Time sequence and spectral analysis - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 184

on changing the integration variable to t =− t . Thus, the Fourier transform of the

time reverse of the window w( t ) is the complex conjugate of W ( f ). Given that the
windoww( t ) is real and even, W ( f ) is real. The convolution satisfies
∞

∞

w n ( s )w ∗ n + m ( s − τ) ds = w n ( s )w n + m (τ − s ) ds

−∞ −∞

∞ ∞

w n + m ( u )w ∗ n ( u = w n + m ( u )w n (τ − u ) du = − τ) du , (2.377)

−∞ −∞

on changing the integration variable to u s , and utilising the real and even

properties of the windows. Using the foregoing properties and the convolutional
relations (2.269) and (2.274), expression (2.374) becomes
E S gg, n ( f ) S gg, n + m ( f )

= τ −

I 2 ∞

f 1 ) W 2 ( f 1 ) df 1 2

1

= −

S gg ( f

−∞

I 2 ∞

b n ) W 2 ( f 1 ) df 1 2

1

f 1 )exp i 2π f 1 ( b n + m −

+ S gg ( f − . (2.378)

−∞

From (2.360), we recognise that

I 2 ∞

f 1 ) W 2 ( f 1 ) df 1 2

E S gg, n ( f ) 2

1

S gg ( f − = , (2.379)

−∞

the square of the expected value of the sample spectral density estimate. Substitut-
ing in (2.365), we find for the covariance,
cov S gg, n ( f ),

S gg, n + m ( f )

I 2 ∞

b n ) W 2 ( f 1 ) df 1 2

1

f 1 )exp i 2π f 1 ( b n + m − = S gg ( f −

−∞

∞

b n ) W 2 ( f ) df 2

S 2

gg ( f )
I 2

exp i 2π f ( b n + m −

≈ , (2.380)

−∞

for spectral densities assumed to vary only slightly over the passband of W 2 ( f ).
The window w n + m ( t ), centred on b n + m , is displaced along the time axis by m Δ s

from the window w n ( t ), centred on b n , where s is the separation of successive

windows. Now consider the convolution of the windoww n ( t ) with the time reverse
of the windoww n + m ( t ),

Δ

∞

w n ( t )w ∗ n + m ( t J m (τ) = − τ) dt . (2.381)

−∞