#    ### 資料紹介

Spectrum Analysis
1 Population Spectrum
A random variable Yt, which follows weak stationary process, can be represented as a weighted sum of
cos(!t) and sin(!t), letting! be a certain frequency. The form of this represantation is
Yt = +
Z
0
(!)cos(!t)d! +
Z
0
(!)sin(!t)d!
The basic idea of spectral analysis is to measure how important diﬀerent frequencies are to explain the
changing of Yt. All of the weak stationary process has the two form of representation; frequency domain
represantation and time domain representation, which do not exclude one other.
LetfYtg
1
t= 1 be weak stationary process with mean and j
th autocovariancej , wherej is absolutely
summable. Then deﬁne the autocovariance generating functiongY(z) to be
gY(z)
def
=
1X
j = 1
j zj
In the equation above, lettingzj = e
i! and dividing by 2 , we obtain
SY(!) =
1
2
gY(e
i! ) =
1
2
1X
j = 1
j e
i!j (1)
Here notice that a spectrum is a function of!, therefore giving a value of! and fj g
1
j = 1 ,SY(!) can be
calculated. According to De Moivre’s theorem, which states that
e
i!j = cos(!j ) sin(!j )
and substituing thie equation into Eq.(1), we get
SY(!) =
1
2
1X
j = 1
j
cos(!j ) i sin(!j )
=
1
2
0(cos0 i sin0) +
1
2
1X
j =1
cos(!j ) + cos( !j ) i sin(!j ) i sin( !j )
=
1
2
h
0 + 2
1X
j =1
j cos(!j )
i
(2)
where the second equality holds from the fact thatj = j in a weak stationary process. Iffj g
1
j = 1 is
absolutely summable then there exists population spectrum by Eq.(2) andSY(!) is a continuous function
of!. Als

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Spectrum Analysis
1 Population Spectrum
A random variable Yt, which follows weak stationary process, can be represented as a weighted sum of
cos(!t) and sin(!t), letting! be a certain frequency. The form of this represantation is
Yt = +
Z
0
(!)cos(!t)d! +
Z
0
(!)sin(!t)d!
The basic idea of spectral analysis is to measure how important diﬀerent frequencies are to explain the
changing of Yt. All of the weak stationary process has the two form of representation; frequency domain
represantation an..

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