MatLab Lecture 11 Lessons Learned from the Fourier Transform Lecture 01 Using MatLab Lecture 02 Looking At Data Lecture 03 Probability and Measurement Error Lecture 04 Multivariate Distributions ID: 408823
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Slide1
Environmental Data Analysis with MatLab
Lecture 11:
Lessons Learned from the Fourier TransformSlide2
Lecture 01
Using
MatLabLecture 02 Looking At DataLecture 03 Probability and Measurement Error Lecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares ProblemsLecture 09 Fourier SeriesLecture 10 Complex Fourier SeriesLecture 11 Lessons Learned from the Fourier Transform Lecture 12 Power SpectraLecture 13 Filter Theory Lecture 14 Applications of Filters Lecture 15 Factor Analysis Lecture 16 Orthogonal functions Lecture 17 Covariance and AutocorrelationLecture 18 Cross-correlationLecture 19 Smoothing, Correlation and SpectraLecture 20 Coherence; Tapering and Spectral Analysis Lecture 21 InterpolationLecture 22 Hypothesis testing Lecture 23 Hypothesis Testing continued; F-TestsLecture 24 Confidence Limits of Spectra, Bootstraps
SYLLABUSSlide3
purpose of the lecture
understand some of the properties of the
Discrete Fourier TransformSlide4
from last week …
time series = sum of
sines and cosinesrememberexp(iωt) = cos(ωt) + i sin(ωt)kSlide5
time series
from last week …
Discrete Fourier Transform of a time seriescoefficientspower spectral density = 2Slide6
d
i
ti Δt
time seriesSlide7
d
i
ti Δt
a time series
is a
discrete representation of
a continuous function
continuous
functionSlide8
d(t)
t
continuous functionWhat happens when to the Discrete Fourier Transform when we switch from discrete to continuous?Slide9
Discrete Fourier Transform
Fourier Transform
turns intoSlide10
note the use of the tilde to distinguish a
the Fourier Transform from the function itself. The two functions are different!
Fourier TransformSlide11
function of time
function of frequency
Fourier Transformpower spectral density = 2Slide12
function of time
function of frequency
the inverse of the Fourier Transform isSlide13
t
recall that an integral
can be approximated by a summation
integral = area under curve =
S
area of rectangle =
S
width
×
height =
Δ
t
S
i
f(
t
i
)
f(t)
f(
t
i
)
Δ
t
t
i
Slide14
then if we use
N
rectangleseach of width Δtandeach of height d(tk) exp(-iωtk)then the Fourier Transform becomesprovided that d(t) is “transient”zero outside of the interval (0,tmax)Slide15
so except for a scaling factor of
Δ
tthe Discrete Fourier Transform is the discrete version of the Fourier Transform of a transient function, d(t)scaling factorSlide16
similarlythe Fourier Series is an approximation of the Inverse Fourier Transform
Inverse Fourier Transform
Fourier Series(up to an overall scaling of Δω)Slide17
Fourier Transform
in some ways
integrals are easier to work with thansummationsSlide18
Property
1
the Fourier Transform of a Normal curve with variance σt2is a Normal curve with variance σω2 =σt-2Slide19
let a2
=
½σt-2[cos(ωt ) + i sin(ωt )] dtcos(ωt ) dt + isin(ωt ) dt
symmetric about zero
antisymmetric
about zero
so integral zero
Normal curve with variance
½a
-2
=
σ
t
2
Slide20
look up in table of integrals
Normal curve with variance
2a2 = σt-2 Slide21
time series with broad features
Fourier Transform with mostly low frequencies
power spectral density with mostly low frequencies time series with narrow featuresFourier Transform with both low and high frequenciespower spectral density with broad range of frequenciesSlide22
increasing variance
time,
tfrequency, fA)increasing varianceB)tmaxfmax00Slide23
Property
2
the Fourier Transform of a spikeis constantSlide24
spike
“Dirac Delta Function”
Normal curve with infinitesimal varianceinfinitely highbut always has unit areaSlide25
δ
(t-t
0)tdepiction of spiket0Slide26
important property of spikeSlide27
t
since the spike is zero everywhere except
t0 t0
t
t
0
f(
t
0
)
f(
t
0
)
this product …
… is equivalent to this oneSlide28
soSlide29
use the previous result when computing the Fourier Transform of a spikeSlide30
A spiky time series
has a “flat” Fourier Transform
and a “flat” power spectral densitySlide31
A) spike function
B) its transform
frequency, ftime, td(t)d(f)^Slide32
Property
3
the Fourier Transform of cos(ω0t )is a pair of spikes at frequencies ±ω0 Slide33
cos(ω
0
t )has Fourier TrnsformSlide34
as is shown by inserting into the Inverse Fourier TransformSlide35
An oscillatory time series
has spiky Fourier Transform
and a power spectral density with spectral peaks Slide36
Property
4
the area under a time seriesis the zero-frequency value of the Fourier TransformSlide37Slide38
A time series with zero mean
has a Fourier Transform
that is zero at zero frequencySlide39
MatLab
dt
=fft(d); area = real(dt(1));Slide40
Property
5
multiplying the Fourier Transform byexp( -i ω t0)delays the time series by t0Slide41
use transformation of variables
t’ = t - t
0and notedt’ = dtandt±∞ corresponds to t’±∞ Slide42
d(t)
time,
ttime, td(t)dshifted(t)Slide43
MatLabt0 = t(16);
ds
=ifft(exp(-i*w*t0).*fft(d)); Slide44
Property
6
multiplying the Fourier Transform byi ωdifferentiates the time seriesSlide45Slide46
use integration by parts
and assume that the times series is zero
as t±∞dvuuv
du
vSlide47
time,
t
A)B) C)d(t)dd/dtdd/dtSlide48
MatLabdddt
=
ifft(i*w.*fft(d)); Slide49
Property
7
dividing the Fourier Transform byi ωintegrates the time seriesSlide50
this is another derivation by
integration by parts
but we’re skipping it hereSlide51
Fourier Transform of integral of
d(t)
note that the zero-frequency value is undefined(divide by zero)this is the “integration constant”Slide52
time,
t
A)B)C)d(t) d(t) dt d(t) dtSlide53
MatLabint2=
ifft
(i*fft(d).*[0,1./w(2:N)']'); set to zero to avoid dividing by zero (equivalent to an integration constant of zero)Slide54
Property
8
Fourier Transform of theconvolution of two time seriesis the product of their transformsSlide55
What’s a convolution ?Slide56
the convolution of
f(t)
and g(t)is the integralwhich is often abbreviated f(t) *g(t)not multiplicationnot complex conjugation(too many uses of the asterisk!)Slide57
uses of convolutions will be presented in the lecture after nextright now, just treat it as a mathematical quantitySlide58Slide59
transformation of variablest’ = t-
τ
so dt’ = dt and t’±∞ when t±reverse order of integrationchange variables: t’ = t-τ use exp(a+b)=exp(a)exp(b)rearrange into the product of two separate Fourier TransformsSlide60
Summary
FT of a Normal is a Normal curve
FT of a spike is constant.FT of a cosine is a pair of spikesMultiplying FT by exp( -i ω t0 ) delays time seriesMultiplying the FT by i ω differentiates the time seriesDividing the FT by i ω integrates the time seriesFT of convolution is product of FT’s