Introduction

Since complex exponentials are eigenfunctions of linear time-invariant (LTI) systems, calculating the output of an LTI system ℋℋ given es⁢t s t as an input amounts to simple multiplcation, where H⁢s∈C H s is a constant (that depends on s). In the figure below we have a simple exponential input that yields the following output:

Using this and the fact that ℋℋ is linear, calculating y⁢t y t for combinations of complex exponentials is also straightforward. This linearity property is depicted in the two equations below - showing the input to the linear system HH on the left side and the output, y⁢t y t , on the right:

The action of HH on an input such as those in the two equations above is easy to explain: ℋℋ independently scales each exponential component e s n ⁢t s n t by a different complex number H⁢ s n ∈C H s n . As such, if we can write a function f⁢t f t as a combination of complex exponentials it allows us to:

Fourier Series

Joseph Fourier demonstrated that an arbitrary T-periodic function f⁢t f t can be written as a linear combination of harmonic complex sinusoids

where ω 0 =2⁢πT ω 0 2 T is the fundamental frequency. For almost all f⁢t f t of practical interest, there exists c n c n to make Equation 2 true. If f⁢t f t is finite energy ( f⁢t∈L20T f t L 0 T 2 ), then the equality in Equation 2 holds in the sense of energy convergence; if f⁢t f t is continuous, then Equation 2 holds pointwise. Also, if f⁢t f t meets some mild conditions (the Dirichlet conditions), then Equation 2 holds pointwise everywhere except at points of discontinuity.

The c n c n - called the Fourier coefficients - tell us "how much" of the sinusoid ei⁢ ω 0 ⁢n⁢t ω 0 n t is in f⁢t f t . Equation 2 essentially breaks down f⁢t f t into pieces, each of which is easily processed by an LTI system (since it is an eigenfunction of every LTI system). Mathematically, Equation 2 tells us that the set of harmonic complex exponentials ∀ n ,n∈Z:ei⁢ ω 0 ⁢n⁢t n n ω 0 n t form a basis for the space of T-periodic continuous time functions. Below are a few examples that are intended to help you think about a given signal or function, f⁢t f t , in terms of its exponential basis functions.

Fourier Coefficients

In general f⁢t f t , the Fourier coefficients can be calculated from Equation 2 by solving for c n c n , which requires a little algebraic manipulation (for the complete derivation see the Fourier coefficients derivation). The end results will yield the following general equation for the fourier coefficients:

The sequence of complex numbers ∀ n ,n∈Z: c n n n c n is just an alternate representation of the function f⁢t f t . Knowing the Fourier coefficients c n c n is the same as knowing f⁢t f t explicitly and vice versa. Given a periodic function, we can transform it into it Fourier series representation using Equation 3. Likewise, we can inverse transform a given sequence of complex numbers, c n c n , using Equation 2 to reconstruct the function f⁢t f t .

Along with being a natural representation for signals being manipulated by LTI systems, the Fourier series provides a description of periodic signals that is convenient in many ways. By looking at the Fourier series of a signal f⁢t f t , we can infer mathematical properties of f⁢t f t such as smoothness, existence of certain symmetries, as well as the physically meaningful frequency content.

Summary: Fourier Series Equations

Our first equation is the synthesis equation, which builds our function, f⁢t f t , by combining sinusoids.

And our second equation, termed the analysis equation, reveals how much of each sinusoid is in f⁢t f t . where we have stated that ω 0 =2⁢πT ω 0 2 T .