The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Magnitude and Phase
The Fourier Transform:
Examples, Properties, Common Pairs
Remember: complex numbers can be thought of as (real,imaginary)
or (magnitude,phase).
Magnitude:
Phase:
CS 450: Introduction to Digital Signal and Image Processing
Bryan Morse
BYU Computer Science
Real part
Imaginary part
Magnitude
Phase
|F |
φ (F )
=
=
<(F )2 + =(F )2
)
tan−1 =(F
<(F )
How much of a cosine of that frequency you need
How much of a sine of that frequency you need
Amplitude of combined cosine and sine
Relative proportions of sine and cosine
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Example: Fourier Transform of a Cosine
Example: Fourier Transform of a Cosine
Z
F (u)
=
∞
−∞
∞
f (t) = cos(2πst)
f (t) e
−i2πut
Z
=
Z
=
=
=
cos(2πst) e
−i2πut
Frequency Domain
1
2 δ(u
cos(2πst)
dt
−∞
∞
−∞
−∞
0 except when u = ±s
1
1
δ(u − s) + δ(u + s)
2
2
0 for all u
0.8
0.5
0.6
0.2
0.4
0.6
1
0.8
0.2
-1
-10
The Fourier Transform: Examples, Properties, Common Pairs
Odd and Even Functions
Sinusoids
∗
for real-valued signals
Odd
f (−t) = −f (t)
Anti-symmetric
Sines
Transform is imaginary∗
0.4
-0.5
The Fourier Transform: Examples, Properties, Common Pairs
Even
f (−t) = f (t)
Symmetric
Cosines
Transform is real∗
− s) + 12 δ(u + s)
1
1
cos(2πst) [cos(−2πut) + i sin(−2πut)] dt
Z ∞
cos(2πst) cos(−2πut) dt + i
cos(2πst) sin(−2πut) dt
−∞
−∞
Z ∞
Z ∞
cos(2πst) cos(2πut) dt − i
cos(2πst) sin(2πut) dt
Z
=
−∞
∞
Spatial Domain
dt
1/2
Spatial Domain
f (t)
-5
Frequency Domain
F (u)
cos(2πst)
1
2
[δ(u + s) + δ(u − s)]
sin(2πst)
1
2i
[δ(u + s) − δ(u − s)]
5
10
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Constant Functions
Delta Functions
Spatial Domain
f (t)
Frequency Domain
F (u)
1
δ(u)
a
a δ(u)
Spatial Domain
f (t)
Frequency Domain
F (u)
δ(t)
1
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Square Pulse
Square Pulse
Spatial Domain
f (t)
1 if −a/2 ≤ t ≤ a/2
0 otherwise
Frequency Domain
F (u)
sinc(aπu) =
sin(aπu)
aπu
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Triangle
Comb
Spatial Domain
f (t)
1 − |t| if −a ≤ t ≤ a
0 otherwise
Frequency Domain
F (u)
Spatial Domain
f (t)
Frequency Domain
F (u)
sinc2 (aπu)
δ(t mod k)
δ(u mod 1/k )
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Gaussian
Differentiation
Spatial Domain
f (t)
e−πt
Frequency Domain
F (u)
2
e−πu
2
Spatial Domain
f (t)
Frequency Domain
F (u)
d
dt
2πiu
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Some Common Fourier Transform Pairs
More Common Fourier Transform Pairs
Spatial Domain
f (t)
Cosine
cos(2πst)
Sine
sin(2πst)
Unit
1
Constant
a
Delta
δ(t)
Comb
δ(t mod k)
Frequency Domain
F (u)
Deltas 12 [δ(u + s) + δ(u − s)]
Deltas 12 i [δ(u + s) − δ(u − s)]
Delta
δ(u)
Delta
aδ(u)
Unit
1
Comb
δ(u mod 1/k )
Spatial Domain
f (t)
1 if −a/2 ≤ t ≤ a/2
Square
0 otherwise
1 − |t| if −a ≤ t ≤ a
Triangle
0
otherwise
2
Gaussian
e−πt
d
Differentiation
dt
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Properties: Notation
Properties: Linearity
Let F denote the Fourier Transform:
F = F(f )
Let F −1 denote the Inverse Fourier Transform:
f = F −1 (F )
Frequency Domain
F (u)
Sinc
sinc(aπu)
Sinc2
sinc2 (aπu)
Gaussian
Ramp
2
e−πu
2πiu
Adding two functions together adds their Fourier Transforms together:
F(f + g) = F(f ) + F(g)
Multiplying a function by a scalar constant multiplies its Fourier
Transform by the same constant:
F(af ) = a F(f )
The Fourier Transform: Examples, Properties, Common Pairs
The Fourier Transform: Examples, Properties, Common Pairs
Properties: Translation
Change of Scale: Square Pulse Revisited
Translating a function leaves the magnitude unchanged and adds a
constant to the phase.
If
f2 = f1 (t − a)
F1 = F(f1 )
F2 = F(f2 )
then
|F2 | = |F1 |
φ (F2 ) = φ (F1 ) − 2πua
Intuition: magnitude tells you “how much”, phase tells you “where”.
The Fourier Transform: Examples, Properties, Common Pairs
Rayleigh’s Theorem
Total “energy” (sum of squares) is the same in either domain:
Z ∞
Z ∞
2
2
|f (t)| dt =
|F (u)| du
−∞
−∞