Power spectral density

The power spectral density $S_{xx}$ is the Fourier transform of the autocorrelation $R_{xx}$ of a wide-sense stationary process $x$.

For CT process $x(t)$:

$$S_{xx}(j\omega) \leftrightarrow R_{xx}(\tau)$$

For DT process $x[n]$:

$$S_{xx}(e^{j\Omega}) \leftrightarrow R_{xx}[m]$$

Instantaneous power is defined:

$$x^2(t)$$

The expectation of instant power is the autocorrelation with zero time shift:

$$E[x^2(t)] = R_{xx}(0)$$

The expectation of instantaneous power can be written in terms of the power spectral density:

$$E[x^2(t)] = R_{xx}(0) = \frac{1}{2\pi} \int_{-\infty}^{\infty} S_{xx}(j\omega) d\omega$$

Therefore, $S_{xx}$ describes how instantaneous power is distributed across frequency.

Consider a process $y$, which is the WSS process $x$ filtered by a function $h$:

$$y(t) = (h \ast x)(t)$$

Then, the PSD of this new process is:

$$S_{yy}(j\omega) = \left|H(j\omega)\right|^2 S_{xx}(j\omega)$$

Fluctuation spectral density is the power spectral density of the fluctuation of a process from its mean. In other words, it is the Fourier transform of autocovariance.

$$C_xx[m] \leftrightarrow D_{xx}(e^{j\Omega})$$ $$C_xx(\tau) \leftrightarrow D_{xx}(j\omega)$$

A white process has a flat power spectral density. For a white process $x(t)$:

$$S_{xx}(j\omega) = k, -\infty \lt \omega \lt \infty$$

Let $x(t)$ be a random process. Window this signal between $-T$ and $T$ to obtain $x_T(t)$. $x_T(t)$ can also be written as:

$$x_T(t) = w_T(t)x(t)$$

where $w_T(t) = 1$ for $|t| < T$ and $0$ otherwise.

The energy spectral density (ESD) is the square of the Fourier transform of the windowed signal $x_T(t)$:

$$\left| X_T(j\omega) \right|^2$$

The ESD has units “energy/Hz.”

The periodogram is defined by the ESD divided by the time interval $2T$.:

$$\frac{1}{2T} \left| X_T(j\omega) \right|^2$$

The periodogram has units “power/Hz.”

The limit of the expectation of the periodogram as $T \to \infty$ is the power spectral density:

$$S_{xx}(j\omega) = \lim_{T \to \infty} \frac{1}{2T} E[|X_T(j\omega)|^2]$$

This result is the Einstein-Wiener-Khinchin theorem.

Spectral estimation is estimating power spectral density $S_{xx}(j\omega)$ or cross spectral density $S_{xy}(j\omega)$ from experimental or simulated data.

To do this, we replace the expectation $E[|X_T(j\omega)|^2]$ in the previous section with the average over many iterations from experiments or simulations.