Spectral Density

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The Spectral Density $S y (f)$ of the real random process $y (t)$ is defined as

$S_y(f) = \begin{matrix}\textrm{lim}\\T\rightarrow\infty\end{matrix} \frac{2}{T}\left|\int^{+T/2}_{-T/2} \left[y(t)-\bar{y}\right] e^{2 \pi i f t} dt\right|^2$

where $\bar{y}$ is the time average of $y (t)$ (note that the above definition of $S y (f)$ assumes that the process is ergodic.) The factor of 2 found in the definition of $S y (f)$ accounts for the fact that the spectral density is defined over only positive frequencies: for a real process $y (t)$ , $S y (f) = S y ( - f)$ , so that negative frequencies contain no new information. The factor of 2 essentially fold the negative frequencies onto the positive ones.

Given the spectral density, the variance of $y (t)$ may be easily computed by using the relation

$\sigma_y^2 = \int_{0}^{\infty}S_y(f) df$ .

For any random process $y (t)$ , the Spectral Density, $S y (f)$ is related to the Correlation Function, $C y (τ)$ , through the Wiener-Khintchine Theorem:

$C_y(\tau) = \int_0^\infty S_y(f) \cos{(2\pi f \tau)} df$

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Categories: Concept | Electronics

Spectral Density

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