1. A method for speech analysis and synthesis comprising the steps of: sampling a short-period power spectrum of speech input into an apparatus with a sampling frequency to obtain sample points, said sampling frequency being controlled so as to trace a basic frequency of input voiced speech: applying a cosine polynomial model to the thus obtained sample points to determine a spectrum envelope; calculating mel cepstrum coefficients from the spectrum envelope; and effecting speech synthesis utilizing the mel cepstrum coefficients as filter coefficients of a mel logarithmic spectrum approximation filter used for speech synthesis.