8.7 Plot conventions  8. Spectral Analysis  8.5 The autocorrelation and

8.6 Relationship of autocovariance function to variance spectrum

The following derivation shows that the variance spectrum and the autocovariance function are Fourier transforms of each other:  

$${rl}\Gamma(\nu) = {\rm lim}_{T\rightarrow\infty}\vert c_k\vert^2 T, \ \ (\nu =k/T)$$

$$= {\rm lim}_{T\rightarrow\infty}\Bigl\{T\bigl({{1}\over{T}}\int_{-T......\int_{-T/2}^{T/2}f^\prime(t^\prime)e^{-i2\pi\nut^\prime}dt^\prime\bigr)\Bigr\}$$ (8.26)

 
where $f^\prime(t)=f(t)-\overline{f(t)}$ is the time series with mean subtracted from all values. If the integration variable is changed from t to $\tau=t-t^\prime$, so that $t=t^\prime+\tau$ and $-T\le\tau\le T$, then

$${rl}\Gamma(\nu) = {\rm lim}_{T\rightarrow\infty} \int_{-T}^T\Bigl[{{1}\over{T}}\int......f^\prime(t^\prime+\tau)f^\prime(t^\prime)dt^\prime\Bigr]e^{-i2\pi\nu\tau}d\tau$$

$$= \int_{-\infty}^{\infty}V_{ff}(\tau)e^{-i2\pi\nu\tau}d\tau \ \ .$$ (8.27)

 
This shows that the variance spectrum is the Fourier transform of the autocovariance function, and similarly the autocovariance function is the Fourier transform of the variance spectrum:

$$V_{ff}(\tau) = \int_{-\infty}^\infty\Gamma(\nu)e^{i2\pi\nu\tau}d\nu .$$ (8.28)

 

Note that the total variance is

$$\sigma^2 = V_{ff}(0) = \int_{-\infty}^\infty\Gamma(\nu)d\nu .$$ (8.29)

 
 

Exercise 8.1: Suppose that the variance spectrum for the vertical wind is given by

$$\Gamma(\nu) = 0.67 {{2\pi}\over{V}}^{-2/3} \epsilon^{2/3} \nu^{-5/3}$$ (8.30)

 

as is expected for observations made in an "inertial subrange" from an aircraft flying at airspeed V. (Here $\epsilon$ characterizes the eddy dissipation rate, and has units of (energy)(mass)^-1(time)^-1.) Find the expected functional form of the autocorrelation function.

   
 8.7 Plot conventions  8. Spectral Analysis  8.5 The autocorrelation and 

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