Environmental Data Analysis with MatLab

Lecture 9:

Fourier Series

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

temporal periodicitiesand their periods

astronomical

rotationdaily

revolutionyearly

other natural

ocean wavesa few seconds

anthropogenic

electric power60 Hz

spatial periodicitiesand their wavelengths

natural

sand duneshundreds of meters

tree ringsa few millimeters

anthropogenic

furrows plowedin a field

few tens of cm

Air temperatureBlack Rock Forest

365 days1 year

Air temperatureBlack Rock Forest

time, days

1 day

Stream FlowNeuse River

disc

harg

e, c

fs

time, days

365 days1 year

amplitude, C

lingo

temporal

f(t) = C cos{ 2π t / T }spatial

f(x) = C cos{ 2π x / λ }amplitude, C

period, T wavelength, λfrequency, f=1/T"cycles per (year,day,s)"

angular frequency, ω=2 π /T"radians per ..."

wavenumber, k=2 π / λ-

f(t) = C cos(ωt) f(x) = C cos(kx)

0 10 20 30 40 50 60 70 80 90-3

-2

-1

0

1

2

3

time, t

d(t)

cosine example

delay, t0

amplitude, C

period, T

d(t)

time, t

sinusoidal oscillationf(t) = C cos{ 2π (t-t0) / T }

pairing sines and cosinesto avoid using time delays

Fourier Serieslinear model containing nothing but

sines and cosines

A’s and B’s aremodel parameters

ω’s are auxiliary variables

Data series: spacing Dt, length T Mean of series in cos(0*t) termLowest frequency is 1 cycle/T

Freq. spacing Dw: 1,2,3,... cycles/T Highest frequency is (1 cycle)/2Dt

(cosine part only – a zigzag)

Nyquist frequency

-2 0 20

5

10

15

20

25

30

col 1

time,

s

-2 0 20

5

10

15

20

25

30

col 2

-2 0 20

5

10

15

20

25

30

col 3

-2 0 20

5

10

15

20

25

30

col 4

-2 0 20

5

10

15

20

25

30

col 5

-2 0 20

5

10

15

20

25

30

col 32cos(Δω t)cos(0t) sin(Δωt) cos(2Δω t) sin(2Δω t)cos(2pt/32 *16)frequency 0 = mean offset cos and sin of (2pt/32 x 1,2,3,...15) (1,2,3...15 cycles/record-length)

32 numbers in physical space 32 numbers in spectral space

problem of aliasing

frequencies higher than Nyquist(that is, periods shorter than 2Dt)masquerade as low frequencies

example

solution:use averages over Dt, not

instantaneous samples spaced Dt apart.

0 5 10 15 20-2

-1

0

1

2

time, t

d1(t)

0 5 10 15 20-2

-1

0

1

2

time, t

d2(t)d2(t)

d1(t)

time, t

time, t

d1 (t) = cos(w1t), with w1=2Dw

d2(t) = cos{w2t}, with w2=(2+N)Dw,

cos(ω t) has same shape as cos(-ω t) and

sin(ω t) has same shape as sin(-ω t) so really only the

0 to ωnypart of the ω-axis is unique

symmetry of sines and cosines

w

-wny wny 2wny 3wny0

equivalent points on the ω-axis

cos(ω t) = cos(ω t +2p) = cos(ω t -2p) = ...sin(ω t) = sin(ω t +2p) = sin(ω t -2p) = ...

so really only the0 to ωny

part of the ω-axis is unique

power spectral density

it is big at a frequency ω whenwhen sine or cosine at that frequency

has a large coefficient

alternatively, plotamplitude spectral density

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

1000

2000

3000

frequency, cycles per day

spec

trum

0 100 200 300 400 500 600 700 800 900 10000

500

1000

1500

2000

period, days

spec

trum

365.2 d = 12 cycles per 4500d

period, days

frequency, cycles per day

ampl

itude

sp

ectra

l den

sity

182.6 days60.0 days

ampl

itude

sp

ectra

l den

sity

Stream FlowNeuse River

all interesting frequencies near origin

can plot period, T=1/f instead

5 cyclesper 4500d

6 per 4500d line is misleading! psd is discrete (should be bar plot)

