Editing Spectral density (section)

=== Cross power spectral density {{anchor|Cross|Cross spectral density|Cross-spectral density}} ===
{{See also|Coherence (signal processing)}}

Given two signals <math>x(t)</math> and <math>y(t)</math>, each of which possess power spectral densities <math>S_{xx}(f)</math> and <math>S_{yy}(f)</math>, it is possible to define a '''cross power spectral density''' ('''CPSD''') or '''cross spectral density''' ('''CSD''').  To begin, let us consider the average power of such a combined signal.
<math display="block">\begin{align}
  P &= \lim_{T\to \infty} \frac{1}{T} \int_{-\infty}^{\infty} \left[x_T(t) + y_T(t)\right]^*\left[x_T(t) + y_T(t)\right]dt
\\
    &= \lim_{T\to \infty} \frac{1}{T} \int_{-\infty}^{\infty} |x_T(t)|^2 + x^*_T(t) y_T(t) + y^*_T(t) x_{T}(t) + |y_T(t)|^2 dt
\\
\end{align}</math>

Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain
<math display="block">\begin{align}
  S_{xy}(f) &= \lim_{T\to\infty} \frac{1}{T} \left[\hat{x}^*_T(f) \hat{y}_T(f)\right] &
  S_{yx}(f) &= \lim_{T\to\infty} \frac{1}{T} \left[\hat{y}^*_T(f) \hat{x}_T(f)\right]
\end{align}</math>
where, again, the contributions of <math>S_{xx}(f)</math> and <math>S_{yy}(f)</math> are already understood.  Note that <math>S^*_{xy}(f) = S_{yx}(f)</math>, so the full contribution to the cross power is, generally, from twice the real part of either individual '''CPSD'''. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit <math>T\to\infty</math> becomes the Fourier transform of a [[cross-correlation]] function.<ref>{{cite web|author=William D Penny|year=2009|title=Signal Processing Course, chapter 7|url=http://www.fil.ion.ucl.ac.uk/~wpenny/course/course.html}}</ref>
<math display="block">\begin{align}
  S_{xy}(f) &= \int_{-\infty}^{\infty} \left[\lim_{T\to\infty} \frac 1 {T} \int_{-\infty}^{\infty} x^*_{T}(t-\tau) y_{T}(t) dt \right] e^{-i 2 \pi f \tau} d\tau= \int_{-\infty}^{\infty} R_{xy}(\tau) e^{-i 2 \pi f \tau} d\tau \\
  S_{yx}(f) &= \int_{-\infty}^{\infty} \left[\lim_{T\to\infty} \frac 1 {T} \int_{-\infty}^{\infty} y^*_{T}(t-\tau) x_{T}(t) dt \right] e^{-i 2 \pi f \tau} d\tau= \int_{-\infty}^{\infty} R_{yx}(\tau) e^{-i 2 \pi f \tau} d\tau,
\end{align}</math>
where <math>R_{xy}(\tau)</math> is the [[cross-correlation]] of <math>x(t)</math> with <math>y(t)</math> and <math>R_{yx}(\tau)</math> is the cross-correlation of <math>y(t)</math> with <math>x(t)</math>.  In light of this, the PSD is seen to be a special case of the CSD for <math>x(t) = y(t)</math>.  If <math>x(t)</math> and <math>y(t)</math> are real signals (e.g. voltage or current), their Fourier transforms <math>\hat{x}(f)</math> and <math>\hat{y}(f)</math> are usually restricted to positive frequencies by convention.  Therefore, in typical signal processing, the full '''CPSD''' is just one of the '''CPSD'''s scaled by a factor of two.
<math display="block">\operatorname{CPSD}_\text{Full} = 2S_{xy}(f) = 2 S_{yx}(f)</math>

For discrete signals {{math|''x<sub>n</sub>''}} and {{math|''y<sub>n</sub>''}}, the relationship between the cross-spectral density and the cross-covariance is
<math display="block">S_{xy}(f) = \sum_{n=-\infty}^\infty R_{xy}(\tau_n)e^{-i 2 \pi f \tau_n}\,\Delta\tau</math>