Using discrete Fourier transforms for signal processing

One of the most useful tools coming from calculus is the Fourier transform (FT). Roughly speaking, the FT changes the representation, in a reversible way, of certain functions. This change of representation is particularly useful in dealing with signals represented as a function of time. In this instance, the FT takes the signal and represents it as a function of frequency; we might describe this as transforming from signal space to frequency space. This can be used to identify the frequencies present in a signal for identification and other processing. In practice, we will usually have a discrete sample of a signal, so we have to use the discrete Fourier transform (DFT) to perform this kind of analysis. Fortunately, there is a computationally efficient algorithm, called the FFT, for applying the DFT to a sample.

We will follow a common process for filtering a noisy signal using the FFT. The first step is to apply the FFT and...