4.4: The Spectrum- Definition
- Page ID
- 47242
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Given an ergodic process \(y(t)\), with mean zero and autocorrelation \(R(\tau)\), the power spectral density of \(y(t)\), or the spectrum, is the Fourier transform of the autocorrelation:
\begin{align} S(\omega) \, &= \int\limits_{-\infty}^{\infty} R(\tau) e^{-i \omega \tau} \, d\tau \\[4pt] R(\tau) \, &= \, \dfrac{1}{2 \pi} \int\limits_{-\infty}^{\infty} S(\omega) e^{i \omega \tau} \, d\omega \end{align}
The spectrum is a real and even function of frequency \(\omega\), because the autocorrelation is real and even. Expanding the above definition,
\[ S(\omega) \, = \int\limits_{-\infty}^{\infty} R(\tau) ( \cos \omega \tau - i \sin \omega \tau) \, d\tau, \]
and clearly only the cosine will create an inner product with \(R(\tau)\).