0 C t U t U t U t U t U t U t U t U t U t U t U t U t U t U t U t U t U t U M M M M M M 2 1 2 2 2 1 2 1 2 1 1 1 M f f f 2 1 0 0 0 0 0 0 M f f f 2 1 0 0 0 2 2 1 2 2 2 2 1 1 2 1 2 2 1 1 1 1 M M M M M M m M m M f t U t U f t U t U f t U t U f t U t U f t U t U f t U t U f t U t U f t U t U f t U t U t U t U C k m mk I is the unit matrix and are the EOFs igenvalue problem corresponding to a linear system: t a f t U M i i im m 1
I is the unit matrix and are the EOFs. Eigenvalue problem corresponding to a linear system:. Sometimes use another complex form of EOFS. Apply the Hilbert Transform of U to understand a bit more about the phase of propagation of the phenomenon. - PowerPoint PPT Presentation
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Transcript
0 C
tUtUtUtUtUtU
tUtUtUtUtUtUtUtUtUtUtUtU
MMMM
M
M
21
22212
12111
Mf
ff
2
1
00
0000
Mf
ff
2
1
0
00
221
2222112
1221111
MMMMMM
mM
mM
ftUtUftUtUftUtU
ftUtUftUtUftUtUftUtUftUtUftUtU
tUtUC kmmk
I is the unit matrix and are the EOFs
Eigenvalue problem corresponding to a linear system:
taftUM
iiimm
1
taftUM
iiimm
1
Sometimes use another complex form of EOFS. Apply the Hilbert Transform of U to understand a bit more about the phase of propagation of the phenomenon
In essence, the Hilbert transform of a signal (time series) produces a complex variable with real part identical to the time series and the imaginary part is shifted 90º from the original (sometimes called the analytical signal):
>> t=[1:1:1000];>> u1=sin(2*pi*t/200);>> plot(t,u1,’LineWidth’,3)>>>> u2=hilbert(u1); >> hold on;>> plot(real(u2),'r--','LineWidth',3)
u
t
>> t=[1:1:1000];>> u1=sin(2*pi*t/200);>> plot(t,u1,’LineWidth’,3)>>>> u2=hilbert(u1); >> hold on>> plot(real(u2),'r--','LineWidth',3)>> plot(imag(u2),'g','LineWidth',3)
u
t
Hilbert Transform of U can be regarded as the convolution of U with h(t) = 1/(t)
dthUPUH
d
tUP1P is the Cauchy principal value
(assigns values to improper integrals)
hilbert uses a four-step algorithm:
1. It calculates the FFT of the input sequence, storing the result in a vector x. 2. It creates a vector h whose elements h(i) have the values:
1 for i = 1, (n/2)+12 for i = 2, 3, ... , (n/2)0 for i = (n/2)+2, ... , n3. It calculates the element-wise product of x and h.4. It calculates the inverse FFT of the sequence obtained in step 3 and returns the first n elements of the result.