6.003: Signals and Systems Applications of Fourier Transforms - - PowerPoint PPT Presentation

6.003: Signals and Systems

Applications of Fourier Transforms

November 17, 2011

1

Filtering

Notion of a filter. LTI systems

• cannot create new frequencies.
• can only scale magnitudes and shift phases of existing components.

Example: Low­Pass Filtering with an RC circuit

+ −

vi + vo − R C

2

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |H(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠H(jω)|

Lowpass Filter

Calculate the frequency response of an RC circuit.

+ −

vi + vo − R C KVL: vi(t) = Ri(t) + vo(t) C: i(t) = Cv ˙o(t) Solving: vi(t) = RC v ˙o(t) + vo(t) Vi(s) = (1 + sRC)Vo(s) Vo(s) 1 H(s) = = Vi(s) 1 + sRC

3

Lowpass Filtering

x(t) = Let the input be a square wave. t

1 2

−1

2

T T π 2 = ω ;

kt jω

e jπk 1

• dd

k

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |X(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠X(jω)|

4

Lowpass Filtering

x(t) = Low frequency square wave: ω0 << 1/RC. t

1 2

−1

2

T T π 2 = ω ;

kt jω

e jπk 1

• dd

k

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |H(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠H(jω)|

5

Lowpass Filtering

x(t) = Higher frequency square wave: ω0 < 1/RC. t

1 2

−1

2

T T π 2 = ω ;

kt jω

e jπk 1

• dd

k

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |H(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠H(jω)|

6

Lowpass Filtering

x(t) = Still higher frequency square wave: ω0 = 1/RC. t

1 2

−1

2

T T π 2 = ω ;

kt jω

e jπk 1

• dd

k

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |H(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠H(jω)|

7

Lowpass Filtering

x(t) = High frequency square wave: ω0 > 1/RC. t

1 2

−1

2

T T π 2 = ω ;

kt jω

e jπk 1

• dd

k

0.01 0.1 1 0.01 0.1 1 10 100 ω 1/RC |H(jω)| −π

2

0.01 0.1 1 10 100 ω 1/RC ∠H(jω)|

8

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 speech throat and nasal cavities

9

Filtering

LTI systems “filter” signals based on their frequency content. Fourier transforms represent signals as sums of complex exponen- tials. ∞ 1

jωtdω

x(t) = X(jω)e 2π

−∞

Complex exponentials are eigenfunctions of LTI systems.

jωt → H(jω)e jωt

e LTI systems “filter” signals by adjusting the amplitudes and phases

• f each frequency component.

∞ ∞ 1 1

jωtdω jωtdω

x(t) = X(jω)e → y(t) = H(jω)X(jω)e 2π

−∞

2π

−∞

10

Filtering

Systems can be designed to selectively pass certain frequency bands. Examples: low­pass filter (LPF) and high­pass filter (HPF). ω LPF HPF LPF HPF t t t

11

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.

10 20 30 40 50 60 −1 1 2

t [s] x(t) [mV]

ECG and analysis by T. F. Weiss

12

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. filter x(t) y(t)

13

Filtering Example: Electrocardiogram

We can identify “noise” using the Fourier transform.

10 20 30 40 50 60 −1 1 2

t [s] x(t) [mV]

0.01 0.1 1 10 100 1000 100 10 1 0.1 0.01 0.001 0.0001 f = ω 2π [Hz] |X(jω)| [µV] low­freq. noise cardiac signal high­freq. noise 60 Hz

14

Filtering Example: Electrocardiogram

Filter design: low­pass flter + high­pass filter + notch. 0.01 0.1 1 10 100 1 0.1 0.01 0.001 f = ω 2π [Hz] |H(jω)|

15

Electrocardiogram: Check Yourself

Which poles and zeros are associated with

• the high­pass filter?
• the low­pass filter?
• the notch filter?

s­plane ( ) ( ) ( )

2 2 2

16

Electrocardiogram: Check Yourself

Which poles and zeros are associated with

• the high­pass filter?
• the low­pass filter?
• the notch filter?

s­plane ( ) ( ) ( )

2 2 2

high­pass low­pass notch notch

17

Filtering Example: Electrocardiogram

Filtering is a simple way to reduce unwanted noise. Unfiltered ECG 10 20 30 40 50 60 1 2 t [s] x(t) [mV ] Filtered ECG 10 20 30 40 50 60 1 t [s] y(t) [mV ]

18

Fourier Transforms in Physics: Diffraction

A diffraction grating breaks a laser beam input into multiple beams. Demonstration.

