Power spectrum

*MAINLY COPY FROM WIKI

 

Energy spectral density

Fourier transform
$$F[x(t)]  = \int_{-\infty}^{\infty} x(t)\exp(-i 2\pi f t) dt$$
inverse transform
$$x(t) = \int_{-\infty}^{\infty} F[x(t)] \exp(i 2\pi f t) df$$

Energy spectral density describes how the energy of a signal or a time series is distributed with frequency. The term energy is used in the generalized sense of signal processing, that is, the energy E of a singal $x(t)$ is
  $$\begin{equation}\label{eq:1}  E = \int_{-\infty}^{\infty} |x(t)|^2 dt. \end{equation}$$

The energy spectral density is most suitable for signal with a finite total energy.

$$\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |\hat{x}(f)|^2 df,$$
where
$$\hat{x}(f) = \int_{-\infty}^{\infty} \exp(-iwt) x(t) dt ,$$
is the Fourier transform of the signal and $w = 2\pi f$ is angular frequency.

Since the  integrate on the right-hand side is the energy of the signal (see eq 1), the integrand $|\hat{x}(f)|^2$ can be interpreted as a density function describing the energy per unit frequency contained in the signal at the frequency $f$.

The energy spectral density of a signal $x(t)$ is defined as
$$S_{xx}(f) = |\hat{x}(f)|^2.$$

The definition to a discrete signal with an infinite number of values $x_n$ such as a signal sampled at discrete times $x_n = x(n \Delta t)$:
$$S_{xx}(f) = | \sum_{n=1}^{N}\exp(-iw n\Delta t) x_n  \Delta t|^2$$