The complex exponential form of Fourier series is a representation of a periodic function (which is usually a signal) with period \( 2\ell \) as infinite series: where a signal's complex Fourier spectrum is provided that this series converges in some sense. The above formula is based on the orthogonality property of exponential functions: Conversion of complex Fourier series into standard trigonometric Fourier series is based on Euler's formulas:

Here, j is the unit vector in positive vertical direction on the complex plane, so \( {\bf j}^2 =-1. \) Mathematica has a default command to calculate complex Fourier series:

Fourier series[ expr, t, n]        (* gives the n-order (complex) Fourier series expansion of expr in t *)

Mathematica has a special command to find complex Fourier coefficent and to determine its numerical approximation:

FourierCoefficient[ expr, t, n]        (* gives the nth coefficient in the exponential Fourier series expansion of expr in t *)

NFourierCoefficient[ expr, t, n]        (* gives a numerical approximation to the nth coefficient in the Fourier exponential series expansion of expr in t *)

One can easily transfer complex form into trigonomtric form vice versa using formulas:

and Note that \( a_{-n} \quad\mbox{and}\quad b_{-n} \) are only defined when n is negative. Then we get

The Fourier series command has an option FourierParameters that involves two parameters and when applied, it looks as FourierParameters->{a,b}
This means that complex Fourier coefficient is evaluated according to the formula:

The complex exponential Fourier form has the following advantages compaired to traditional trigonometric form:

• only need to perform one integration;
• a single exponential can be manipulated more easily than a sum of sinusoids, and
• it provides a logical transition into a further discussion of the Fourier Transform.

A signal's Fourier series spectrum \( \alpha_k \) has interesting properties.

• If f(t) is a real function, then \( \alpha_{-k} = \overline{\alpha_k} ; \)
• if f(t) is an even function, then \( \alpha_{-k} = \alpha_k ; \) if f(t) is an odd function, then \( \alpha_{-k} = -\alpha_k , \) , and
• the Fourier series obeys Parseval's Theorem
  \[ \int_0^T f^2 (t) \,{\text d}t = T\,\sum_k \left\vert \alpha_k \right\vert^2 . \]

Example: Consider a piecewise constant function on the interval [-2 , 2]:

There is no need to define the function at the points of discontinuity \( x=-2, -1, 0, 2 \) because the corresponding Fourier series will specify the values at these points to be the averages of left and right limit values. Therefore, \( f(-2) = 3/2, \ f(-1) = 1/2, \ f(0) = 1, \ f(2) =3/2 . \) We can find the Fourier coefficients either by evaluating integrals or using standard comamnd: The corresponding real trigonometric series is Correctness of calculations could be checked numerically: because \( (-1)^5 + \cos \frac{5\pi}{2} -2 = -3 . \)

We plot partial sums with 10 and 50 terms, respectively:

When the complex Fourier series is used to represent a periodic function, then the amplitude spectrum, sketched below, is two-sided. It consists of the points \( \left( \frac{k\pi}{\ell} , \left\vert \alpha_k \right\vert \right) , \quad k= 0, \pm 1, \pm 2, \ldots \) that in our case become The two-sided amplitude spectrum of the function.

The power spectrum for f is also two-sided, consisting of the points \( \left( \frac{k\pi}{\ell} , \left\vert \alpha_k \right\vert^2 \right) , \quad k= 0, \pm 1, \pm 2, \ldots \) that in our case become Note that here we used Mathematica's command FourierCoefficient that gives the kth coefficient in the Fourier series expansion of f. The two-sided power spectrum of the function.

Example: Find complex and regular Fourier series expansion of the function \( f(x) = \frac{1-a\,\cos x}{1- 2a\,\cos x + a^2} , \) where real number a has absolute value less than 1: \( |a| <1 . \)

First, we substitute instead of \( \cos x \) its Euler representation: \( \cos x = \frac{1}{2}\,e^{{\bf j}x} + \frac{1}{2}\,e^{-{\bf j}x} . \) Then

Since the numerator can be written as we simplify the given function as Using geometric series formula \( \frac{1}{1-q} = \sum_{n\ge 0} q^n \) twice, first time with \( q= a\,e^{-{\bf j}x} \) and second time with \( q= a\,e^{{\bf j}x} , \) we obtain the required complex Fourier series: which we rewrite in symmetric way Next, we unite these two sums into one to obtain real trigonometric series:

Example: Our next example deals with the periodic pulse function shown below

The function is a pulse function with amplitude 1, and pulse width h, and period T. Using Mathematica, we can define the pulse function in many ways; however, we demonstrate application of command Which. The Which command provides a logical expression that allows us to evaluate a function in only one statement like the one given in the equation, definding the pulse function. The "Which" command has the general form :
Which[condition1, value1, condition2, value2 ...]
The command will return the value that is true; let' s see how this works in practice in an example of the pulse function: We expand the pulse function into exponential Fourier series: Note that the value of \( \alpha_0 = h/T \) does not follow from the general formula directly. The above Fourier series defines the pulse functions at the points of discontinuity as the mean values from left and right: \( \Pi (0,h,T) = \Pi (h,h,T) = 1/2 . \) We can also convert the complex Fourier form to a regular real trigonometric form: Then we plot partial sums with 10 and 50 terms for particular numerical values \( h=1 \quad\mbox{and}\quad T=3: \) We can calculate Fourier coefficients directly: Then we check numerical values of the partial sum with 30 terms at the points of discontinuity: which is closed to the true value \( \Pi (1,1,3)=1/2. \)

 

Examples of Fourier Series

Gibbs Phenomenon

Even and Odd Functions

Cesàro Approximation

Chebyshev expantion

Legendre expansion

Bessel--Fourier Series

Hermite Expansion

Laguerra Expansion

Motivated examples