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Fourier Transforms and
Frequency Domain analysis:
Application to Solar Spectra
(It’s very challenging for
undergraduate students, but you
can have a big extra credit.
Note that it’s an extra credit. So
it’s not required; it’s optional.)
However, whatever you do in
future, getting acquainted with
Fourier Transformation is very
useful in any field of science
and engineering.
It’s an extremely powerful
technique!
How to Represent (any) Signal (in mathematical form)?
• Option 1: Taylor series represents any function using
polynomials.
• Polynomials are not the best - unstable and not very
physically meaningful.
• Easier to talk about “signals” in terms of its “frequencies”
(how fast/often signals change, etc).
Credit: S. Narasimhan
Jean Baptiste Joseph Fourier (1768-1830)
• Had crazy idea (1807):
• Any periodic function can be
rewritten as a weighted sum
of Sines (sin) and Cosines
(cos) of different frequencies.
• Don’t believe it?
– Neither did Lagrange,
Laplace, Poisson and
other big wigs
– Not translated into
English until 1878!
• But it’s true!
– called Fourier Series
– Possibly the greatest tool
used in Engineering
Credit: S. Narasimhan
A Sum of Sinusoids
• Our building block:
• Add enough of them to
get any signal f(x) you
want!
• How many degrees of
freedom?
• What does each control?
• Which one encodes the
coarse vs. fine structure of
the signal?
xAsin(
Credit: S. Narasimhan
Fourier Transform
• We want to understand the frequency of our signal. So, let’s
reparametrize the signal by instead of x:
xAsin(
f(x) F()Fourier
Transform
F() f(x)Inverse Fourier
Transform
• For every from 0 to infinite, F() holds the amplitude A and
phase of the corresponding sine
– How can F hold both? Complex number trick!
)()()( iIRF
22 )()( IRA )(
)(tan 1
R
I
Credit: S. Narasimhan
Time and Frequency
• example : g(t) = sin(2πf t) + (1/3)sin(2π (3f ) t)
Credit: S. Narasimhan
Time and Frequency
= +
• example : g(t) = sin(2πf t) + (1/3)sin(2π (3f ) t)
Credit: S. Narasimhan
Frequency Spectra
• example : g(t) = sin(2πf t) + (1/3)sin(2π(3f ) t)
= +
Credit: S. Narasimhan
Fourier Transform – more formally
Arbitrary function Single Analytic Expression
Spatial Domain (x) Frequency Domain (u)
Represent the signal as an infinite weighted sum
of an infinite number of sinusoids
dxexfuF uxi 2
(Frequency Spectrum F(u))
1sincos ikikeikNote:
Inverse Fourier Transform (IFT)
dxeuFxf uxi 2
Credit: S. Narasimhan
• Also, defined as:
dxexfuF iux
1sincos ikikeikNote:
• Inverse Fourier Transform (IFT)
dxeuFxf iux
2
1
Fourier Transform
(Relates five most
important numbers.)
Fourier Transform Pairs (Examples)
angular frequency ( )iuxe
Note that these are derived using
Credit: S. Narasimhan
angular frequency ( )iuxe
Note that these are derived using
Fourier Transform Pairs (Examples)
Credit: S. Narasimhan
Properties of Fourier Transform
Spatial Domain (x) Frequency Domain (u)
Linearity xgcxfc 21 uGcuFc 21
Scaling axf
a
uF
a
1
Shifting 0xxf uFeuxi 02
Symmetry xF uf
Conjugation xf uF
Convolution xgxf uGuF
Differentiation n
n
dx
xfd uFui
n2
frequency ( )uxie 2
Note that these are derived using
Convolution
As a mathematical formula:
Convolutions are commutative:
(Symbols of ∗ or ⊗ often used for convolution)
Fourier Transform and Convolution
hfg
dxexguG uxi 2
dxdexhf uxi 2
dxexhdef xuiui 22
'' '22 dxexhdef uxiui
Let
Then
uHuF
Convolution in spatial (e.g., time series) domain
Multiplication in frequency domain
“Convolution”
Fourier Transform and Convolution
hfg FHG
fhg HFG
Spatial Domain (x) Frequency Domain (u)
So, we can find g(x) by Fourier transform
g f h
G F H
FT FTIFT
Convolution
Examples
Convolution Theorem
• The Fourier transform of a convolution is the product of the
Fourier transforms
• The Fourier transform of a product is the convolution of the
Fourier transforms
(Symbols of ∗ or ⊗ often used for convolution)
Cross Correlation
(Note: This is convolution.)
(Pay attention to
differences from
convolution.)
Auto Correlation
Power (density) spectrum!
But the data are not continuous.
Discrete Fourier
Transformation
(DFT)
Directional cosines
Directional cosines
forms orthonormal
basis for a vector.
And inner vector
product gives an
amplitude for each
direction (= each
directional cosine
vector).
Discrete Fourier Transformation
[cos(2πijk/N) & i sin(2πijk/N)]
are orthonormal basis functions
of DFT. Note i is for the
imaginary part of complex
number and j determines
frequency of each base. For
given spatial data (Xk), DTF
finds an amplitude (xj) for each
base (= different frequency).
Think about DFT as inner
product.
Fast Fourier Transformation
(FFT) belongs to DFT.
Frequency Spectra
Credit: S. Narasimhan
Discrete Fourier Transformation
Discrete Fourier Transformation
Discrete Fourier Transformation
Can use Python functions
• Use scipy functions fft(), ifft(), and conj()
• The FFT cross correlation will give you a
lag curve, but because the offset is close
to zero, you will see the correlation peak at
either end of the curve
– Hint: Try rolling the lag curve a small amount
to properly see the correlation peak
