Embed Size (px)
DESCRIPTION
Power spectral density . frequency-side, , vs. time-side, t /2 : frequency (cycles/unit time). Non-negative Unifies analyses of processes of widely varying types. Examples. Spectral representation . stationary increments - Kolmogorov. - PowerPoint PPT Presentation
Citation preview
Power spectral density. frequency-side, , vs. time-side, t
/2 : frequency (cycles/unit time)
|| largefor 21
~
)(}exp{21
21
)]()(}[exp{21
)(
N
NNN
NNNNN
p
duuquip
duuqpuuif
Non-negative
Unifies analyses of processes of widely varying types
Examples.
Spectral representation. stationary increments - Kolmogorov
)(}exp{/)(
)(1}exp{
)(
N
N
dZitdttdN
dZiit
tN
})(){(},cov{
increments orthogonal
)()()}(),(cov{
order of spectrumcumulant
...),...,()...()}(),...,({
)()}({
)()(dZ valued,-complex random, :
111...11
N
YX
NNNN
KKNNKKNN
N
NN
YXEYX
ddfdZdZ
K
ddfdZdZcum
ddZE
dZZ
Frequency domain approach. Coherency, coherence
Cross-spectrum.
duuquif MNMN )(}exp{21
)(
Coherency.
R MN() = f MN()/{f MM() f NN()}
complex-valued, 0 if denominator 0
Coherence
|R MN()|2 = |f MN()| 2 /{f MM() f NN()|
|R MN()|2 1, c.p. multiple R2
where
A() = exp{-iu}a(u)du
fOO () is a minimum at A() = fNM()fMM()-1
Minimum: (1 - |RMN()|2 )fNN()
0 |R MN()|2 1
AAfAfAfff MMNMMNNNOO
Proof. Filtering. M = {j }
a(t-v)dM(v) = a(t-j )
Consider
dO(t) = dN(t) - a(t-v)dM(v)dt, (stationary increments)
Proof.
0
Take
0
sderivative second andfirst Consider
1
1
MNMMNMNN
MMNM
OO
MMNMMNNNOO
ffff
ffA
f
AAfAfAfff
Coherence, measure of the linear time invariant association of the components of a stationary bivariate process.
Empirical examples.
sea hare
Muscle spindle
Spectral representation approach.
b.v. of ,)()()}(),(cov{
)(}exp{/)(
)(}exp{/)(
NMMNNM
N
M
FddFdZdZ
dZitdttdN
dZitdttdM
Filtering.
dO(t)/dt = a(t-v)dM(v) = a(t-j )
= exp{it}dZM()
Partial coherency. Trivariate process {M,N,O}
]}||1][||1{[/][ 22
| ONMOONMOMNOMN ffffff
“Removes” the linear time invariant effects of O from M and N
