Multiscale Adaptive Gabor Expansion (MAGE): Improved Detection of Transient Oscillatory Burst Amplitude and Phase

Since Denis Gabor’s pioneering paper on the discrete Gabor Expansion (Gabor, 1946), time-frequency signal analysis has proven to be an important tool for many fields. In neurophysiology, time-frequency analysis has often been used to characterize and describe transient bursts in local field potential data. However, these transient bursts have a wide range of variable durations, suggesting that a time-frequency-scale dictionary composed of elementary signal “atoms” may prove useful to more accurately match recorded bursts. While overcomplete multiscale dictionaries are useful, generating a sparse code over such dictionaries is a difficult computational problem. Existing adaptive algorithms for discovering a sparse description are slow and computationally intensive. Here we describe the Multiscale Adaptive Gabor Expansion (MAGE), which uses an implicit dictionary of parametric time-frequency-scale Gabor atoms to perform fast parameter reassignment to accelerate discovery of a sparse decomposition. Using analytic expressions together with numerical computation, MAGE is a greedy pursuit algorithm similar to Matching Pursuit, restricted to a dictionary of multiscale Gaussian-envelope Gabor atoms. MAGE parameter reassignment is robust in the presence of moderate noise. By expressing a unknown signal as a weighted sum of Gabor atoms, MAGE permits a more accurate estimate of the amplitude and phase of transient bursts than existing methods.

Projected Gabor deconvolution

Gabor deconvolution consists of approximating a nonstationary problem as several stationary subproblems via the Gabor frame, solving each subproblem independently, and then recombining/projecting the subsolutions into an approximate solution to the original nonstationary problem. The approximations, however, cause inherent instability due to systematic errors that prevent the algorithm from converging to the true solution even for noise-free cases. Furthermore, the method will be time consuming when a nonlinear optimization is used for shaping the reflectivity structure. An alternative projected Gabor deconvolution is considered, which is based on the Gabor expansion. However, in contrast to the conventional form, in the new method, first the problem is projected into the time domain, and then an inversion is performed to obtain the final reflectivity. Compared with the Gabor deconvolution, the projected alternative (1) exhibits an improved convergence property, (2) is more efficient for sparse deconvolution because only a single optimization is required to be solved, and (3) is more flexible for incorporating prior information about noise and reflectivity structure via a least-squares method. Numerical tests using simulated and field data are presented showing that the new method generates more accurate and stable reflectivity models compared with the Gabor deconvolution.

Application of Time-Frequency Distribution and FRFT in Modal Parameter Identification

A combination of fractional Fourier transform (FRFT) and time-frequency method is presented to identify ambient excited modes. In this method, Gabor expansion is applied to identify natural frequencies and damping ratios. The signal autocorrelation can enhance energy distribution of each channel signal, which can help to identify parameter. FRFT is a good tool to estimate excitation signal. A simulation example is presented to demonstrate the performance of this method. The results have shown that the proposed method gives a reasonable estimation of modal parameters and excitation signal from response signals.