Product-function frames in l2(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 l²(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 l²(Z). 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 l²(Z), which we call the "scalemodulation 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.