Finite real-valued discrete Gabor transform with multi-windows

To efficiently analyze the dynamic time-frequency contents of signals that contain a wide range of time and frequency components, a new real-valued discrete Gabor transform with multi-windows (M-RDGT) based on the bi-orthogonal analysis approach is presented in this paper, which permits a computationally faster implementation as an alternate formulation of the complex-valued discrete Gabor transform with multi-windows (M-CDGT). The completeness condition of the M-RDGT is proved to be equivalent to its bi-orthogonality constraint between analysis windows and synthesis windows. The M-RDGT is defined by replacing the complex-valued Gabor basis functions of the M-CDGT with real-valued Gabor basis functions. The real-valued Gabor basis functions of the M-RDGT contain the Hartley's cas function which allows the M-RDGT to utilize the fast discrete Hartley transform algorithms for fast computation. In addition, the M-RDGT has a simple relationship with the M-CDGT such that the M-CDGT coefficients can be directly computed from the M-RDGT coefficients. Therefore, the M-RDGT also offers a faster and more efficient method to compute the M-CDGT.