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

purpose of the lecture

detect and quantify periodicities in data

importance of periodicities

Stream FlowNeuse River

disc

harg

e, c

fs

time, days

365 days1 year

Air temperatureBlack Rock Forest

365 days1 year

Air temperatureBlack Rock Forest

time, days

1 day

temporal periodicitiesand their periods

astronomical

rotationdaily

revolutionyearly

other natural

ocean wavesa few seconds

anthropogenic

electric power60 Hz

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 }

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

angular frequency, ω=2 π /T wavenumber, k=2 π / λ-

f(t) = C cos(ωt) f(x) = C cos(kx)

spatial periodicitiesand their wavelengths

natural

sand duneshundreds of meters

tree ringsa few millimeters

anthropogenic

furrows plowed

in a fieldfew tens of cm

pairing sines and cosinesto avoid using time delays

derived using trig identity

A B

A BA=C cos(ωt0)B=C sin(ωt0) A2=C cos2 (ωt0)B2=C sin2 (ωt0)A2+B2=C2 [cos2 (ωt0)+sin2 (ωt0)]= C2

Fourier Serieslinear model containing nothing but

sines and cosines

A’s and B’s aremodel parameters

ω’s are auxiliary variables

two choices

values of frequencies?

total number of frequencies?

surprising fact about time series with evenly sampled data

Nyquist frequency

values of frequencies?

evenly spaced, ωn = (n-1)Δ ω

minimum frequency of zero

maximum frequency of fnytotal number of frequencies? N/2+1

number of model parameters, M= number of data, N

implies

Number of Frequencieswhy N/2+1 and not N/2 ?

first and last sine are omitted from the Fourier Series since they are

identically zero:

-2 0 20

5

10

15

20

25

30

col 1

tim

e,

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(½NΔω t)

Nyquist Sampling Theorem

when m=n+Nanother way of stating it

note evenly sampled times

ωn = (n-1)Δ ω and tk = (k-1) Δt

Step 1: Insert discrete frequencies and times into l.h.s. of equations.

ωn = (n-1+N)Δ ω and tk = (k-1) Δt

Step 2: Insert discrete frequencies and times into r.h.s. of equations.

same as l.h.s.

same as l.h.s.

Step 3: Note that l.h.s equals r.h.s.

when m=n+Nor when

ωm=ωn+2ωny

only a 2ωny interval of the ω -axis is uniquesay from-ωny to +ωny

Step 4: Identify unique region of ω-axis

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

Step 5: Apply symmetry of sines and cosines

w

-wny wny 2wny 3wny0

equivalent points on the ω-axis

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,

problem of aliasing

high frequenciesmasquerading as low frequencies

solution:pre-process data to remove high

frequenciesbefore digitizing it

Discrete Fourier Series

d = Gm

Least Squares Solution mest = [GTG]-1 GTdhas substantial simplification

… since it can be shown that …

% N = number of data, presumed even% Dt is time sampling intervalt = Dt*[0:N-1]’;

Df = 1 / (N * Dt );Dw = 2 * pi / (N * Dt);

Nf = N/2+1;Nw = N/2+1;

f = Df*[0:N/2];w = Dw*[0:N/2];

frequency and time setupin MatLab

% set up G G=zeros(N,M); % zero frequency column G(:,1)=1; % interior M/2-1 columns for i = [1:M/2-1] j = 2*i; k = j+1; G(:,j)=cos(w(i+1).*t); G(:,k)=sin(w(i+1).*t); end % nyquist column G(:,M)=cos(w(Nw).*t);

Building G in MatLab

gtgi = 2* ones(M,1)/N; gtgi(1)=1/N; gtgi(M)=1/N; mest = gtgi .* (G'*d);

solving for model parameters in MatLab

how to plot the model parameters?A’s and B ’splot

against frequency

power spectral density

big at frequency ω when

when sine or cosine at the frequency

has a large coefficient

alternatively, plot

amplitude 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 days

period, days

frequency, cycles per day

ampl

itud

e sp

ectr

al d

ensi

ty

182.6 days60.0 days

ampl

itud

e sp

ectr

al d

ensi

ty

Stream FlowNeuse River

all interesting frequencies near origin,so plot period, T=1/f instead