Orthogonalization of circular stationary vector sequences and its application to the Gabor decomposition

N. Polyak, W.A. Pearlman, Y.Y. Zeevi
1995 IEEE Transactions on Signal Processing  
Certain vector sequences in Hermitian or in Hilbert spaces, can be orthogonalized by a Fourier transf'orm. In the Anite-dimensional case, the discrete Fourier transform OFT) accomplishes the orthogonalization. The property of a vector sequence which allows the orthogonalization of the sequence by the DFT, called circular stationarity (CS), is discusped inthis paper. Applying the DFT to a given CS vector sequence results in an orthogonal vector sequence, which has the same span as the original
more » ... e. In order to obtain coefficients of the decomposition of a vector upon a particular nonorthogonal CS vector sequence, the decomposition is first found upon the equivalent DFT-orthogonalized one and then the required c d c i e n t s are found through the DFT. It is shown that the sequence of discrete Gabor basis functions with periodic kernel and with a certain inner product on the space of N-periodic discrete 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.
doi:10.1109/78.403337 fatcat:pp3tslx32vgqzc7apni5sqoeyi