8.8 Examples  8. Spectral Analysis  8.6 Relationship of autocovariance

8.7 Plot conventions

In a linear plot of the spectral density estimate $\hat\Gamma(\nu)$ vs frequency, the area under the curve represents the contribution of each frequency interval to the variance. However, it is usually the case that the range of frequencies and amplitudes is better displayed with a logarithmic display. When the frequency is shown with a logarithmic scale, it is common practice to show the spectrum multiplied by the frequency because the plot then shows the contribution to the variance from each logarithmic interval in frequency rather than each linear interval in frequency.

To see this, let $F(\nu)$ be the cumulative variance spectrum giving the variance arising from all frequencies smaller than $\nu$. Then

$$\Gamma(\nu) = {{dF(\nu)}\over{d\nu}} = {{dF}\over{d\log \nu}} \times{{d\log \nu}\over{d\ln \nu}} \times {{d \ln\nu}\over{d\nu}}$$

$$= {{dF(\nu)}\over{d\log\nu}} \times {{1}\over{\ln(10)}} \times {{1}\over{\nu}}$$ (8.31)

 

so

$${{dF(\nu)}\over{d\log\nu}} = \ln (10) \nu \Gamma(\nu) \ .$$ (8.32)

 

Another advantage of this plot format is that the spectral density is the same whether plotted in terms of frequency $\nu$ or wavenumber $k=2\pi/\lambda$, where $\lambda$ is the wavelength, for spectra where $\nu$ and $\lambda$ are related to a speed of motion through the field V by $V=\nu\lambda$:

$$\nu{{dF}\over{d\nu}} = \nu{{dF}\over{dk}} {{dk}\over{d\nu}} =k{{dF}\over{dk}} \ .$$ (8.33)

 

For variance spectra measured from an aircraft, for example, representing the spectrum in terms of the wavenumber is preferable because the measured frequency is dependent on the flight speed through the field while the wavenumber is not. Equation (8.33) shows that all that is required to transform from one representation to the other is a relabeling of the frequency axis to the corresponding wavenumber.

   
 8.8 Examples  8. Spectral Analysis  8.6 Relationship of autocovariance 

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