Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 174

giving

∞

h ∗ ( t H ∗ ( f ) e − i 2π f ( t − τ) df . − τ) = (2.323)

−∞

Thus, the autocorrelation of g( t ) becomes

∞

h ( t ) ∞
−∞

1
T

H ∗ ( f ) e − i 2π f ( t − τ) dfdt φ gg (τ) = lim

T →∞

−∞

∞ ∞

1
T

H ∗ ( f ) e i 2π f τ h ( t ) e − i 2π ft dt = lim

T →∞

· · df , (2.324)

−∞ −∞

on reversing the order of integration. Now,
∞

h ( t ) e − i 2π ft dt = H ( f ), (2.325)

−∞

the Fourier transform of h ( t ). Hence,

∞

1
T

H ∗ ( f ) H ( f ) e i 2π f τ df φ gg (τ) = lim

T →∞

−∞

∞
−∞ |

∞

1
T

2 e i 2π f τ df

S gg ( f ) e i 2π f τ df ,

= lim

T →∞

H ( f ) | = (2.326)

−∞

indicating that the power spectral density is

1
T |

2

S gg ( f ) = lim

T

H ( f ) |

→∞

∞

h ( t ) e − i 2π ft dt

2

1
T

= lim

T →∞

−∞

− T /2 g( t ) e − i 2π ft dt

T /2 2

1
T

= lim

T →∞

. (2.327)

In any practical case, we are confined to records of finite length and have only
the sample power spectral density estimator ,

− T /2 g( t ) e − i 2π ft dt

T /2

2

1
T

S gg ( f )

=

∞

h ( t ) e − i 2π ft dt

2

1
T

= (2.328)

−∞

1
T |

2

= H ( f ) | , (2.329)

and the sample autocorrelation estimator ,

∞
−∞ |

1
T

˜

2 e i 2π f τ df .

φ gg (τ) = H ( f ) | (2.330)