Spectrum Analysis

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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 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

    資料の原本内容( この資料を購入すると、テキストデータがみえます。 )

    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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