Environmental Data Analysis with MatLab
2nd Edition
Lecture 11:
Lessons Learned from the Fourier Transform
SYLLABUS
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Lecture 25
Lecture 26
Using MatLab
Looking At Data
Probability and Measurement Error
Multivariate Distributions
Linear Models
The Principle of Least Squares
Prior Information
Solving Generalized Least Squares Problems
Fourier Series
Complex Fourier Series
Lessons Learned from the Fourier Transform
Power Spectra
Filter Theory
Applications of Filters
Factor Analysis
Orthogonal functions
Covariance and Autocorrelation
Cross-correlation
Smoothing, Correlation and Spectra
Coherence; Tapering and Spectral Analysis
Interpolation
Linear Approximations and Non Linear Least Squares
Adaptable Approximations with Neural Networks
Hypothesis testing
Hypothesis Testing continued; F-Tests
Confidence Limits of Spectra, Bootstraps
Goals of the lecture
understand some of the properties of the
Discrete Fourier Transform
from last week …
time series = sum of sines and cosines
k
remember
exp(iωt) = cos(ωt) + i sin(ωt)
from last week …
Discrete Fourier Transform of a time series
coefficients
time series
power spectral density =
2
time series
di
ti
Δt
continuous function
di
ti
Δt
a time series is a discrete representation of a
continuous function
continuous function
d(t)
t
What happens when to the Discrete Fourier Transform
when we switch from discrete to continuous?
Discrete Fourier Transform
turns into
Fourier Transform
Fourier Transform
note the use of the tilde to distinguish a the
Fourier Transform from the function itself.
The two functions are different!
Fourier Transform
function of frequency
function of time
power spectral density =
2
the inverse of the Fourier Transform is
function of time
function of frequency
recall that an integral can be approximated by a
summation
f(t)
f(ti)
t
ti Δt
integral = area under curve =
S area of rectangle = S width × height = Δt Si f(ti)
then if we use N rectangles
each of width Δt
and
each of height d(tk) exp(-iωtk)
then the Fourier Transform becomes
provided that d(t) is “transient”
zero outside of the interval (0,tmax)
so except for a scaling factor of
Δt
the Discrete Fourier Transform is the discrete
version of the Fourier Transform of a
transient function, d(t)
scaling factor
similarly
the Fourier Series is an approximation of
the Inverse Fourier Transform
Inverse Fourier Transform
Fourier Series
(up to an overall scaling of Δω)
Fourier Transform
in some ways
integrals are easier to work with than
summations
Property 1
the Fourier Transform of a Normal curve with
variance σt2
is a Normal curve with variance σω2 =σt-2
let a2= ½σt-2
Normal curve with variance ½a-2 = σt2
[cos(ωt ) + i sin(ωt )] dt
cos(ωt ) dt + i
symmetric about zero
sin(ωt ) dt
antisymmetric about zero
so integral zero
look up in table of integrals
Normal curve with variance 2a2 = σt-2
time series with broad features
Fourier Transform with mostly low frequencies
power spectral density with mostly low frequencies
time series with narrow features
Fourier Transform with both low and high frequencies
power spectral density with broad range of frequencies
A)
0
time, t
frequency, f
0
tmax
B)
increasing variance
fmax
increasing variance
Property 2
the Fourier Transform of a spike
is constant
spike
“Dirac Delta Function”
Normal curve with infinitesimal variance
infinitely high
but always has unit area
depiction of spike
δ(t-t0)
t0
t
important property of spike
since the spike is zero everywhere except t0
this product …
f(t0)
t0
t
… is equivalent
to this one
f(t0)
t0
t
so
use the previous result when computing the Fourier
Transform of a spike
A spiky time series
has a “flat” Fourier Transform
and a “flat” power spectral density
A) spike function
1
d(t)
0.5
d(t)
0
-0.5
-1
2
0
50
B) its transform
100
time, t
150
200
250
time, t
1.5
d(f)
^
d(f)
1
0.5
0
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
frequency, f
frequency, f
0.2
0.3
0.4
0.5
Property 3
the Fourier Transform of cos(ω0t )
is a pair of spikes at frequencies ±ω0
cos(ω0t )
has Fourier Trnsform
as is shown by inserting into the Inverse
Fourier Transform
An oscillatory time series
has spiky Fourier Transform
and a power spectral density with spectral peaks
Property 4
the area under a time series
is the zero-frequency value of the Fourier Transform
A time series with zero mean
has a Fourier Transform
that is zero at zero frequency
MatLab
dt=fft(d);
area = real(dt(1));
Property 5
multiplying the Fourier Transform by
exp( -i ω t0)
delays the time series by t0
use transformation of variables
t’ = t - t0
and note
dt’ = dt
and
t±∞ corresponds to t’±∞
1
d(t)
d(t)
0.5
0
-0.5
-1
0
50
100
150
200
250
200
250
time, t
time, t
0.5
0
s
d(t)
d
hifted(t)
shifted
(t)
1
d
-0.5
-1
0
50
100
150
time, t
time, t
MatLab
t0 = t(16);
ds=ifft(exp(-i*w*t0).*fft(d));
Property 6
multiplying the Fourier Transform by
iω
differentiates the time series
use integration by parts
and assume that the times series is zero
as t±∞
u
dv
v
u
v
du
A)
d(t)
d(t)
1
0
-1
0
50
100
150
200
250
150
200
250
150
200
250
time, t
B)
dd/dt(t)
0.02
dd/dt
0
-0.02
0
50
100
time, t
C)
dd/dt(t)
0.02
dd/dt
0
-0.02
0
50
100
time, t
time, t
MatLab
dddt=ifft(i*w.*fft(d));
Property 7
dividing the Fourier Transform by
iω
integrates the time series
this is another derivation by
integration by parts
but we’re skipping it here
Fourier Transform of
integral of d(t)
note that the zero-frequency value is undefined
(divide by zero)
this is the “integration constant”
A)
d(t)
d(t)
1
0
integral
d(t) dt
-1
0
50
100
150
200
250
150
200
250
150
200
250
time, t
B)
100
0
-100
0
50
100
integral
 d(t)
dt
time, t
C)
100
0
-100
0
50
100
time, t
time, t
MatLab
int2=ifft(i*fft(d).*[0,1./w(2:N)']');
set to zero to avoid dividing
by zero (equivalent to an
integration constant of zero)
Property 8
Fourier Transform of the
convolution of two time series
is the product of their transforms
What’s a convolution ?
the convolution of f(t) and g(t)
is the integral
which is often abbreviated f(t) *g(t)
not multiplication
not complex conjugation
(too many uses of the asterisk!)
uses of convolutions will be presented in the
lecture after next
right now, just treat it as a mathematical quantity
reverse order of integration
change variables: t’ = t-τ
use exp(a+b)=exp(a)exp(b)
rearrange into the product of two
separate Fourier Transforms
transformation of variables
t’ = t-τ so dt’ = dt and t’±∞ when t±
Summary
1.
2.
3.
4.
5.
FT of a Normal is a Normal curve
FT of a spike is constant.
FT of a cosine is a pair of spikes
Multiplying FT by exp( -i ω t0 ) delays time series
Multiplying the FT by i ω differentiates the time
series
6. Dividing the FT by i ω integrates the time series
7. FT of convolution is product of FT’s