Convolution¶

FFT mode¶

if \(|A|\) similar to \(|B|\), the function is evaluated by first transforming \(R\) and \(F\) to the Fourier domain, multiplying the transforms, then transforming back to the original domain. The GSL FFT routines are used to do the actual transformations.

It should be noted that the two functions (\(R\) and \(F\)) are evaluated on different intervals. \(F\) is computed on \([A,B]\) while \(R\) is computed on \([-\Delta/2, \Delta/2]\), where \(\Delta=B-A\).

In the following example a Convolution is convolved with a box function:

Note that the box function is defined on interval [-5, 5]:

Direct mode¶

If \(|A|\) and \(|B|\) differ, the convolution is performed with the direct formula. \(F\) is computed on \([A-B,B-A]\) and \(R\) is computed on \([A,B]\). This setting guarantees that \(F\) overlaps completely \(R\) in the domain \([A,B]\) when performing the convolution.

In the following example a QENS signal is fitted to a two-Lorentzian model, convolved with the experimental resolution, in the asymmetric energy range \([A,B]=[-0.12, 0.52]\).