  We study the standing periodic waves in the semi-discrete integrable system
modelled by the Ablowitz-Ladik equation. We have related the stability spectrum
to the Lax spectrum by separating the variables and by finding the
characteristic polynomial for the standing periodic waves. We have also
obtained rogue waves on the background of the modulationally unstable standing
periodic waves by using the end points of spectral bands and the corresponding
eigenfunctions. The magnification factors for the rogue waves have been
computed analytically and compared with their continuous counterparts. The main
novelty of this work is that we explore a non-standard linear Lax system, which
is different from the standard Lax representation of the Ablowitz-Ladik
equation.
