Discrete Fourier Transform: Introduction

This lecture introduces the discrete Fourier transform (DFT), a crucial tool for analyzing signals digitally. The DFT allows for the calculation of a signal's Fourier transform in a fully digital way, enabling the analysis of signal spectra and the convolution of signals. By sampling the spectrum at discrete frequencies, the DFT simplifies the calculation process and provides insights into the relationship between the original signal and its Fourier transform. The lecture covers the periodicity of the Fourier transform of digital signals, the sampling process, and the properties of the DFT. Special attention is given to the notation and properties of complex exponentials, which play a key role in understanding the DFT. The lecture concludes by setting the stage for exploring the connection between the inverse Fourier transform of the original signal and the discrete version of it.