Series Fourier

44
Emery & Thomson (2005) : Chapter 5: Time Series Analysis Methods Basic Concepts (375-378) Correlation Function (p. 378-384) Fourier Analysis (p.384-388)

description

Bahan kuliah

Transcript of Series Fourier

  • Emery & Thomson (2005) : Chapter 5: Time Series Analysis Methods Basic Concepts (375-378) Correlation Function (p. 378-384) Fourier Analysis (p.384-388)

  • Fourier Series

    . any function could be written as an infinite sum of the trigonometric functions cosine and sine..

  • purpose of the lecture

    detect and quantify periodicities in data

  • importance of periodicities

  • Time Series SSH & SST (2007-2012)

    Annual Fundamental period Strength of amplitude ?

  • Time Series Zonal & Merid Current (2007-2012)

    Fundamental period ? Strength of amplitude ?

  • Decomposition of SST time series

    Periodicity: Inter-annual, Annual, Semi-annual, Intra-seasonal

  • Temporal Periodicities and their periods

    astronomical

    rotation daily

    revolution

    yearly

    ..

    other natural

    ocean waves a few seconds

    ocean currents days months

    .

    anthropogenic

    electric power 60 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 oscillation f(t) = C cos{ 2 (t-t0) / T }

  • 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)

  • 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

  • pairing sines and cosines to avoid using time delays

  • derived using trigonometri identity

    A B

  • 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

  • Fourier Series linear model containing nothing but

    sines and cosines

  • As and Bs are model parameters

    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, 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

  • implies

  • 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:

  • -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( t)cos(0t) sin(t) cos(2 t) sin(2 t)cos(N t)

  • Nyquist Sampling Theorem

    when m=n+Nanother way of stating itnote evenly

    sampled times

  • problem of aliasing

    high frequencies

    masquerading as low frequencies

    solution: pre-process data to remove high

    frequencies before digitizing it

  • Discrete Fourier Series

    d = Gm

  • Least Squares Solution mest = [GTG]-1 GTd has substantial simplification

    since it can be shown that

  • % 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

  • % 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? As and B s plot

    against frequency

  • power spectral density

    big at frequency when when sine or cosine at the frequency

    has a large coefficient

  • alternatively, plot amplitude spectral density

  • Tambahan . Transformasi Fourier

  • 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)

  • Transformasi Fourier menghubungkan fungsi domain waktu f (ditampilkan dalam warna merah)

    Visualisasi dari Transformasi Fourier

    time

  • terhadap fungsi domain frekuensi (ditampilkan dalam warna biru)

    Visualisasi dari Transformasi Fourier

  • Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi

    Visualisasi dari Transformasi Fourier

  • Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi

    Visualisasi dari Transformasi Fourier

  • Komponen-komponen frekuensi tersebar di seluruh spektrum frekuensi

    Visualisasi dari Transformasi Fourier

  • Yang ditampilkan sebagai puncak dalam domain frekuensi

    Visualisasi dari Transformasi Fourier

    frequency

  • Visualisasi dari Transformasi Fourier

    Fungsi f dalam domain waktu

    Hasil transformasi Fourier dalam domain frekuensi

    Transformasi Fourier

    frequency

    time

  • 14 day

    27 day

    27 day

    14 day

    Wattimena et al. (2014)

    Time-series arus zonal

    Time-series arus zonal