![Page 1: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/1.jpg)
6.003: Signals and Systems
Applications of Fourier Transforms
November 17, 2011
![Page 2: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/2.jpg)
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
+−
vi
+
vo
−
R
C
![Page 3: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/3.jpg)
Lowpass Filter
Calculate the frequency response of an RC circuit.
+−
vi
+
vo
−
R
C
KVL: vi(t) = Ri(t) + vo(t)C: i(t) = Cv̇o(t)Solving: vi(t) = RCv̇o(t) + vo(t)
Vi(s) = (1 + sRC)Vo(s)
H(s) = Vo(s)Vi(s)
= 11 + sRC
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
![Page 4: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/4.jpg)
Lowpass Filtering
Let the input be a square wave.
t
12
−12
0 T
x(t) =∑k odd
1jπk
e jω0kt ; ω0 = 2πT
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|X(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠X
(jω
)|
![Page 5: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/5.jpg)
Lowpass Filtering
Low frequency square wave: ω0 << 1/RC.
t
12
−12
0 T
x(t) =∑k odd
1jπk
e jω0kt ; ω0 = 2πT
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
![Page 6: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/6.jpg)
Lowpass Filtering
Higher frequency square wave: ω0 < 1/RC.
t
12
−12
0 T
x(t) =∑k odd
1jπk
e jω0kt ; ω0 = 2πT
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
![Page 7: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/7.jpg)
Lowpass Filtering
Still higher frequency square wave: ω0 = 1/RC.
t
12
−12
0 T
x(t) =∑k odd
1jπk
e jω0kt ; ω0 = 2πT
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
![Page 8: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/8.jpg)
Lowpass Filtering
High frequency square wave: ω0 > 1/RC.
t
12
−12
0 T
x(t) =∑k odd
1jπk
e jω0kt ; ω0 = 2πT
0.01
0.1
1
0.01 0.1 1 10 100ω
1/RC
|H(jω
)|
0
−π20.01 0.1 1 10 100
ω
1/RC
∠H
(jω
)|
![Page 9: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/9.jpg)
Source-Filter Model of Speech Production
Vibrations of the vocal cords are “filtered” by the mouth and nasal
cavities to generate speech.
buzz fromvocal cords
speechthroat and
nasal cavities
![Page 10: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/10.jpg)
Filtering
LTI systems “filter” signals based on their frequency content.
Fourier transforms represent signals as sums of complex exponen-
tials.
x(t) = 12π
∫ ∞−∞
X(jω)e jωtdω
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) = 12π
∫ ∞−∞
X(jω)e jωtdω → y(t) = 12π
∫ ∞−∞
H(jω)X(jω)e jωtdω
![Page 11: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/11.jpg)
Filtering
Systems can be designed to selectively pass certain frequency bands.
Examples: low-pass filter (LPF) and high-pass filter (HPF).
0ω
LPF HPF
LPF
HPF
t
t
t
![Page 12: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/12.jpg)
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.
0 10 20 30 40 50 60−1
012
t [s]
x(t) [mV]
ECG and analysis by T. F. Weiss
![Page 13: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/13.jpg)
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.
filterx(t) y(t)
![Page 14: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/14.jpg)
Filtering Example: Electrocardiogram
We can identify “noise” using the Fourier transform.
0 10 20 30 40 50 60−1
012
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
)|[µ
V]
low-freq.noise cardiac
signal high-freq.noise
60 Hz
![Page 15: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/15.jpg)
Filtering Example: Electrocardiogram
Filter design: low-pass flter + high-pass filter + notch.
0.01 0.1 1 10 100
1
0.1
0.01
0.001
f = ω
2π [Hz]
|H(jω
)|
![Page 16: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/16.jpg)
Electrocardiogram: Check Yourself
Which poles and zeros are associated with
• the high-pass filter?
• the low-pass filter?
• the notch filter?s-plane
( )( )( )222
![Page 17: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/17.jpg)
Electrocardiogram: Check Yourself
Which poles and zeros are associated with
• the high-pass filter?
• the low-pass filter?
• the notch filter?s-plane
( )( )( )222 high-passlow-pass
notch
notch
![Page 18: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/18.jpg)
Filtering Example: Electrocardiogram
Filtering is a simple way to reduce unwanted noise.
Unfiltered ECG
0 10 20 30 40 50 600
1
2
t [s]
x(t
)[mV
]
Filtered ECG
0 10 20 30 40 50 600
1
t [s]
y(t
)[mV
]
![Page 19: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/19.jpg)
Fourier Transforms in Physics: Diffraction
A diffraction grating breaks a laser beam input into multiple beams.
Demonstration.
