Lecture 1, 20/10/2009: Basics in Fourier series In this lecture we learn how to represent a function as the infinite sum of complex exponentials; we compute the expression for the coefficients. We meet Gibbs' phenomenon and see how the latter is related to the speed of convergence to 0 of the coefficients. We present the standard model for recurrent neural networks of rate-based neurons, and compute the frequency-dependent amplification factor using Fourier series. Lecture, Exercises Lecture 2, 3/11/2009: Geometric theory Fourier series We show how to construct an infinite-dimensional vector space for which the complex exponentials form a basis, in an appropriate sense. We show that all standard geometric tools work in this setting. We introduce the Haar wavelet and the fast Haar transform to encode signals. We finally continue the discussion of recurrent neural networks, comparing the model by Somers et al. (1995) and by Carandini and Ringach (1997) with respect to the realism and to the possibility of performing a mathematical analysis of them. Lecture, Exercises Lecture 3, 10/11/2009: Fourier series tricks Lecture Lecture 4, 10/11/2009: Fourier transform basics Lecture, Exercises |