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 $x_n = x(n \Delta t)$:
\[S_{xx}(f) = | \sum_{n=1}^{N}\exp(-iw n\Delta t) x_n  \Delta t|^2 \]

 power spectral density

 The power spectral density describes how power of a signal or time series is distribute over frequency. Here, power can be the actual physical power or more often for convenience with abstract signals is simply identified with the squared value of the signal.

 The average power P of a signal $x(t)$ over all time is given by the following time average:
\begin{equation}
P = \lim_{T \to  \infty} \frac{1}{T} \int_0^T |x(t)|^2 dt.
\end{equation}

Then the power spectral density can be  defined as
\[ S_{xx}(w) = \lim_{T \to \infty} E[|\hat{x}(w)|^2], \]
where $\hat{x}(w)$ is defined as,
\[ \hat{x}(w) = \frac{1}{\sqrt(T)} \int_0^T x(t) \exp(-iwt) dt .\]
Here E denotes the expected value; explicitly, we have
\[ E[|\hat{x}(w)|^2]  = E[\hat{x}^\ast(w)  \hat{x}(w)] .\]

The auto-correlation  function of a signal is defined as
\[ \gamma(\tau) = E[X(t)X(t+\tau)] \].

Provided that $\gamma(\tau)$ is absolutely integral,
\[ S_{xx}(w) = \int_{-\infty}^{\infty} \gamma(\tau) \exp(-iw\tau) d\tau = \hat{\gamma}(\tau)  .\]

The definition of the power spectral density can be generalized to discrete time variable $x_n$. As above we can consider a finite window of $1\leq n \leq N$ with the signal sampled at discrete times $x_n = x(n\Delta t)$ for a total measurement period $T=n\Delta t$. Then a single estimate of the PSD can be obtained through summation rather than integration:
\[ S_{xx}(w) = \frac{1}{T} \sum_{n=1}^{N} |x_n \exp(-iwn\Delta t)|^2 \Delta t\].



***
A series is stationary if it does not change qualitatively over the time.
Definition: Divide a series into an arbitrary number of section (of arbitrary lengths). If statistical distributions of values in every section are identical, the series is stationary.
Another definition: A series is stationary if its autocorrelation function (1a) depends only on offsets
FROM http://th-www.if.uj.edu.pl/zfs/gora/timeseries09/lecture02.pdf


RECOMMEND REFERENCE:
Power Spectra Estimation https://www.dcs.warwick.ac.uk/~feng/teaching/PowerSpectrum.pdf