Gabor-type matrices and discrete huge Gabor transforms

A Gabor family is obtained from a Gabor atom (or Gabor window, or basic building block) by time-frequency shifts along some discrete TF-lattice. Such a family is usually not orthogonal. Therefore the determination of appropriate coefficients in order to obtain a series representation of a given signal in terms of this family has been considered a computational intensive task for a long time. We introduce a class of matrices, called Gabor-type matrices and show that the product of two Gabor-type matrices is again a Gabor-type matrix of the same type. The key point for applications is based on the observation that the multiplication of Gabor-type matrices can be replaced by some special “multiplication” of associated small block matrices. We propose an efficient algorithm, which we call the block-multiplication, and which makes explicit use of the sparsity of those Gabor-type matrices. As an interesting consequence, we show that Gabor operators corresponding to Gabor triples (g_k,a,b) commute for arbitrary signals g_k (k=1,2) provided that ab divides the signal length