The spectrum of a pure tone is not a perfect vertical line

[Qianchu L]

I don’t quite understand the spectrum representation in Praat. Why do we see spikes rather than straight lines in spectrums? Even for a simple sin wave (a pure tone), we will see a peak at the fundamental frequency and energy spread around the peak. Why?

[Simon]

That’s a good question, but one with a rather technical answer.

First, it’s worth remembering that we usually view the spectrum on a log scale, and this exaggerates this effect.

The short answer is that this is a consequence of analysing a short region of the signal that – in general – will not contain a perfect integer number of complete cycles of the waveform. Therefore, we have to multiply the waveform by a tapered window to avoid discontinuities at the start and end (see my blog post about what happens without a tapered window).

Fading the signal in and out with the tapered window effectively changes its frequency content: for example, our pure sine wave would not be precisely a pure sine wave anymore (i.e., will now contain some other frequencies, caused by the application of the window function).

This article gives a good, and longer answer. Scroll down to “Windowing” and Figure 10, then read onwards to Figure 13. After that, it becomes a “my window function is better than your window function” competition.

The Wikipedia entry “Window function” has a long shopping list of slightly different window functions. Otherwise, I think that article is long but not very illuminating.

[Author]

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