8.5 The autocorrelation and  8. Spectral Analysis  8.3 The Fourier integral

8.4 The variance spectrum

The Fourier line spectrum represents the contribution of a set of discrete frequencies to the variance. However, that set of frequencies may be an arbitrary result of the sampling strategy (i.e., the selection of $\Delta T$ and T) rather than a reflection of physical reality. An example is measurement of atmospheric turbulence, where the frequencies contributing to the variance probably cover a continuous range. The line spectrum calculated from measurements of atmospheric turbulence is probably an artificially discrete representation of a spectrum that is really continuous.

For this reason, the line spectrum is often considered an estimator for a continuous spectrum. The formal relationship can be obtained by considering the limit at large T, by analogy with the Fourier integral. The limit for the variance in the time series is

$$= {\rm lim}_{T\rightarrow\infty} {{1}\over{T}}\int_{-T/2}^{T/2} \bigl(f(t)-\overline{f(t)}\bigr)^2 dt$$  V_ff

$$= {\rm lim}_{T\rightarrow\infty} \sum_{k\ne 0}\vert c_k\vert^2T{{1}\over{T}}$$ (8.15)

 

so with the substitutions $\nu=k/T$, $d\nu=1/T$, and $\Gamma(\nu)=\vert c_k\vert^2T$,

$$V_{ff} = \int_{-\infty}^\infty \Gamma(\nu)d\nu \ .$$ (8.16)

 

The variance spectrum $\Gamma(\nu)$ is thus the limit of |c_k|²T as T becomes infinite.

Because all real measurement sequences are finite, this limiting function is never measured directly, but it can still serve as the model underlying estimation of the variance spectrum: we use $\hat \Gamma(\nu)=T\vert c_k\vert^2$, which can be determined from a finite measured series, as an estimator for $\Gamma(\nu)$. Expressions (8 .16) and (8 .6) for the Fourier transform can be used to show that

$$\hat\Gamma(\nu) = {{1}\over{T}} \vert\hat f(\nu)\vert^2$$ (8.17)

 

where $\hat f(\nu)$ is an estimator for the Fourier transform obtained over the available finite interval:

$$\hat f(\nu) = \int_{-T/2}^{T/2} f(t)e^{-i2\pi\nu t} dt \ .$$ (8.18)

 

For N discrete samples $\{f_k\}$ separated by time intervals $\Delta T$, and if N=2n with n an integer, the result is

$$\hat \Gamma(\nu) = {{\Delta T}\over{N}} \Bigl\vert\sum_{k=-n}...... , \ \ -{{1}\over{2\DeltaT}}\le\nu <{{1}\over{2\Delta T}} \ .$$ (8.19)

 

In terms of real functions, this is equivalent to

$$\hat \Gamma(\nu) = {{\Delta T}\over{N}} \left\{\left(\sum_{k=......\sum_{k=-n}^{n-1}f_k\sin(2\pi\nu k\Delta T)\right)^2\right\}$$ (8.20)

 

With the preceding equation, an estimator for the power spectrum can be obtained from a set of measurements $\{f_k\}$. Usually, this form is recombined into a spectrum $\hat \Gamma^\prime (\nu)$ that varies in frequency from 0 to $\nu_n$ instead of from $-\nu_n$ to $\nu_n$; except for the extreme points, this spectrum is $2\Gamma(\nu)$ for $\nu >0$.

A problem with this estimator of the variance spectrum is that it has high variance about the true spectrum, so plots of $\Gamma(\nu)$ vs $\nu$ will show high scatter. Smoothing or averaging of values is usually necessary, as discussed later in this chapter.

   
 8.5 The autocorrelation and  8. Spectral Analysis  8.3 The Fourier integral 

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