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I have a thirty-second speech signal that was sampled at 44.1 kHz. Now, I would like to show what frequencies the speech has. However, I'm not sure what would be the best way to do that. It seems sometimes one calculates the absolute value of a Fourier transform, and sometimes power spectral density. If I understand correctly, the latter works so that I divide my signal into parts, do FFT part-by-part and somehow sum these. Window functions are somehow involved. Can you clarify this a bit for me? I'm new to DSP.

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    $\begingroup$ Breaking the signal up into segments, finding the spectrum of each, and then averaging the spectra can help to reduce noise, but reduces resolution, too. see en.wikipedia.org/wiki/Welch%27s_method $\endgroup$ – endolith Jul 25 '13 at 19:43
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Now, I would like to show what frequencies the speech has. However, I'm not sure what would be the best way to do that. It seems sometimes one calculates the absolute value of a Fourier transform, and sometimes power spectral density.

If you want to attach physical meaning to your analysis, then go with the power spectral density, (PSD). This is because this will simply give you the power of your signal, in each frequency band. On the other hand if you do not want/care about a physical meaning, but want to know how the fourier amplitudes of each band vary relative to each other, you can stick to absolute magnitude.

In practice, you can compute the PSD as simply the absolute magnitude of the fourier transform squared. For example, if your signal is $x[n]$, and its DFT is $X(f)$, then the absolute magnitude of the DFT is $|X(f)|$, while the PSD is $|X(f)|^2$.

If I understand correctly, the latter works so that I divide my signal into parts, do FFT part-by-part and somehow sum these. Window functions are somehow involved. Can you clarify this a bit for me? I'm new to DSP.

No, this is not true. What you are talking about here refers to the Short Time Fourier Transform, (STFT). This is simply chopping up your time domain signal, widowing it, and then taking the fourier trnasform. At the end of the day though, you will still have a complex matrix. If you choose to take its absolute magnitude, you will have an absolute magnitude fourier transform matrix. If you take its absolute magnitude squared, you will have a power spectral density matrix.

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The important thing to understand about something like a speech signal is that its frequency components are time-varying. In order to represent speech in the frequency domain we usually take a short enough window of the signal within which we can assume that the spectrum of the speech does not vary significantly (typically 10 ms). So we calculate the power spectrum for each successive 10 ms window using the STFT (often with some overlap between windows) and treat each successive spectrum as a "snapshot" of the frequency components of the speech at that particular time. Often the successive spectra are plotted in a 3D plot such as a spectrogram, with time on the X axis, frequency on the Y axis, and magnitude plotted as a false colour or grey scale intensity at each X,Y location.

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