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The problem of estimating the exact shape of the glottal flow derivative (GFD) using reweighted 1-norm minimization of the second derivative of the GFD is addressed in this paper. By using physiological models of the glottal flow derivative, such as the Liljencrants-Fant (LF) and Rosenberg models, it is intuitively found that the second derivative of those models is highly sparse. Based on this observation an iteratively reweighted 1-norm minimization algorithm is proposed to accurately estimate the vocal tract of the speech signal by exploiting the sparsity of the second derivative of the GFD (the residual of the linear prediction model). An experimental study using a data set of 40 vowels /a/ and /e/, 20 for each, is conducted, showing the efficiency, in terms of the number of iterations and the total run-time reduction, of the proposed algorithm. Furthermore, the results of estimating the GFD of two vowels /a/ & /e/ using Joint Source-Filter Model Optimization and our proposed method, demonstrate the accuracy, in terms of similarity to the physiological model and precise synthesis, of our proposed algorithm.