Why do I need to apply a window function to samples when building a power spectrum of an audio signal?

Question

I have found for several times the following guidelines for getting the power spectrum of an audio signal:

• collect N samples, where N is a power of 2

Answer 1

As @cyco130 says, your samples are already windowed by a rectangular function. Since a Fourier Transform assumes periodicity, any discontinuity between the last sample and the repeated first sample will cause artefacts in the spectrum (e.g. "smearing" of the peaks). This is known as spectral leakage. To reduce the effect of this we apply a tapered window function such as a Hann window which smooths out any such discontinuity and thereby reduces artefacts in the spectrum.