The objective of the present work is to propose a method to automatically detect polarity of the speech signals by estimating instants of significant excitation of the vocaltract and the cosine phase of the analytic signal representation. The phase changes in the analytic signal around the Hilbert envelope (HE) peaks are found to vary according to the polarity of the given speech signal. The relevant HE peaks for the Hilbert phase analysis are selected by estimating the instants of significant excitation in speech. The speech polarity identification rate obtained for the proposed method is almost equal to the state of the art residual skewness method for speech polarity detection. The proposed method also provides the same results for the polarity detection in electro-glottogram signals. Finally, the robustness of the proposed method is confirmed from the reduced detection error rates obtained in noisy environments with various signal to noise ratios (SNRs). The MATLAB codes used for implementing the proposed method are available for download from the following link: http://nlp.amrita.edu:8080/TTS/polarityprograms.zip