Series Fourier
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Transcript of Series Fourier
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Emery & Thomson (2005) : Chapter 5: Time Series Analysis Methods Basic Concepts (375-378) Correlation Function (p. 378-384) Fourier Analysis (p.384-388)
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Fourier Series
. any function could be written as an infinite sum of the trigonometric functions cosine and sine..
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purpose of the lecture
detect and quantify periodicities in data
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importance of periodicities
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Time Series SSH & SST (2007-2012)
Annual Fundamental period Strength of amplitude ?
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Time Series Zonal & Merid Current (2007-2012)
Fundamental period ? Strength of amplitude ?
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Decomposition of SST time series
Periodicity: Inter-annual, Annual, Semi-annual, Intra-seasonal
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Temporal Periodicities and their periods
astronomical
rotation daily
revolution
yearly
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other natural
ocean waves a few seconds
ocean currents days months
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anthropogenic
electric power 60 Hz
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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 oscillation f(t) = C cos{ 2 (t-t0) / T }
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amplitude, C
Periodicities
temporal f(t) = C cos{ 2 t / T } spatial f(x) = C cos{ 2 x / }
amplitude, Cperiod, T wavelength,
frequency, f=1/Tangular frequency, =2 /T wavenumber, k=2 / -
f(t) = C cos(t) f(x) = C cos(kx)
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spatial periodicities and their wavelengths
natural
sand dunes hundreds of meters
tree rings a few millimeters
anthropogenic
furrows plowed in a field
few tens of cm
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pairing sines and cosines to avoid using time delays
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derived using trigonometri identity
A B
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A BA=C cos(t0)B=C sin(t0)
A2=C cos2 (t0)B2=C sin2 (t0)
A2+B2=C2 [cos2 (t0)+sin2 (t0)]= C2
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Fourier Series linear model containing nothing but
sines and cosines
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As and Bs are model parameters
s are auxiliary variables
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two choices
values of frequencies?
total number of frequencies?
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surprising fact about time series with evenly sampled data
Nyquist frequency
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values of frequencies? evenly spaced, n = (n-1) minimum frequency of zero maximum frequency of fny
total number of frequencies? N/2+1
number of model parameters, M = number of data, N
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implies
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Number of Frequencies why N/2+1 and not N/2 ?
first and last sine are omitted from the
Fourier Series since they are identically zero:
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-2 0 20
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col 1
time,
s
-2 0 20
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col 2-2 0 20
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col 3-2 0 20
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col 4-2 0 20
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col 5-2 0 20
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col 32cos( t)cos(0t) sin(t) cos(2 t) sin(2 t)cos(N t)
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Nyquist Sampling Theorem
when m=n+Nanother way of stating itnote evenly
sampled times
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problem of aliasing
high frequencies
masquerading as low frequencies
solution: pre-process data to remove high
frequencies before digitizing it
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Discrete Fourier Series
d = Gm
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Least Squares Solution mest = [GTG]-1 GTd has substantial simplification
since it can be shown that
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% N = number of data, presumed even % Dt is time sampling interval t = 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 setup in MatLab
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% 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
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gtgi = 2* ones(M,1)/N; gtgi(1)=1/N; gtgi(M)=1/N; mest = gtgi .* (G'*d);
solving for model parameters in MatLab
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how to plot the model parameters? As and B s plot
against frequency
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power spectral density
big at frequency when when sine or cosine at the frequency
has a large coefficient
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alternatively, plot amplitude spectral density
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Tambahan . Transformasi Fourier
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2. Fourier Transform
Transformasi Fourier adalah transformasi matematika dengan banyak aplikasi dalam fisika dan teknik.
Mengubah fungsi matematika waktu f(t) menjadi fungsi baru F atau dengan argumen frekuensi dg satuan siklus/detik (hertz).
Fungsi baru ini dikenal sebagai Transformasi Fourier atau spektrum frekuensi dari fungsi f.
f disebut domain waktu, dan F adalah domain frekuensi. Transformasi Fourier dinyatakan sebagai :
Variabel bebas x mewakili waktu (detik) dan variabel transformasi merupakan frekuensi (hertz).
untuk setiap bilangan riil (zeta)
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Transformasi Fourier menghubungkan fungsi domain waktu f (ditampilkan dalam warna merah)
Visualisasi dari Transformasi Fourier
time
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terhadap fungsi domain frekuensi (ditampilkan dalam warna biru)
Visualisasi dari Transformasi Fourier
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Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
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Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
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Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi
Visualisasi dari Transformasi Fourier
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Yang ditampilkan sebagai puncak dalam domain frekuensi
Visualisasi dari Transformasi Fourier
frequency
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Visualisasi dari Transformasi Fourier
Fungsi f dalam domain waktu
Hasil transformasi Fourier dalam domain frekuensi
Transformasi Fourier
frequency
time
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14 day
27 day
27 day
14 day
Wattimena et al. (2014)
Time-series arus zonal
Time-series arus zonal