We quantize a compactified version of the trigonometric
Ruijse\-naars-Schneider particle model with a phase space that is
symplectomorphic to the complex projective space CP^N. The quantum Hamiltonian
is realized as a discrete difference operator acting in a finite-dimensional
Hilbert space of complex functions with support in a finite uniform lattice
over a convex polytope (viz., a restricted Weyl alcove with walls having a
thickness proportional to the coupling parameter). We solve the corresponding
finite-dimensional (bispectral) eigenvalue problem in terms of discretized
Macdonald polynomials with q (and t) on the unit circle. The normalization of
the wave functions is determined using a terminating version of a recent
summation formula due to Aomoto, Ito and Macdonald. The resulting eigenfunction
transform determines a discrete Fourier-type involution in the Hilbert space of
lattice functions. This is in correspondence with Ruijsenaars' observation
that---at the classical level---the action-angle transformation defines an
(anti)symplectic involution of CP^N. From the perspective of algebraic
combinatorics, our results give rise to a novel system of bilinear summation
identities for the Macdonald symmetric functions.