6.1 Continuous time fourier series (ctfs)

Introduction

In this module, we will derive an expansion for continuous-time, periodic functions, and in doing so, derive the Continuous Time Fourier Series (CTFS).

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems , calculating the output of an LTI system $\mathscr{H}$ given $e^{st}$ as an input amounts to simple multiplication, where $H(s)\in \mathbb{C}$ is the eigenvalue corresponding to s. As shown in the figure, a simple exponential input would yield the output

Using this and the fact that $\mathscr{H}$ is linear, calculating $y(t)$ for combinations of complex exponentials is also straightforward.

$$c_{1}e^{s_{1}t}+c_{2}e^{s_{2}t}\to c_{1}H(s_1)e^{s_{1}t}+c_{2}H(s_2)e^{s_{2}t}$$ $$\sum c_{n}e^{s_{n}t}\to \sum c_{n}H(s_n)e^{s_{n}t}$$

The action of $H$ on an input such as those in the two equations above is easy to explain. $\mathscr{H}$ independently scales each exponential component $e^{s_{n}t}$ by a different complex number $H(s_n)\in \mathbb{C}$ . As such, if we can write a function $f(t)$ as a combination of complex exponentials it allows us to easily calculate the output of a system.

Fourier series synthesis

Joseph Fourier demonstrated that an arbitrary $f(t)$ can be written as a linear combination of harmonic complex sinusoids

The $c_n$ - called the Fourier coefficients - tell us "how much" of the sinusoid $e^{j{\omega }_{0}nt}$ is in $f(t)$ . The formula shows $f(t)$ as a sum of complex exponentials, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, it tells us that the set ofcomplex exponentials $\{\forall n, n\in \mathbb{Z}\colon e^{j{\omega }_{0}nt}\}$ form a basis for the space of T-periodic continuous time functions.

Synthesis with sinusoids demonstration

Fourier series analysis

Finding the coefficients of the Fourier series expansion involves some algebraic manipulation of the synthesis formula. First of all we will multiply both sides of the equation by $e^{-(j{\omega }_{0}kt)}$ , where $k\in \mathbb{Z}$ .

Fourier series summary

Because complex exponentials are eigenfunctions of LTI systems, it is often useful to represent signals using a set of complex exponentials as a basis. The continuous time Fourier series synthesis formula expresses a continuous time, periodic function as the sum of continuous time, discrete frequency complex exponentials.