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Discrete Systems and Signals on Phase Space |
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PP: 141-181 |
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Author(s) |
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Kurt Bernardo Wolf,
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Abstract |
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| 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 f?j;?j+1; : : : ; jg 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 twodimensional
signals (pixellated images) on square and round screens, and their unitary
transformations. |
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