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For a highschool assessment I am conducting a research on vowel inventories of numerous Turkish speakers. My question is in two parts: Firstly I would like to ask if it is possible (if yes, how so?) to deduce the formula of a periodical signal wave of a vowel. I know that many programs can do that however for the sake of the assessment I should show at least one example by hand. Secondly, by the which formula can I dissect the signal into its sinusoids. I tried to google this one but people advised me to put the input in a program. Due to the same reason I need to conduct DFT/FFT by hand. Thank you for your kindness to answer, I am asking questions which could be answered by google as well. I ask because I do not understand googled answers. I will major in humanities anyways.

For example, here is the waveform of the vowel /e/, how can I find its formula

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closed as off-topic by Namaste, Gibbs, Cesareo, Brahadeesh, Shailesh Sep 23 '18 at 13:19

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  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Namaste, Gibbs, Cesareo, Brahadeesh, Shailesh
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  • $\begingroup$ Suggestion, not an answer. You should use a program to find the Fourier transforms you need. Don't try to do one by hand. You might want to work out a simple artificial example by hand in your exposition - or just show how a few sinusiods combine to produce a periodic signal. $\endgroup$ – Ethan Bolker Sep 23 '18 at 0:48
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Speech analysis in applications (generally) involves taking the short time Fourier transform (STFT) of the audio signal which beforehand is converted from its encoded audio file form (e.g. .mp3, .aiff, .wav) to a sequence of real numbers (a vector). The STFT is generally used for vowel analysis because it makes clear the changes in frequency that occur over time. STFT's can be computed with a variety of software (WaveSurfer is for example free software for vowel analysis).

The STFT is essentially a Fourier transform of the vector representing the air pressure changes that an audio signal is measured by. However the STFT is different in that the signal is "windowed" before Fourier transforms are computed (i.e. the vector representing your audio is pointwise multiplied by a function which tapers off from the time of interest).

Vowel analysis is best done with the STFT which builds on knowledge of the discrete Fourier transform (DFT). Using software is realistically the only way to get very meaningful information for vowel analysis.

A good reference for speech analysis is Rabiner and Schafer's book.

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