On the time-bandwidth concentration of signal functions forming given geometric vector configurations

Landau, Pollak, and Slepian, [4]-[6] have shown that the prelate spheroidal wave functions play an important role in determining the approximate dimensionality of a space of functions whose energies are concentrated in a given time bandwidth
WT
. They have also shown the extent to which this space may be assumed
2 WT
dimensional. The function space which they consider is actually infinite dimensional and a subset of
{\cal L}_{\infty}^{2}
, but it is not a {\em linear subspace} of
{\cal L}_{\infty}^{2}
, nor in general does it necessarily contain any linear subspace of dimensionality
2 WT
. However, in the problem of the discrete
M
-nary channel with additive Gaussian noise and perhaps other types of noise, one is mainly concerned with given
n
-dimensional linear subspaces of
{\cal L}_{\infty}^{2}
and given geometric configurations of vectors in those subspaces. Thus to be conveniently applied to this problem, the results of Landau, Pollak and Slepian should be reformulated in terms of arbitrary given finite dimensional linear subspaces of
{\cal L}_{\infty}^{2}
, with given geometric configurations therein. This paper undertakes such a reformulation for some important special cases. In particular, for the cases of orthogonal, biorthogonal and simplex configurations, it is shown that one can orient the configuration such that the time-bandwidth concentration of the least concentrated vector in the configuration is maximized. The maxi-min criterion is chosen because, as is also shown, the average concentration for these three configurations is always independent of orientation.