  In this letter, we generalize the convolutional NMF by taking the
$\beta$-divergence as the contrast function and present the correct
multiplicative updates for its factors in closed form. The new updates unify
the $\beta$-NMF and the convolutional NMF. We state why almost all of the
existing updates are inexact and approximative w.r.t. the convolutional data
model. We show that our updates are stable and that their convergence
performance is consistent across the most common values of $\beta$.
