6.003: Signals and Systems Lecture 20 November 17, 2011

Transcription

1 6.3: Signals and Sysems Lecure November 7, 6.3: Signals and Sysems Applicaions of Fourier ransforms 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 R + v + i v o C November 7, Lowpass Filering Higher frequency square wave: < /RC. x() = k odd jπk e jk ; = π Source-Filer Model of Speech Producion Vibraions of he vocal cords are filered by he mouh and nasal caviies o generae speech. H(j). H(j)... π.. /RC /RC buzz from vocal cords hroa and nasal caviies speech Filering LI sysems filer signals based on heir frequency conen. Fourier ransforms represen signals as sums of complex exponenials. x() = X(j)e j d π 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 Filering Sysems can be designed o selecively pass cerain frequency bands. Examples: low-pass filer (LPF) and high-pass filer (HPF). LPF HPF LPF HPF

2 6.3: Signals and Sysems Lecure November 7, 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. Filering Example: Elecrocardiogram In addiion o elecrical responses of hear, elecrodes on he skin also pick up oher elecrical signals ha we regard as noise. We wish o design a filer o eliminae he noise. x() filer y() x() [mv] [s] ECG and analysis by. F. Weiss Filering Example: Elecrocardiogram We can idenify noise using he Fourier ransform. x() [mv] [s] Hz Filering Example: Elecrocardiogram Filer design: low-pass fler + high-pass filer + noch. H(j).. X(j) [µv]... low-freq. noise cardiac signal high-freq. noise... f = π [Hz]... f = π [Hz] Elecrocardiogram: Check Yourself Which poles and zeros are associaed wih he high-pass filer? he low-pass filer? he noch filer? s-plane ( ) ( )( ) Filering Example: Elecrocardiogram Filering is a simple way o reduce unwaned noise. Unfilered ECG x() [mv ] Filered ECG [s] y() [mv ] [s]

3 6.3: Signals and Sysems Lecure November 7, Fourier ransforms in Physics: iffracion A diffracion graing breaks a laser beam inpu ino muliple beams. Fourier ransforms in Physics: iffracion Muliple beams resul from periodic srucure of graing (period ). graing λ sin = λ emonsraion. Viewed a a disance from angle, scaerers are separaed by sin. Consrucive inerference if sin = nλ, i.e., if sin = nλ periodic array of dos in he far field Check Yourself Check Yourself C demonsraion. 3 fee V demonsraion. fee laser poiner λ = 5 nm fee laser poiner λ = 5 nm fee C screen Wha is he spacing of he racks on he C?. 6 nm. 6 nm 3. 6µm 4. 6µm V screen Wha is rack spacing on V divided by ha for C? Fourier ransforms in Physics: iffracion Macroscopic informaion in he far field provides microscopic (invisible) informaion abou he graing. Fourier ransforms in Physics: Crysallography Wha if he arge is more complicaed han a graing? λ arge sin = λ image? 3

4 6.3: Signals and Sysems Lecure November 7, Fourier ransforms in Physics: Crysallography Par of image a angle has conribuions for all pars of he arge. Fourier ransforms in Physics: Crysallography he phase of ligh scaered from differen pars of he arge undergo differen amouns of phase delay. arge x sin x image? Phase a a poin x is delayed (i.e., negaive) relaive o ha a : φ = π x sin λ Fourier ransforms in Physics: Crysallography oal ligh F () a angle is inegral of ligh scaered from each par of arge f(x), appropriaely shifed in phase. jπ x sin F () = f(x) e λ dx Fourier ransforms in Physics: iffracion Fourier ransform relaion beween srucure of objec and far-field inensiy paern. 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)! graing impulse rain wih pich far-field inensiy impulse rain wih reciprocal pich λ π Impulse rain he Fourier ransform of an impulse rain is an impulse rain. x() = δ( k ) k= wo imensions emonsraion: graing. X(j) = a k = k k= π π δ( k ) π π k 4

5 6.3: Signals and Sysems Lecure November 7, An Hisoric Fourier ransform aken by Rosalind Franklin, his image sparked Wason and Crick s insigh ino he double helix. 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 repeaing srucure of base pairs. An Hisoric Fourier ransform Low-frequency bands indicae a lower frequency repeaing srucure. b /b h /h An Hisoric Fourier ransform il of low-frequency bands indicaes il of low-frequency repeaing srucure: he double helix! Simulaion Easy o calculae relaion beween srucure and Fourier ransform. 5

6 6.3: Signals and Sysems Lecure November 7, 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. 6