1. Purpose:

In this lab, you will investigate the properties of orthogonal functions and the Fourier series. This is a long lab, so plan accordingly.

2. Background:

Orthogonal basis functions are central to the analysis of linear systems. The most important of these functions is the complex exponential e^jωt, where ω is the angular frequency, which is real. In this laboratory exercise, we are specifically interested in the family of complex exponential basis functions, φ_k(t), that are related to each other by having angular frequencies that are multiples of common fundamental frequency, ω_o

φ_k(t) = e^jkωot

Each function can be viewed a vector of length 1 rotating at different frequency in the complex plane.

φ₁(t) rotates at the functional frequency ωo = 2π/T_o, where T_o is the fundamental period. φ_k(t) rotates at a frequency kω_o radians/sec, where k is an integer that ranges from - ∞ < k < ∞.

Complex exponential is so important for the study of linear systems for two reasons: 1) They are orthogonal and 2) they are eigenfunctions of linear systems. In this lab we will look at the orthogonality property. In the next lab, we will look at the eigenfunction property.