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In this paper, we propose a new method for accurate detection, estimation and tracking of formants in speech signals using time-varying quasi-closed phase analysis (TVQCP). The proposed method combines two different methods of analysis namely, the time-varying linear prediction (TVLP) and quasi-closed phase (QCP) analysis. TVLP helps in better tracking of formant frequencies by imposing a time-continuity constraint on the linear prediction (LP) coefficients. QCP analysis, a type of weighted LP (WLP), improves the estimation accuracies of the formant frequencies by using a carefully designed weight function on the error signal that is minimized. The QCP weight function emphasizes the closed-phase region of the glottal cycle, and also weights down the regions around the main excitations. This results in reduced coupling of the subglottal cavity and the excitation source. Experimental results on natural speech signals show that the proposed method performs considerably better than the detect-and-track approach used in popular tools like Wavesurfer or Praat.