  Many signal processing problems--such as analysis, compression, denoising,
and reconstruction--can be facilitated by expressing the signal as a linear
combination of atoms from a well-chosen dictionary. In this paper, we study
possible dictionaries for representing the discrete vector one obtains when
collecting a finite set of uniform samples from a multiband analog signal. By
analyzing the spectrum of combined time- and multiband-limiting operations in
the discrete-time domain, we conclude that the information level of the sampled
multiband vectors is essentially equal to the time-frequency area. For
representing these vectors, we consider a dictionary formed by concatenating a
collection of modulated Discrete Prolate Spheroidal Sequences (DPSS's). We
study the angle between the subspaces spanned by this dictionary and an optimal
dictionary, and we conclude that the multiband modulated DPSS dictionary--which
is simple to construct and more flexible than the optimal dictionary in
practical applications--is nearly optimal for representing multiband sample
vectors. We also show that the multiband modulated DPSS dictionary not only
provides a very high degree of approximation accuracy in an MSE sense for
multiband sample vectors (using a number of atoms comparable to the information
level), but also that it can provide high-quality approximations of all sampled
sinusoids within the bands of interest.
