6.003: Signals and Systems. Applications of Fourier Transforms

Transcription

1 6.003: Signals and Systems Applications of Fourier Transforms November 7, 20

2 Filtering Notion of a filter. LTI systems cannot create new frequencies. can only scale magnitudes and shift phases of existing components. Example: Low-Pass Filtering with an RC circuit R + v + i v o C

3 Lowpass Filter Calculate the frequency response of an RC circuit. R + v + i v o C KVL: v i (t) = Ri(t) + v o (t) C: i(t) = C v o (t) Solving: v i (t) = RC v o (t) + v o (t) 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

4 Lowpass Filtering Let the input be a square wave. 2 x(t) = k odd 0 T 2 jπk e jω 0kt ; ω 0 = 2π T t X(jω) X(jω) π ω /RC ω /RC

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

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

7 Lowpass Filtering Still higher frequency square wave: ω 0 = /RC. 2 x(t) = k odd 0 T 2 jπk e jω 0kt ; ω 0 = 2π T t H(jω) H(jω) π ω /RC ω /RC

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

9 Source-Filter Model of Speech Production Vibrations of the vocal cords are filtered by the mouth and nasal cavities to generate speech. buzz from vocal cords throat and nasal cavities speech

10 Filtering LTI systems filter signals based on their frequency content. Fourier transforms represent signals as sums of complex exponentials. x(t) = 2π X(jω)e jωt dω Complex exponentials are eigenfunctions of LTI systems. e jωt H(jω)e jωt LTI systems filter signals by adjusting the amplitudes and phases of each frequency component. x(t) = X(jω)e jωt dω y(t) = H(jω)X(jω)e jωt dω 2π 2π

11 Filtering Systems can be designed to selectively pass certain frequency bands. Examples: low-pass filter (LPF) and high-pass filter (HPF). LPF HPF 0 ω LPF t t HPF t

12 Filtering Example: Electrocardiogram An electrocardiogram is a record of electrical potentials that are generated by the heart and measured on the surface of the chest. x(t) [mv] 2 0 t [s] ECG and analysis by T. F. Weiss

13 Filtering Example: Electrocardiogram In addition to electrical responses of heart, electrodes on the skin also pick up other electrical signals that we regard as noise. We wish to design a filter to eliminate the noise. x(t) filter y(t)

14 Filtering Example: Electrocardiogram We can identify noise using the Fourier transform. x(t) [mv] 2 0 t [s] Hz 00 X(jω) [µv] low-freq. noise cardiac signal high-freq. noise f = ω 2π [Hz]

15 Filtering Example: Electrocardiogram Filter design: low-pass flter + high-pass filter + notch. H(jω) f = ω 2π [Hz]

16 Electrocardiogram: Check Yourself Which poles and zeros are associated with the high-pass filter? the low-pass filter? the notch filter? s-plane 2 ( ) 2 2 ( )( )

17 Electrocardiogram: Check Yourself Which poles and zeros are associated with the high-pass filter? the low-pass filter? the notch filter? notch s-plane low-pass high-pass ( ) ( )( ) notch

18 Filtering Example: Electrocardiogram Filtering is a simple way to reduce unwanted noise. Unfiltered ECG 2 x(t) [mv ] t [s] Filtered ECG y(t) [mv ] t [s]

19 Fourier Transforms in Physics: Diffraction A diffraction grating breaks a laser beam input into multiple beams. Demonstration.

20 Fourier Transforms in Physics: Diffraction Multiple beams result from periodic structure of grating (period D). grating λ D θ sin θ = λ D Viewed at a distance from angle θ, scatterers are separated by D sin θ. Constructive interference if D sin θ = nλ, i.e., if sin θ = nλ D periodic array of dots in the far field

21 Fourier Transforms in Physics: Diffraction CD demonstration.

22 Check Yourself CD demonstration. 3 feet laser pointer λ = 500 nm feet CD screen What is the spacing of the tracks on the CD?. 60 nm nm 3. 6µm 4. 60µm

23 Check Yourself What is the spacing of the tracks on the CD? grating tan θ θ sin θ D = 500 nm sin θ manufacturing spec. CD nm 600 nm

24 Check Yourself Demonstration. 3 feet laser pointer λ = 500 nm feet CD screen What is the spacing of the tracks on the CD? nm nm 3. 6µm 4. 60µm

25 Fourier Transforms in Physics: Diffraction DVD demonstration.

26 Check Yourself DVD demonstration. feet laser pointer λ = 500 nm feet DVD screen What is track spacing on DVD divided by that for CD?

27 Check Yourself What is spacing of tracks on DVD divided by that for CD? grating tan θ θ sin θ D = 500 nm sin θ manufacturing spec. CD nm 600 nm DVD nm 740 nm

28 Check Yourself DVD demonstration. feet laser pointer λ = 500 nm feet DVD screen What is track spacing on DVD divided by that for CD?

29 Fourier Transforms in Physics: Diffraction Macroscopic information in the far field provides microscopic (invisible) information about the grating. λ sin θ = λ D D θ

30 Fourier Transforms in Physics: Crystallography What if the target is more complicated than a grating? target image?

31 Fourier Transforms in Physics: Crystallography Part of image at angle θ has contributions for all parts of the target. target θ image?

32 Fourier Transforms in Physics: Crystallography The phase of light scattered from different parts of the target undergo different amounts of phase delay. x sin θ θ x Phase at a point x is delayed (i.e., negative) relative to that at 0: φ = 2π x sin θ λ

33 Fourier Transforms in Physics: Crystallography Total light F (θ) at angle θ is integral of light scattered from each part of target f(x), appropriately shifted in phase. j2π x sin θ F (θ) = f(x) e λ dx Assume small angles so sin θ θ. Let ω = 2π λ θ, then the pattern of light at the detector is F (ω) = f(x) e jωx dx which is the Fourier transform of f(x)!

34 Fourier Transforms in Physics: Diffraction Fourier transform relation between structure of object and far-field intensity pattern. grating impulse train with pitch D 0 D t far-field intensity impulse train with reciprocal pitch λ D 0 2π D ω

35 Impulse Train The Fourier transform of an impulse train is an impulse train. x(t) = X(jω) = k= δ(t kt ) 0 T a k = T k T k= 2π 2π δ(ω k T T ) 2π T 0 2π T t k ω

36 Two Dimensions Demonstration: 2D grating.

37 An Historic Fourier Transform Taken by Rosalind Franklin, this image sparked Watson and Crick s insight into the double helix.

38 An Historic Fourier Transform This is an x-ray crystallographic image of DNA, and it shows the Fourier transform of the structure of DNA.

39 An Historic Fourier Transform High-frequency bands indicate repeating structure of base pairs. b /b

40 An Historic Fourier Transform Low-frequency bands indicate a lower frequency repeating structure. h /h

41 An Historic Fourier Transform Tilt of low-frequency bands indicates tilt of low-frequency repeating structure: the double helix! θ θ

42 Simulation Easy to calculate relation between structure and Fourier transform.

43 Fourier Transform Summary Represent signals by their frequency content. Key to filtering, and to signal-processing in general. Important in many physical phenomenon: x-ray crystallography.