Bilinear time-frequency distributions for ultrasonic signal processing and NDE applications

In this paper, we introduce Singular Value Decomposition (SVD) of Wigner Distribution (WD). The WD is a special and widely adapted method of generalized bilinear time-frequency (GBTF) representation that reveals frequency contents of ultrasonic signals as function of time. The WD offers the best time-frequency (t-f) resolution but significantly suffers because of cross-term artifacts generated due to the inherent bilinear operation of GBTF. In this study, we use SVD to reduce the adverse impact of cross-terms. This approach has been extended to the SVD of Choi-Williams Distribution (CWD), a well-adapted version of GBTF distributions. Gaussian echoes are used to model the ultrasonic backscattered signals and this model is examined to reveal the pitfalls of WD and CWD. For CWD, the range of the exponential kernel parameter is attained based on the impetus to sustain desirable auto-terms, while suppressing the superfluous cross-terms. The SVD of WD and CWD generates basis functions uniquely corresponding to auto-terms and cross-terms. After identifying and discarding the basis functions corresponding to the cross-terms and noise, the desirable basis functions of auto-terms and their singular values are used to reconstruct the t-f distribution. We have applied SVD to WD and CWD of two overlapping echoes with different arrival times and frequency distributions corrupted with noise. We further used this method to detect flaw echoes masked by microstructure scattering noise. The flaw echo in the t-f plane clearly reveals lower frequency distribution at the location of the defect. These numerical results and experimental evaluations confirm that the application of SVD is a desirable approach to improve the accuracy of GBTF.