Environmental Data Analysis with - PowerPoint Presentation

Slide1

Environmental Data Analysis with MatLab2nd Edition

Lecture 11:

Lessons Learned from the Fourier TransformSlide2

Lecture 01 Using MatLab

Lecture 02 Looking At Data

Lecture 03 Probability and Measurement ErrorLecture 04 Multivariate DistributionsLecture 05 Linear ModelsLecture 06 The Principle of Least SquaresLecture 07 Prior InformationLecture 08 Solving Generalized Least Squares Problems Lecture 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 Interpolation Lecture 22 Linear Approximations and Non Linear Least Squares Lecture 23 Adaptable Approximations with Neural NetworksLecture 24 Hypothesis testing Lecture 25 Hypothesis Testing continued; F-TestsLecture 26 Confidence Limits of Spectra, Bootstraps

SYLLABUSSlide3

Goals of the lectureunderstand some of the properties of theDiscrete 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 rectangles

each 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 ofΔ

t

the 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= ½σ

t

-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 featuresFourier 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 spike

is 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 serieshas 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(ω0t )

has Fourier

TrnsformSlide34

as is shown by inserting into the Inverse Fourier TransformSlide35

An oscillatory time serieshas spiky Fourier Transformand a power spectral density with spectral peaks Slide36

Property 4

the area under a time series

is the zero-frequency value of the Fourier TransformSlide37
Slide38

A time series with zero meanhas a Fourier Transformthat is zero at zero frequencySlide39

MatLabdt=fft(d); area = real(

dt

(1));Slide40

Property 5

multiplying the Fourier Transform by

exp( -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 by

i ωdifferentiates the time seriesSlide45
Slide46

use integration by partsand assume that the times series is zeroas

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 by

i ωintegrates the time seriesSlide50

this is another derivation byintegration 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)

d

tSlide53

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 the

convolution 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 quantitySlide58
Slide59

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

SummaryFT 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