3.3 The Power Spectrum
Definition. Power spectrum
The power spectrum or power spectral density (PSD) of a W.S.S. random process $ \mathbf{X}\left(t\right) $ , real or complex, is the Fourier transform of the autocorrelation function:
$ S_{\mathbf{XX}}\left(\omega\right)\triangleq\int_{-\infty}^{\infty}R_{\mathbf{XX}}\left(\tau\right)e^{-i\omega\tau}d\tau $
where $ R_{\mathbf{XX}}\left(\tau\right)=E\left[\mathbf{X}\left(t+\tau\right)\mathbf{X}^{*}\left(t\right)\right]. $
Note
1. Because $ R_{\mathbf{XX}}\left(-\tau\right)=R_{\mathbf{XX}}^{*}\left(\tau\right) $ , $ S_{\mathbf{XX}}\left(\omega\right) $ is a real function.
2. $ R_{\mathbf{XX}}\left(\tau\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_{\mathbf{XX}}\left(\omega\right)e^{i\omega\tau}d\omega $ . (Fourier inversion formula)
3. In order to consider $ S_{\mathbf{XX}}\left(\omega\right) $ , we assume $ \mathbf{X}\left(t\right) $ is at least W.S.S.
4. The PSD of $ \mathbf{X}\left(t\right) $ is a non-negative valued function of $ \omega $ . $ (\because R_{\mathbf{XX}}\left(\tau\right) $ is non-negative definite.)
Note
The PSD gives the average distribution of power in frequency for a random process.
Key result
If $ \mathbf{X}\left(t\right) $ is a W.S.S. random process and it is the input to a stable L.T.I. system with impulse response $ h\left(t\right) $ , then the output $ \mathbf{Y}\left(t\right) $ has PSD
$ S_{\mathbf{YY}}\left(\omega\right)=S_{\mathbf{XX}}\left(\omega\right)\left|H\left(\omega\right)\right|^{2} $
where $ H\left(\omega\right)=\int_{-\infty}^{\infty}h\left(t\right)e^{-i\omega t}dt $ .
Definition. Cross-power spectral density
The cross-power spectral density of jointly-distributed W.S.S. random processes $ \mathbf{X}\left(t\right) $ and $ \mathbf{Y}\left(t\right) $ is the Fourier transform of their cross-correlation:
$ S_{\mathbf{XY}}\left(\omega\right)\triangleq\int_{-\infty}^{\infty}R_{\mathbf{XY}}\left(\tau\right)e^{-i\omega\tau}d\tau $
where $ R_{\mathbf{XY}}\left(\tau\right)=E\left[\mathbf{X}\left(t+\tau\right)\mathbf{Y}^{*}\left(t\right)\right] $ .
Note
The cross-power spectral density need not be real or non-negative.
Note
$ R_{\mathbf{XY}}\left(\tau\right)=\frac{1}{2\pi}\int_{-\infty}^{\infty}S_{\mathbf{XY}}\left(\omega\right)e^{i\omega\tau}d\omega. $
