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: Low-Pass Filtering with an RC circuit R + vi + − C vo − 2 Lowpass Filter Calculate the frequency response of an RC circuit. R + vi (t) = Ri(t) + vo (t) C: i(t) = Cv̇o (t) Solving: vi (t) vo C = RC v̇o (t) + vo (t) Vi (s) = (1 + sRC)Vo (s) 1 Vo (s) H(s) = = 1 + sRC Vi (s) − 1 |H(jω)| + − 0.1 0.01 ∠H(jω)| vi KVL: 0.01 0.1 1 10 ω 100 1/RC 10 ω 100 1/RC 0 − π2 0.01 0.1 1 3 Lowpass Filtering Let the input be a square wave. 1 2 0 − 12 |X(jω)| 1 jω0 kt e ; jπk k odd 1 ω0 = 2π T 0.1 0.01 ∠X(jω)| x(t) = t T 0.01 0.1 1 10 ω 100 1/RC 10 ω 1/RC 100 0 − π2 0.01 0.1 1 4 Lowpass Filtering Low frequency square wave: ω0 << 1/RC. 1 2 0 − 12 1 jω0 kt e ; jπk k odd 1 |H(jω)| X ω0 = 2π T 0.1 0.01 ∠H(jω)| x(t) = t T 0.01 0.1 1 10 ω 100 1/RC 10 ω 1/RC 100 0 − π2 0.01 0.1 1 5 Lowpass Filtering Higher frequency square wave: ω0 < 1/RC. 1 2 0 − 12 1 jω0 kt e ; jπk k odd 1 |H(jω)| X ω0 = 2π T 0.1 0.01 ∠H(jω)| x(t) = t T 0.01 0.1 1 10 ω 100 1/RC 10 ω 1/RC 100 0 − π2 0.01 0.1 1 6 Lowpass Filtering Still higher frequency square wave: ω0 = 1/RC. 1 2 0 − 12 1 jω0 kt e ; jπk k odd 1 |H(jω)| X ω0 = 2π T 0.1 0.01 ∠H(jω)| x(t) = t T 0.01 0.1 1 10 ω 100 1/RC 10 ω 1/RC 100 0 − π2 0.01 0.1 1 7 Lowpass Filtering High frequency square wave: ω0 > 1/RC. 1 2 0 − 12 1 jω0 kt e ; jπk k odd 1 |H(jω)| X ω0 = 2π T 0.1 0.01 ∠H(jω)| x(t) = t T 0.01 0.1 1 10 ω 100 1/RC 10 ω 1/RC 100 0 − π2 0.01 0.1 1 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 throat and nasal cavities 9 speech Filtering LTI systems “filter” signals based on their frequency content. Fourier transforms represent signals as sums of complex exponen­ tials. ∞ 1 x(t) = X(jω)e jωt dω 2π −∞ 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. ∞ ∞ 1 1 x(t) = X(jω)e jωt dω → y(t) = H(jω)X(jω)e jωt dω 2π −∞ 2π −∞ 10 Filtering Systems can be designed to selectively pass certain frequency bands. Examples: low-pass filter (LPF) and high-pass filter (HPF). LPF 0 HPF ω LPF t HPF 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. x(t) [mV] 2 1 0 −1 t [s] 0 ECG and analysis by T. F. Weiss 12 10 20 30 40 50 60 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 13 y(t) Filtering Example: Electrocardiogram We can identify “noise” using the Fourier transform. x(t) [mV] 2 1 0 −1 t [s] 0 10 20 30 40 50 60 1000 60 Hz |X(jω)| [µV] 100 10 1 0.1 0.01 low-freq. noise 0.001 cardiac signal 0.0001 0.01 0.1 f= 14 10 1 ω [Hz] 2π high-freq. noise 100 Filtering Example: Electrocardiogram Filter design: low-pass flter + high-pass filter + notch. |H(jω)| 1 0.1 0.01 0.001 0.01 0.1 1 10 ω [Hz] f= 2π 15 100 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 ( )( ) 16 Electrocardiogram: Check Yourself Which poles and zeros are associated with • the high-pass filter? • the low-pass filter? • the notch filter? s-plane notch low-pass 2 2 2 ( ) ( )( ) notch 17 high-pass Filtering Example: Electrocardiogram Filtering is a simple way to reduce unwanted noise. x(t) [mV ] Unfiltered ECG 2 1 t [s] 0 0 10 20 30 40 50 60 y(t) [mV ] Filtered ECG 1 t [s] 0 0 10 20 30 40 50 18 60 Fourier Transforms in Physics: Diffraction A diffraction grating breaks a laser beam input into multiple beams. Demonstration. 19 Fourier Transforms in Physics: Diffraction grating Multiple beams result from periodic structure of grating (period D). λ sin θ = D λ D θ 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 20 Fourier Transforms in Physics: Diffraction CD demonstration. 