Multiwindow Real-Valued Discrete Gabor Transform and Its Fast Algorithms

Based on the biorthogonal analysis approach, a multiwindow real-valued discrete Gabor transform (M-RDGT) for periodic sequences is presented to efficiently analyze the dynamic time-frequency content of a signal containing components with multiple and/or time-varying frequencies. The M-RDGT offers a computationally efficient implementation as well as a real-valued formulation of the multiwindow complex-valued discrete Gabor transform (M-CDGT). The completeness condition of the M-RDGT is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows. 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 its coefficients can be directly computed from the M-RDGT coefficients. Therefore, the M-RDGT offers an efficient method to compute the M-CDGT. Since the analyzed sequence, analysis and synthesis windows in the existing M-CDGT must have an equal period, if the period of a sequence is very long, solving its windows requires a huge amount of computation and memory and could lead to numerical instability. To overcome this problem, a modified M-RDGT for long-periodic (or even infinite) sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified, in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long-periodic (or even in finite) sequence. Finally, the multirate-based parallel implementation of the M-RDGT is presented, which has shown to be effective and fast for time-frequency analysis.