Multi-window real-valued discrete Gabor transform for long and infinite sequences

The length of windows used in the existing multiwindow complex-valued discrete Gabor transform (M-CDGT) is restricted to be the length of analyzed sequences. Consequently, if the lengths of analyzed sequences are long, to solve the windows requires high computation burden and memory and even leads to numerical instability. To overcome this problem, a multi-window real-valued discrete Gabor transform (M-RDGT) for long and infinite sequences is presented in this paper based on the biorthogonal analysis approach, and its corresponding biorthogonality constraint between analysis windows and synthesis windows is derived. The lengths of the analysis and synthesis windows are independent of the lengths of analyzed sequences so that one can apply finite (or short) windows to process any long (or even infinite) sequences. The completeness condition of the M-RDGT is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows. In addition, the M-RDGT can utilize the fast discrete Hartley transform algorithms for fast computation and has a simple relationship with the M-CDGT such that the M-CDGT coefficients can be directly computed from the M-RDGT coefficients.