Product-function frames in l(2)(Z)
Frames are the most widely used tool for signal representation in different domains. In this correspondence, we introduce the concept of product=function frames for l(2),(Z). The frame elements of these frames are represented as products of two or more sequences. This forms a generalized structure for many currently existing. transforms. We define necessary and sufficient conditions on the frame elements so that they form a frame for 12(T). We obtain windowed transforms as a special case and derive the biorthogonal-like condition: Finally, we introduce a new family of transforms for finite-dimensional subspaces of l(2) (Z), which we call the "scale-modulation transforms." The frame elements of these transforms can be obtained via scaling and modulating a "mother" window. This transform, thus, complements the shift-modulation structure of the discrete-time Gabor transform and the shift-scale structure of the discrete wavelet transform.
frames windowed transforms scale-modulation transform gabor transform signal reconstruction