Fourier transform

[Sandy F]

Can you do Fourier analysis on non-periodic sounds like fricatives? If not, how do you extract feature vectors for fricatives?

[Simon]

Great question! Fourier analysis decomposes any signal into a sum of simple signals (called base functions): sine waves, each with a frequency, magnitude and phase.

Since sine waves are periodic, Fourier analysis can surely only be applied to periodic signals, can’t it? Correct. At least, only to signals that we assume are periodic.

Short-term analysis

For a signal such as speech, where the spectral envelope changes over time, we must always use short-term analysis techniques. That means taking a frame of the signal (typically 25ms) and making some assumptions about the signal within that frame.

We will assume that the spectrum doesn’t change at all within the frame: the signal is “stationary“.

Assumption that the signal is periodic

To apply Fourier analysis, we make another assumption: the signal is periodic. In the case of short term analysis, the Fourier analysis effectively assumes that the frame of signal is repeated over and over before and after the frame.

for sounds like fricatives, we effectively turn them into signals that repeat with a period of one frame. Since the frequency resolution of the Fourier transform is limited by the duration of the frame, we don’t actually see this “assumed periodicity” in the resulting spectrum: it’s at a frequency lower than we can resolve.

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