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 (section 8.5.3). 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 $N$) and outputs the complex-valued amplitudes of its $N$ possible sinusoidal components. These $N$ 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.