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Fourier analysis and resynthesis

Among the applications of filters discussed in Chapter 8, we saw how to use heterodyning, combined with a low-pass filter, to find the amplitude and phase of a sinusoidal component of a signal (Page ). In this chapter we will refine this technique into what is called Fourier analysis. In its simplest form, Fourier analysis takes as input any periodic signal (of period ) and outputs the complex-valued amplitudes of its possible sinusoidal components. These complex amplitudes can theoretically be used to reconstruct the original signal exactly. This reconstruction is called Fourier resynthesis.

In this chapter we will start by developing the theory of Fourier analysis and resynthesis of periodic sampled signals. Then we will go on to show how to apply the same techniques to arbitrary signals, whether periodic or not. Finally, we will develop some standard applications such as the phase vocoder.

Subsections

- Fourier analysis of periodic signals
  - Periodicity of the Fourier transform
  - Fourier transform as additive synthesis
- Properties of Fourier transforms
  - Fourier transform of DC
  - Shifts and phase changes
  - Fourier transform of a sinusoid
- Fourier analysis of non-periodic signals
- Fourier analysis and reconstruction of audio signals
  - Narrow-band companding
  - Timbre stamping (classical vocoder)
- Phase
  - Phase relationships between channels
- Phase bashing
- Examples
  - Fourier analysis and resynthesis in Pd
  - Narrow-band companding: noise suppression
  - Timbre stamp ("vocoder")
  - Phase vocoder time bender
- Exercises