21 Check Yourself CD demonstration. 3 feet 1 feet laser pointer λ = 500 nm CD screen What is the spacing of the tracks on the CD? 1. 160 nm 2. 1600 nm 3. 16µm 22 4. 160µm Check Yourself What is the spacing of the tracks on the CD? grating tan θ θ sin θ CD 1 3 0.32 0.31 D= 500 nm sin θ 1613 nm 23 manufacturing spec. 1600 nm Check Yourself Demonstration. 3 feet 1 feet laser pointer λ = 500 nm CD screen What is the spacing of the tracks on the CD? 1. 160 nm 2. 1600 nm 3. 16µm 24 2. 4. 160µm Fourier Transforms in Physics: Diffraction DVD demonstration. 25 Check Yourself DVD demonstration. 1 feet 1 feet laser pointer λ = 500 nm DVD screen What is track spacing on DVD divided by that for CD? 1. 4× 3. 12 × 2. 2× 26 4. 14 × Check Yourself What is spacing of tracks on DVD divided by that for CD? D= 500 nm sin θ grating tan θ θ sin θ CD 1 3 0.32 0.31 1613 nm 1600 nm DVD 1 0.78 0.71 704 nm 740 nm 27 manufacturing spec. Check Yourself DVD demonstration. 1 feet 1 feet laser pointer λ = 500 nm DVD screen What is track spacing on DVD divided by that for CD? 3 1. 4× 3. 12 × 2. 2× 28 4. 14 × Fourier Transforms in Physics: Diffraction Macroscopic information in the far field provides microscopic (invis­ ible) information about the grating. λ sin θ = D θ 29 λ D 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. θ target image? 31 Fourier Transforms in Physics: Crystallography The phase of light scattered from different parts of the target un­ dergo 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 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. Z x sin θ F (θ) = f (x) e−j2π λ dx Assume small angles so sin θ ≈ θ. θ , then the pattern of light at the detector is Let ω = 2π λ Z F (ω) = f (x) e−jωx dx which is the Fourier transform of f (x) ! 33 Fourier Transforms in Physics: Diffraction Fourier transform relation between structure of object and far-field intensity pattern. grating ≈ impulse train with pitch D ··· ··· t 0 D λ far-field intensity ≈ impulse train with reciprocal pitch ∝ D ··· ··· ω 0 34 2π D Impulse Train The Fourier transform of an impulse train is an impulse train. x(t) = ∞ X δ(t − kT ) k=−∞ 1 ··· ··· t 0 ak = T1 ··· X(jω) = T ∀ k 1 T ··· k ∞ X 2π 2π δ(ω − k ) T T k=−∞ 2π T ··· 0 35 2π T ··· ω 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. 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. 37 An Historic Fourier Transform This is an x-ray crystallographic image of DNA, and it shows the Fourier transform of the structure of DNA. 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. 38 An Historic Fourier Transform High-frequency bands indicate repeating structure of base pairs. b 1/b 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. 39 An Historic Fourier Transform Low-frequency bands indicate a lower frequency repeating structure. 1/h h 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. 40 An Historic Fourier Transform Tilt of low-frequency bands indicates tilt of low-frequency repeating structure: the double helix! θ θ 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. 41 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 signal-processing in general. Important in many physical phenomenon: x-ray crystallography. 43 MIT OpenCourseWare http://ocw.mit.edu 6.003 Signals and Systems Fall 2011 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.