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
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6.003 Signals and Systems
Fall 2011
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