The display of energy of ultrasonic backscattered echoes simultaneously on a joint time-frequency (t-f) plane reveals critical information pertaining to time of arrival and frequency of echoes. The quadratic t-f distributions play important role in displaying the energy of the signal on a joint t-f plane. The t-f energy distribution of the signal is dependent on a weighting function, kernel, of generalized quadratic t-f distribution. This kernel, a function of product of time lag and frequency lag variables, controls the t-f concentration of the signal and the suppression of artifacts generated by the quadratic t-f distribution. A generalized exponential product (GEP) kernel function is explored in this paper, Exponential (i.e., Choi-Williams) distribution is a special case of this generalized exponential distribution. A whole family of Quadratic exponential distributions can be generated by varying the parameters of the generalized exponential product kernel. We evaluate these parameters on the basis of optimum concentration of the ultrasonic backscattered echoes, resolution of defect echoes, suppression of the cross-terms artifacts, and performance in the presence of noise. These parameters are evaluated by reducing the cross-terms and keeping auto-terms on the ambiguity plane close to the ideal. It is shown that by controlling the parameters of the generalized exponential product kernel we can achieve better performance in the form of time-frequency concentration, and resolution for multiple echoes as compared to exponential distribution. The application of GEP kernel to ultrasonic experimental data, with properly chosen parameters, not only discern the defect echo embedded in grain echoes but diminish the cross-terms generated by the bilinear structure of the t-f distribution.
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