N. Polyak et al., ORTHOGONALIZATION OF CIRCULAR STATIONARY VECTOR SEQUENCES AND ITS APPLICATION TO THE GABOR DECOMPOSITION, IEEE transactions on signal processing, 43(8), 1995, pp. 1778-1789
Certain vector sequences in Hermitian or in Hilbert spaces, can be ort
hogonalized by a Fourier transform, In the finite-dimensional case, th
e discrete Fourier transform (DFT) accomplishes the orthogonalization.
The property of a vector sequence which allows the orthogonalization
of the sequence by the DFT, called circular stationarity (CS), is disc
ussed in this paper. Applying the DFT to a given CS vector sequence re
sults in an orthogonal vector sequence, which has the same span as the
original one, In order to obtain coefficients of the decomposition of
a vector upon a particular nonorthogonal CS vector sequence, the deco
mposition is first found upon the equivalent DFT-orthogonalized one an
d then the required coefficients are found through the DFT, It is show
n that the sequence of discrete Gabor basis functions with periodic ke
rnel and with a certain inner product on the space of N-periodic discr
ete functions, satisfies the CS condition, The theory of decomposition
upon CS vector sequences is then applied to the Gabor basis functions
to produce a fast algorithm for calculation of the Gabor coefficients
.