Environmental Data Analysis with MatLab

Lecture 11:

Lessons Learned from the Fourier Transform

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 TransformLecture 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

SYLLABUS

purpose of the lecture

understand some of the properties of the

Discrete Fourier Transform

from last week …

time series = sum of sines and cosines

rememberexp(iωt) = cos(ωt) + i sin(ωt)

k

time series

from last week …

Discrete Fourier Transform of a time series

coefficients

power spectral density = 2

di

ti Δt

time series

di

ti Δta time series is a discrete representation of a

continuous function

continuous function

d(t)

t

continuous function

What happens when to the Discrete Fourier Transform when we switch from discrete to continuous?

Discrete Fourier Transform

Fourier Transform

turns into

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 timefunction of frequency

Fourier Transform

power spectral density = 2

function of time function of frequency

the inverse of the Fourier Transform is

t

recall that an integral can be approximated by a summation

integral = area under curve =S area of rectangle = S width × height = Δt Si f(ti)

f(t)f(ti)

Δtti

then if we use N rectangleseach of width Δt

andeach 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 ofΔtthe Discrete Fourier Transform is the discrete

version of the Fourier Transform of a transient function, d(t)

scaling factor

similarlythe 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 waysintegrals 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

[cos(ωt ) + i sin(ωt )] dtcos(ωt ) dt + i sin(ωt ) dt

symmetric about zero antisymmetric about zeroso integral zero

Normal curve with variance a½ -2 = σt2

look up in table of integrals

Normal curve with variance 2a2 = σt-2

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 frequencies

increasing variance

time,

t

freq

uenc

y, f

A)

increasing variance

B)

tmax fmax

0 0

Property 2

the Fourier Transform of a spike

is constant

spike“Dirac Delta Function”

Normal curve with infinitesimal variance

infinitely highbut always has unit area

δ(t-t0)

t

depiction of spike

t0

important property of spike

t

since the spike is zero everywhere except t0

t0

tt0

f(t0)

f(t0)

this product …

… is equivalent to this one

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

0 50 100 150 200 250-1

-0.5

0

0.5

1

time, t

d(t)

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

2

frequency, f

d(f)

A) spike function

B) its transform

frequency, f

time, t

d(t)

d(f)^

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 Transformand 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 Transformthat is zero at zero frequency

MatLab

dt=fft(d); area = real(dt(1));

Property 5

multiplying the Fourier Transform byexp( -i ω t0)delays the time series by t0

use transformation of variablest’ = t - t0and notedt’ = dtandt±∞ corresponds to t’±∞

0 50 100 150 200 250-1

-0.5

0

0.5

1

time, t

d(t)

0 50 100 150 200 250-1

-0.5

0

0.5

1

time, t

d shifte

d(t)d(t)

time, t

time, t

d(t)

dshift

ed(t)

MatLab

t0 = t(16); ds=ifft(exp(-i*w*t0).*fft(d));

Property 6

multiplying the Fourier Transform byi ωdifferentiates the time series

use integration by partsand assume that the times series is zeroas t±∞

dvu uv duv

0 50 100 150 200 250-1

0

1

time, t

d(t)

0 50 100 150 200 250-0.02

0

0.02

time, t

dd/d

t(t)

0 50 100 150 200 250-0.02

0

0.02

time, t

dd/d

t(t)

time, t

A)

B)

C)

d(t)

dd/dt

dd/dt

MatLab

dddt=ifft(i*w.*fft(d));

Property 7

dividing the Fourier Transform byi ωintegrates the time series

this is another derivation byintegration 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”

0 50 100 150 200 250-1

0

1

time, t

d(t)

0 50 100 150 200 250-100

0

100

time, t

inte

gral

0 50 100 150 200 250-100

0

100

time, t

inte

gral

time, t

A)

B)

C)

d(t)

d

(t) d

t

d(t)

dt

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 theconvolution 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

transformation of variablest’ = t-τ so dt’ = dt and t’±∞ when t±

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

Summary1. FT of a Normal is a Normal curve2. FT of a spike is constant.3. FT of a cosine is a pair of spikes4. Multiplying FT by exp( -i ω t0 ) delays time

series5. Multiplying the FT by i ω differentiates the time

series6. Dividing the FT by i ω integrates the time series7. FT of convolution is product of FT’s