## 1. Cross Spectrum

### 1.2. Description

To compute the cross spectrum \(S_{xy}(f)\) of two time series \(x(t)\) and \(y(t)\), the series are blocked into \(N\) parts (default \(N=1\)) consisting of \(2^n\) points, and the cross spectra of these parts of the time series are computed and averaged:

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 \(B\):

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

The cross spectrum may be smoothed with a weight function

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.