourier ransform his is an x-ray crysallographic image of NA, and i shows he Fourier ransform of he srucure of NA. An Hisoric Fourier ransform High-frequency bands indicae repreaing srucure of base pairs. b /b Source unknown. All righs reserved. his conen is excluded from our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/fairuse. Source unknown. All righs reserved. his conen is excluded from our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/fairuse. An Hisoric Fourier ransform Low-frequency bands indicae a lower frequency repeaing srucure. An Hisoric Fourier ransform il of low-frequency bands indicaes il of low-frequency repeaing srucure: he double helix! h /h Source unknown. All righs reserved. his conen is excluded from our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/fairuse. Source unknown. All righs reserved. his conen is excluded from our Creaive Commons license. For more informaion, see hp://ocw.mi.edu/fairuse. Simulaion Easy o calculae relaion beween srucure and Fourier ransform. Fourier ransform Summary Represen signals by heir frequency conen. Key o filering, and o signal-processing in general. Imporan in many physical phenomenon: x-ray crysallography. Images removed due o copyrigh resricions. Lef: double helix drawing. Righ: x-ray diffracion image. 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.