Lab1: Discrete Complex Exponentials
Where shall we begin? From the very beginning. And in the beginning there was nothing. But then there was time. Our exploration of information processing starts from an exploration of the notion of time. More precisely, from the notion of rate of change. This is because different rates of change are often associated with different types of information. In a picture of a face, no change means you are looking at hair, foreheads, or cheeks. Rapid change means you are looking at a hairline, a pair of eyes, or a pair of lips. Different vowels are associated to different frequencies. So are musical tones. The lull of the ocean spreads across all frequencies. Weather changes occur over days. Season changes over months. Climate changes over years.
The Fourier transform is the mathematical tool that uncovers the relationship between different rates of change and different types of information. In order to define Fourier transforms, which we will do in Lab 2, we first have to study discrete complex exponentials and some of their properties. This is the objective of this Lab.