1. Auto Spectrum

1.1. Purpose

Compute the auto spectrum of a signal.

1.2. Description

To compute the auto spectrum \(S_{xx}(f)\) of a time series \(x(t)\), the series is blocked into \(N\) parts (default \(N=1\)) consisting of \(2^n\) points, and the Fourier transform of these parts of the time series are computed and averaged:

\[S_{xx}(f) = \frac{1}{N} \sum_{j=1}^N \left| \int x_j(t)\exp(–2\pi i f t)\, dt \right|^2\]

To avoid leakage to neighboring frequencies due to end-effects, each part of the time series is multiplied with a window function. This is usually called tapering. By performing this multiplication in the time domain, some of the energy in the signal is lost, and the computed spectrum has to be corrected for this. If the relative length of each cosine half-wave is called \(r\), all spectral densities have to be multiplied with a factor \(B\):

\[B = \frac{1}{1-\frac{10r}{8}}, \quad 0\leq r \leq 0.5\]

If the faded overlap option is chosen, the blocking is performed twice, where the second blocking is shifted with half a block.

The auto spectrum may be smoothed with a weight function

\[w = 1 - \cos\left(\frac{\pi k}{m+1} \right), \quad k=1,2,…,2m+1\]

The smoothing parameter \(m\) is by default set to 3.

1.3. Input

Any equidistant signal (see Signal Types ) with time on the \(x\)-axis.

1.4. Output

The output is the auto spectrum as a function of period of frequency, depending on the user’s unit preferences.