Fourier series

Introduction

In the module on the spectrum of sinusoidal signals we studied the spectral behavior of the sum of sinusoidal signals. In this module we will study signals which are composed of a sum of harmonic related sine waves.

Screencast video [⯈]

Module overview

This module covers the following topics:

1. Periodic signals - Signals can have a repetitive structure which is inherently related to the frequency structure of the signal. This section will discuss when a signal is periodic and how this period can be determined from its frequency content.
2. Fourier series synthesis - This section will give the representation of the Fourier series, which is a weighted sum of harmonic related phasor components.
3. Fourier series analysis - Similarly, a signal can be decomposed into these harmonic related phasor components. This section will describe the procedure for calculating the corresponding weights.
4. Examples [⯈] - In order to make sure that the reader gets properly acquainted with the topic, some additional examples are provided.

Exercises

In this section several exercises are available, including their answers. The exercises marked in blue are explained by means of more extensive pencast videos. These exercises also include topics relating to the spectrum of sinusoidal signals.

Video quiz

Test your knowledge (click to expand)

Question 1

Define the Fundamental Frequency of the signal $x(t) = x_1 + x_2 + x_3$, where $x_1= 4 + \sin(500\pi t - \frac{\pi}{3})$, $x_2 = \cos(2.8\pi t - \frac{\pi}{8})$ and $x_3 = \sin(2\pi\cdot 22t - \frac{\pi}{4})$.
$F_0 = 0.2$ Hz
$F_0 = 0.4$ Hz
$F_0 = 0.8$ Hz
None of the answers is correct.

Question 2

With $F_0 = 1/T_0$ and integer $k=0,\pm 1, \pm 2 , \ldots$, what holds for the following expression? $\int_{-T_0/2}^{T_0/2}e^{jk2\pi F_0 t}dt = ?$

$=0$ for $k=0$
$=0$ for all values of $k$
$\neq 0$ for $k=0$
$\neq 0$ for all values of $k$
None of the answers is correct.

Question 3

For what value of $\alpha$ does the following integral with $F_0=1/T_0$ hold? $\int_{0}^{\alpha T_0}e^{j2\pi F_0 t}dt = 0$

$\alpha = 1/4$
$\alpha = 2/4$
$\alpha = 3/4$
$\alpha = 4/4$
None of the answers is correct.

Question 4

With $F_0 = 1/T_0$ and positive integers $k$ and $l$, when does the following hold? $\int_{T_0/2}^{T_0/2} \cos(k\cdot 2\pi F_0 t) \cdot \cos(l\cdot 2\pi F_0 t) \neq 0$

$k>l$
$k < l$
$k=l$
$k\neq l$

Question 5

Given the following periodic signal $x(t)$ and considering the fundamental frequency $F_0$, which of the following figures represent the spectral plot? $x(t) = 3 + 0.5\cos(7.8\pi t + \frac{\pi}{4}) + 2\sin(130\pi t)$

None of the answers is correct.

Exercise bundle

Answers

Download the answers here.

Pencast videos [⯈]

The above video player contains a playlist of all pencast videos which can be expanded by clicking the playlist icon in the upper-right corner.

MATLAB lab

Accompanied to this modules are some exercises in MATLAB, which will test your knowledge of the module and will help improve your MATLAB skills. These lab exercises also include topics relating to the spectrum of sinusoidal signals.

Lab assignment

MATLAB demo [⯈]

Summary

• A periodic signal has the following property:
  A real signal $x(t)$, which consists of a DC component and a sum of $N$ different frequencies $f_1, f_2,\ldots, f_N$, with $f_1 < f_2 < \ldots < f_N$, is a periodic signal with Fundamental period $T_0 = 1/F_0$ and has a Fundamental frequency $F_0 = \text{gcd}\{f_1,f_2,\ldots,f_N\} $.
• The integral over one period $T_0 = 1/F_0$ of a phasor with a harmonic related frequency $k \cdot F_0$ results always in zero except for the case $k = 0$. Mathematically this can be written as follows:

  $$\bbox[5px,border:1px solid black]{ \frac{1}{T_0}\int_0^{T_0}e^{j2\pi kF_0 t}\mathrm{d}t = \begin{cases} 1 &\text{for }k=0\newline 0 &\text{for }k\neq 0 \end{cases}} $$

• Fourier analysis and synthesis equations for periodic signal $x(t)=x(t+T_0)$ with $T_0=1/F_0$: $$\bbox[5px,border:1px solid black]{ \alpha_k = \frac{1}{T_0}\int_0^{T_0} x(t)e^{-j2\pi F_0 kt}\mathrm{d}t \ \circ \hspace{-1.7mm} - \hspace{-1.7mm} \circ \ x(t) = \sum_{k=-\infty}^\infty \alpha_k e^{j2\pi F_0 kt }\ } $$