Discrete Systems and Signals on Phase Space

The analysis of discrete signals in particular finite N-point signals is done in terms of the eigenstates of discrete Hamiltonian systems, which are built in the context of Lie algebras and groups. These systems are in correspondence, through a 'discrete-quantization' process, with the quadratic potentials in classical mechanics: the harmonic oscillator, the repulsive oscillator, and the free particle. Discrete quantization is achieved through selecting the position operator to be a compact generator within the algebra, so that its eigenvalues are discrete. The discrete harmonic oscillator model is contained in the 'rotation' Lie algebra so(3), and applies to finite discrete systems, where the positions are {-j, -j+1, ..., j} in a representation of dimension N = 2j + 1. The discrete radial and the repulsive oscillator are contained in the complementary and principal representation series of the Lorentz algebra so(2,1), while the discrete free particle leads to the Fourier series in the Euclidean algebra iso(2). For the finite case of so(3) we give a digest of results in the treatment of aberrations as unitary U(N) transformations of the signals on phase space. Finally, we show two-dimensional signals (pixellated images) on square and round screens, and their unitary transformations.