HRFK

High resolution frequency wavenumber Toolbox (Capon's method) - overview

The high resolution frequency wavenumber (hrfk) algorithm follows the ideas of Capon (1969) . It can be viewed as a generalized beamforming algorithm resulting in an auto-adaptive (optimal) complex spatial weighting scheme. for analysis of narrowband stationary signals, it is one of the most common and preferred frequency wavenumber techniques applied to ambient vibration analysis.

The Capon beamformer relies on a very simple formula:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle BP_{Capon}(\omega,\vec{k}) = \frac{1}{\vec{e}^\dagger(\omega,\vec{k})\underline{R}^{-1}(\omega)\vec{e}(\omega,\vec{k}) }

where ${\underline {R}}^{-1}(\omega )$ is simply the inverse of the cross spectral matrix estimate and ${\vec {e}}$ is the so-called steering vector summarizing the shift times for the harmonic plane wave with wavenumber vector ${\vec {k}}$ for each station within the array.

The corresponding toolbox that can be called using the following plugin icon can be used straightforward in nearly the same way as the conventional f-k toolbox.

Parameter settings

There is just one more (optional!) parameter that can be set. The calcuation for the hrfk algorithm involves the inversion of the cross spectral matrix estimate from the data. In order to guarantee a numerically stable solution for the Matrix inversion, a regularization parameter (damping constant) can be provided to load the diagonal of the cross spectral matrix before inversion.

Note: by doing so (using a damping constant), you will smooth out your wavenumber power spectra and eventually not obtain much better results compared to the conventional beamforming approach.

Contents of output file

The output files for the high resolution f-k approach are equivalent and compatible to the output files produced with the conventional approach. For details check the following page.

Graphical display of high resolution f-k results using max2curve

The usage of max2curve is equivalent to using this tool with .max files produced with the conventional f-k approach. Check also the following page for details.