![Page 20: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/20.jpg)
Fourier Transforms in Physics: Diffraction
Multiple beams result from periodic structure of grating (period D).
gra
tin
g
θ
λ
D
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 = nλD
→ periodic array of dots in the far field
![Page 21: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/21.jpg)
Fourier Transforms in Physics: Diffraction
CD demonstration.
![Page 22: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/22.jpg)
Check Yourself
CD demonstration.
laser pointer
λ = 500 nm
CDscreen
3 feet
1 feet
What is the spacing of the tracks on the CD?
1. 160 nm 2. 1600 nm 3. 16µm 4. 160µm
![Page 23: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/23.jpg)
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 13 0.32 0.31 1613 nm 1600 nm
![Page 24: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/24.jpg)
Check Yourself
Demonstration.
laser pointer
λ = 500 nm
CDscreen
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
![Page 25: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/25.jpg)
Fourier Transforms in Physics: Diffraction
DVD demonstration.
![Page 26: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/26.jpg)
Check Yourself
DVD demonstration.
laser pointer
λ = 500 nm
DVDscreen
1 feet
1 feet
What is track spacing on DVD divided by that for CD?
1. 4× 2. 2× 3. 12× 4. 1
4×
![Page 27: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/27.jpg)
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 13 0.32 0.31 1613 nm 1600 nm
DVD 1 0.78 0.71 704 nm 740 nm
![Page 28: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/28.jpg)
Check Yourself
DVD demonstration.
laser pointer
λ = 500 nm
DVDscreen
1 feet
1 feet
What is track spacing on DVD divided by that for CD? 3
1. 4× 2. 2× 3. 12× 4. 1
4×
![Page 29: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/29.jpg)
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
![Page 30: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/30.jpg)
Fourier Transforms in Physics: Crystallography
What if the target is more complicated than a grating?
target
image?
![Page 31: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/31.jpg)
Fourier Transforms in Physics: Crystallography
Part of image at angle θ has contributions for all parts of the target.
target
image?
θ
![Page 32: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/32.jpg)
Fourier Transforms in Physics: Crystallography
The phase of light scattered from different parts of the target un-
dergo different amounts of phase delay.
θx sin θ
x
Phase at a point x is delayed (i.e., negative) relative to that at 0:
φ = −2πx sin θλ
![Page 33: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/33.jpg)
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.
F (θ) =∫f(x) e−j2π
x sin θλ dx
Assume small angles so sin θ ≈ θ.
Let ω = 2π θλ , then the pattern of light at the detector is
F (ω) =∫f(x) e−jωxdx
which is the Fourier transform of f(x) !
![Page 34: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/34.jpg)
Fourier Transforms in Physics: Diffraction
Fourier transform relation between structure of object and far-field
intensity pattern.
· · ·· · ·
grating ≈ impulse train with pitch D
t0 D
· · ·· · ·
far-field intensity ≈ impulse train with reciprocal pitch ∝ λD
ω0 2π
D
![Page 35: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/35.jpg)
Impulse Train
The Fourier transform of an impulse train is an impulse train.
· · ·· · ·
x(t) =∞∑
k=−∞δ(t− kT )
t0 T
1
· · ·· · ·
ak = 1T ∀ k
k
1T
· · ·· · ·
X(jω) =∞∑
k=−∞
2πTδ(ω − k2π
T)
ω0 2π
T
2πT
![Page 36: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/36.jpg)
Two Dimensions
Demonstration: 2D grating.
![Page 37: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/37.jpg)
An Historic Fourier Transform
Taken by Rosalind Franklin, this image sparked Watson and Crick’s
insight into the double helix.
![Page 38: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/38.jpg)
An Historic Fourier Transform
This is an x-ray crystallographic image of DNA, and it shows the
Fourier transform of the structure of DNA.
![Page 39: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/39.jpg)
An Historic Fourier Transform
High-frequency bands indicate repeating structure of base pairs.
b1/b
![Page 40: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/40.jpg)
An Historic Fourier Transform
Low-frequency bands indicate a lower frequency repeating structure.
h 1/h
![Page 41: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/41.jpg)
An Historic Fourier Transform
Tilt of low-frequency bands indicates tilt of low-frequency repeating
structure: the double helix!
θ
θ
![Page 42: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/42.jpg)
Simulation
Easy to calculate relation between structure and Fourier transform.
![Page 43: 6.003: Signals and Systemsweb.mit.edu/6.003/F11/www/handouts/lec20.pdf · Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T x(t) = X k odd 1 jπk ejω0kt; ω0 = 2π](https://reader036.vdocuments.site/reader036/viewer/2022070710/5ec6df422dfa2635894081da/html5/thumbnails/43.jpg)
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.