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:

$BP_{Capon}(\omega ,{\vec {k}})={\frac {1}{{\vec {e}}^{\dagger }(\omega ,{\vec {k}}){\underline {R}}^{-1}(\omega ){\vec {e}}(\omega ,{\vec {k}})}}$

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 $\dagger$ symbol presents the conjugate transpose operation. In the same notation, the conventional f-k algorithm would be formulated as:

$BP_{conv.}(\omega ,{\vec {k}})={\vec {e}}^{\dagger }(\omega ,{\vec {k}}){\underline {R}}(\omega ){\vec {e}}(\omega ,{\vec {k}})$

For further details, we refer you to the array signal processing page of this wiki. Here we only emphasize, that the inverse of the cross-spectral matrix has to be computed for the high-resolution technique, making the estimate potentially unstable.

The corresponding toolbox for the high-resolution frequency wavenumber technique can be opened using the following plugin icon . The toolbox is nearly equivalent to the conventional frequency wavenumber toolbox and the processing flow is exactly equivalent to the conventional technique.

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.