19

Fourier Transforms in Physics: Diffraction

Multiple beams result from periodic structure of grating (period D). grating θ λ D sin θ = λ D Viewed at a distance from angle θ, scatterers are separated by D sin θ.

nλ

Constructive interference if D sin θ = nλ, i.e., if sin θ = D → periodic array of dots in the far field

20

Fourier Transforms in Physics: Diffraction

CD demonstration.

21

Check Yourself

CD demonstration. laser pointer λ = 500 nm CD screen 3 feet 1 feet What is the spacing of the tracks on the CD?

• 1. 160 nm
• 2. 1600 nm
• 3. 16µm
• 4. 160µm

22

Check Yourself

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

1

CD 0.32 0.31 1613 nm 1600 nm

3

23

Check Yourself

Demonstration. laser pointer λ = 500 nm CD screen 3 feet 1 feet What is the spacing of the tracks on the CD? 2.

• 1. 160 nm
• 2. 1600 nm
• 3. 16µm
• 4. 160µm

24

Fourier Transforms in Physics: Diffraction

DVD demonstration.

25

Check Yourself

DVD demonstration. laser pointer λ = 500 nm DVD screen 1 feet 1 feet What is track spacing on DVD divided by that for CD?

• 1. 4×
• 2. 2×

3.

1 2×

4.

1 4×

26

Check Yourself

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

1

CD 0.32 0.31 1613 nm 1600 nm

3

DVD 1 0.78 0.71 704 nm 740 nm

27

Check Yourself

DVD demonstration. laser pointer λ = 500 nm DVD screen 1 feet 1 feet What is track spacing on DVD divided by that for CD? 3

• 1. 4×
• 2. 2×

3.

1 2×

4.

1 4×

28

Fourier Transforms in Physics: Diffraction

Macroscopic information in the far field provides microscopic (invis- ible) information about the grating. θ λ D sin θ = λ D

29

Fourier Transforms in Physics: Crystallography

What if the target is more complicated than a grating? target image?

30

Fourier Transforms in Physics: Crystallography

Part of image at angle θ has contributions for all parts of the target. target image? θ

31

Fourier Transforms in Physics: Crystallography

The phase of light scattered from different parts of the target un- dergo different amounts of phase delay. θ x sin θ x Phase at a point x is delayed (i.e., negative) relative to that at 0: x sin θ φ = −2π λ

32

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 θ , then the pattern of light at the detector is

λ −jωxdx

F (ω) = f(x) e which is the Fourier transform of f(x) !

33

Fourier Transforms in Physics: Diffraction

Fourier transform relation between structure of object and far­field intensity pattern.

· · · · · · grating ≈ impulse train with pitch D t D · · · · · · far­field intensity ≈ impulse train with reciprocal pitch ∝ λ

D

ω

2π D

34

Impulse Train

The Fourier transform of an impulse train is an impulse train.

· · · · · · x(t) =

∞ k=−∞

δ(t − kT) t T 1 · · · · · · ak = 1

T

∀ k k

1 T

· · · · · · X(jω) =

∞ k=−∞

2π T δ(ω − k2π T ) ω

2π T 2π T

35

Two Dimensions

Demonstration: 2D grating.

36

An Historic Fourier Transform

Taken by Rosalind Franklin, this image sparked Watson and Crick’s insight into the double helix.

37 Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): 740-741. (c) 1953.

An Historic Fourier Transform

This is an x­ray crystallographic image of DNA, and it shows the Fourier transform of the structure of DNA.

38 Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): 740-741. (c) 1953.

An Historic Fourier Transform

High­frequency bands indicate repeating structure of base pairs. b 1/b

39 Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): 740-741. (c) 1953.

An Historic Fourier Transform

Low­frequency bands indicate a lower frequency repeating structure. h 1/h

40 Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): 740-741. (c) 1953.

An Historic Fourier Transform

Tilt of low­frequency bands indicates tilt of low­frequency repeating structure: the double helix! θ θ

41 Reprinted by permission from Macmillan Publishers Ltd: Nature. Source: Franklin, R., and R. G. Gosling. "Molecular Configuration in Sodium Thymonucleate." Nature 171 (1953): 740-741. (c) 1953.

Simulation

Easy to calculate relation between structure and Fourier transform.

42

Fourier Transform Summary

Represent signals by their frequency content. Key to “filtering,” and to signal­processing in general. Important in many physical phenomenon: x­ray crystallography.

43

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6.003 Signals and Systems

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