Filter a piecewise-constant function with a sinc filter and resample it again.
- Parameters
pwc (Pwc) – The piecewise-constant function \(\alpha(t)\) to be filtered.
cutoff_frequency (float) – Upper limit \(\omega_c\) of the range of angular frequencies that you want to
preserve in your function.
segment_count (int) – The number of segments of the resampled filtered function.
name (str, optional) – The name of the node.
- Returns
The filtered and resampled piecewise-constant function.
- Return type
Pwc
Notes
The resulting filtered function is
\[\int_{-\infty}^\infty \alpha(\tau)
\frac{\sin[\omega_c (t-\tau)]}{\pi (t-\tau)} \mathrm{d}\tau
= \frac{1}{2\pi} \int_{-\omega_c}^{\omega_c}
e^{i \omega t} \hat\alpha(\omega) \mathrm{d}\omega\]
where
\[\hat\alpha(\omega)
=\int_{-\infty}^\infty e^{-i \omega \tau}\alpha(\tau) \mathrm{d}\tau\]
is the Fourier transform of \(\alpha(t)\). Hence the filter eliminates components of
the signal that have angular frequencies greater than \(\omega_c\).