CS168: The Modern Algorithmic Toolbox Lecture #11: The Fourier Transform and Convolution

Thus far, we have seen a number of different approaches to extracting information from data. This week, we will discuss the Fourier transform, and other related transformations that map data into a rather different space, while preserving the information. The Fourier transformation is usually presented as an operation that transforms data from the “time” or “space” domain, into “frequency” domain. That is, given a vector of data, with the ith entry representing some value at time i (or some value at location i), the Fourier transform will map the vector to a different vector, w, where the ith entry of w represents the “amplitude” of frequency i. Here, the amplitude will be some complex number. We will revisit the above interpretation of the Fourier transformation. First, a quick disclaimer: one of the main reasons people occasionally get confused by Fourier transforms is because there are a number of seemingly different interpretations of the Fourier transformation. In general, it is helpful to keep all of the interpretations in mind, since in different settings, certain interpretations will be more natural than others.