Editing Spectral density (section)

==== Properties of the power spectral density ====

Some properties of the PSD include:{{sfn | Miller | Childers | 2012 | p=431}}
{{bulleted list
| The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an [[even function]] of frequency: <math>S_{xx}(-f) = S_{xx}(f)</math>.
| For a continuous [[stochastic process]] x(t), the autocorrelation function ''R''<sub>''xx''</sub>(''t'') can be reconstructed from its power spectrum S<sub>xx</sub>(f) by using the [[inverse Fourier transform]]
| Using [[Parseval's theorem]], one can compute the [[variance]] (average power) of a process by integrating the power spectrum over all frequency:
<math display="block">P = \operatorname{Var}(x) = \int_{-\infty}^{\infty}\! S_{xx}(f) \, df</math>
| For a real process ''x''(''t'') with power spectral density  <math>S_{xx}(f)</math>, one can compute the ''integrated spectrum'' or ''power spectral distribution'' <math>F(f)</math>, which specifies the average ''bandlimited'' power contained in frequencies from DC to f using:{{sfn | Davenport | Root | 1987}}
<math display="block">F(f) = 2 \int _0^f S_{xx}(f')\, df'. </math>
Note that the previous expression for total power (signal variance) is a special case where&nbsp;{{math|''f'' → ∞}}.
}}