7

purpose of the lecture

switch from Fourier Series containingsines and cosines

to Fourier Series containingcomplex exponentials

purpose of the lectureswitch from Fourier Series containing

sin(ωt) and cos(ωt)

to

Fourier series containing

exp(-iωt) and exp(+iωt)

review of complex numbers

imaginary unit

i

such thati2 = -1

complex number

a = ar + i ai

real part

imaginary part

adding complex numbers

a = ar + i ai b = br + i bic = a+b = (ar + i ai )+ (br + i bi ) = (ar + br )+ i(ai +bi ) cr ci … just add real and imaginary parts, separately

subtracting complex numbers

a = ar + i ai b = br + i bic = a-b = (ar + i ai )-(br + i bi ) =

(ar - br )+ i(ai - bi ) cr ci … just subtract real and imaginary parts, separately

multiplying complex numbersa = ar + i ai b = br + i bic = ab = (ar + i ai )(br + i bi ) == ar br + i ar bi + i ai br + i2 ai bi = (ar br - ai bi )+ i(ar bi + ai br ) cr ci … like multiplying polynomials

complex conjugate, a*a = ar + i ai a* = ar - i ai

absolute value, |a ||a | = [ ar2 + ai2 ]½

note |a |2 = a* a

MatLabhandles complex numbers completely transparently

a = 2 + 3*i;b = 4 + 6*i;

c = a+b;

works just fine

Warning!accidentally resetting i to something

other than i is so easy i=100;

(and then you get nonsense)

so execute aclear i;

at the top of your script if you plan to use i

or use the alternate notation

a = complex(2,3);b = complex(4,6);

c = a+b;

which is safer

end of review

Euler’s Formula

exp(iz) = cos(z) + i sin(z)

… where does that come from ???

any complex number can be written

z = r exp(iθ) where r = |z | and θ=tan-1(zi/zr)

Euler’s Formulascomplex exponentials can be written as sines and cosines

or reverse themsine and cosines can be written as complex exponentials

so a Fourier Seriesalternatively can be written

as a sum of sines and cosinesor a sum of complex exponentials

old-style Fourier Series

non-negative frequencies only,

from0 to ωny

sin=0 for 0 and ωny

new-style Fourier Seriesor “Inverse Discrete Fourier Transform”

non-negativefrequencies

negativefrequencies

added for compatibility with MatLab

why the weird ordering of frequencies?ωn = ( 0, Δω, 2Δω, …, N½ Δω, -( N½ -1) Δω, …, -2 Δω, -Δω )

same asωn = ( 0, Δω, 2Δω, …, N½ Δω, ( N½ +1) Δω, …, (N-1)Δω, NΔω )

non-negativefrequencies

negativefrequencies

non-negative frequencies up to

Nyquist

non-negative frequencies above the Nyquist

least-squares solution for the Fourier coefficients, Cn or “Discrete Fourier Transform”

… derivation requires complex version of least-

squares. See text for details …

MatLab

Fourier Coefficients Cj from time series dn

c = fft(d);

vector of N datavector of N complex Fourier

coefficients

MatLab

time series dn from Fourier Coefficients Cj d = ifft(c);

vector of N data

vector of N complex Fourier

coefficients

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

same number M of Fourier Coefficients as number N of data

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

maximum time, for N data sampled at Dt

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

time column-vector

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

Nyquist frequency, fny

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

frequency sampling, Δf

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

frequency column- vector

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

number of frequencies between 0 and Nyquist

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

angular frequency sampling, Δω

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf;

angular frequency

column- vector

standard setupM=N; tmax=Dt*(N-1); t=Dt*[0:N-1]'; fmax=1/(2.0*Dt); df=fmax/(N/2); f=df*[0:N/2,-N/2+1:-1]'; Nf=N/2+1; dw=2*pi*df; w=dw*[0:N/2,-N/2+1:-1]'; Nw=Nf; number of angular frequencies between 0 and Nyquist

Computing Power Spectral Density

% compute Fourier coefficients mest = fft(d);

% compute amplitude spectral density s=abs(mest(1:Nw));

% compute power spectral density s2=s^2;

Spectral Density

% compute Fourier coefficients mest = fft(d);

% compute amplitude spectral density s=abs(mest(1:Nw));

% compute power spectral density s2=s^2;

note that negative frequencies are discardedsince they are redundant

Why The Switch to Complex?

One function, exp(), is better than two, cos() & sin()exp() is algebraically simpler than cos() & sin(),

e.g. exp(a) / exp(b) = exp( a – b)which substantially simplifies formulas

Tradition – that’s the way everyone does it! Or more importantly, that’s the way MatLab does it.

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

Example of Fourier transform

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 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

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