It is trivial to construct valid time-frequency dictionaries for
Hilbert spaces, the difficulty lies rather in designing such
dictionaries to efficiently yield a salient representation for the
class of signals most likely to be encountered. Here we propose
and formalise the concept of reduced multi-Gabor systems and show
that they constitute a frame for L2 and other Hilbert spaces of
interest. We then apply this concept, in the form of a multi-Gabor
dictionary, to the atomic decomposition of music and speech signals
observed in noise. Results indicate the potential of such a
system to yield a salient representation of audio signals while at
the same time reducing computational costs associated with the
decomposition.
