Part 5 - Fourier series | rezarezvan.com

Introduction

In this part we’ll try to understand the concept and power with Fourier series.

Let’s firstly define the what, why and how with Fourier series.

Definition 1 (Fourier series)

Fourier series, is an expansion of a periodic function, into a sum of trigonometric functions.

So the power is really, given a complex periodic function, we can easily represent it as a sum of simple trigonometric functions!

However, before we mathematically define the Fourier series, let’s take a different approach to this.

Signal representation by orthogonal signal set

We can see that, if we only use $c_1 x_1$ and $c_2 x_2$ , we get something similar for $f$ .

We can say that:

f \sim c_1 x_1 + c_2 x_2.

Let’s call our error, $e$ .

e = f - (c_1 x_1 + c_2 x_2).

Therefore:

f \sim c_1 x_1 + c_2 x_2 + e = c_1 x_1 + c_2 x_2 + c_3 x_3, \quad \text{in this case}.

From linear algebra we know that basis vectors are orthogonal, this means:

c_i = \dfrac{f \cdot x_i}{x_i \cdot x_i} = \dfrac{1}{|x_i|^2} f \cdot x_i.

This is really useful, so, from what we know, we can say that.

Given a set of orthogonal real signals, $x_1(t), x_2(t), \ldots, x_N(t)$ over the interval, $[t_1, t_2]$ :

\int_{t_1}^{t_2} x_m(t) x_n(t) \, dt = \begin{cases} 0 & m \neq n \newline E_n & m = n \end{cases}

We can approximate, $f(t)$ over the interval, $[t_1, t_2]$ as:

f(t) \sim c_1 x_1(t) + c_2 x_2(t) + \ldots + c_N x_N(t) = \sum_{n = 1}^N c_n x_n(t).

For any $c_n$ we can express it as an integral:

c_n = \dfrac{\int_{t_1}^{t_2} f(t) x_n(t) \, dt}{\int_{t_1}^{t_2} x_n^2(t) \, dt} = \dfrac{1}{E_n} \int_{t_1}^{t_2} f(t) x_n(t) \, dt.

As before, we will have an error, $e(t)$ as well:

e(t) = f(t) - \sum_{n = 1}^N c_n x_n(t).

We’ll also have an error in the energy, defined as:

E_e = \int_{t_1}^{t_2} f^2(t) \, dt - \sum_{n = 1}^N c_n^2 E_n.

As, $N \to \infty$ , we hope that both of these go to zero.

This is the generalized Fourier series!

Let’s define it properly

Definition 2 (Generalized Fourier series)

f(t) = c_1 x_1(t) + c_2 x_2(t) + \ldots + c_N x_N(t) = \sum_{n = 1}^N c_n x_n(t).

If, $E_e \to 0$ as $N \to \infty$ , the set $\{x_n(t)\}$ is a complete set on the interval $[t_1, t_2]$ , for that class of $f(t)$ .

$\{x_n(t)\}$ are called basis functions or basis signals.

Theorem 1 (Parseval’s Theorem)

The error energy can approach zero even though $e(t)$ is non-zero at some isolated instants.

\int_{t_1}^{t_2} f^2(t) \, dt = c_1^2 E_1 + c_2^2 E_2 + \ldots = \sum_{n = 1}^{\infty} c_n^2 E_n.

Generalization to complex signals

In the most general case, signals are considered to be complex functions.

Therefore, let’s generalize further to the complex world.

\int_{t_1}^{t_2} x_m(t) x_n^{*}(t) \, dt = \begin{cases} 0 & m \neq n \newline E_n & m = n \end{cases}

Where, $x_n^{*}(t)$ is the complex conjugate of, $x_n(t)$ . All equations are the same essentially:

f(t) = c_1 x_1(t) + c_2 x_2(t) + \ldots + c_i x_i(t) + \ldots,

c_n = \dfrac{1}{E_n} \int_{t_1}^{t_2} f(t) x_n^{*}(t) \, dt.

Trigonometric Fourier series

If the set, is the following: $\{1, \cos \omega_0 t, \cos 2\omega_0 t, \ldots, \cos n\omega_0 t, \ldots; \sin \omega_0 t, \sin 2\omega_0 t, \ldots, \sin n\omega_0 t, \ldots\}$

This is the orthogonal complete, trigonometric set.

$\omega_0$ is called the fundamental frequency

$n \omega_0$ is called the $n$ th harmonic.

We can call the fundamental period for, $T_0$ :

T_0 = \dfrac{2\pi}{\omega_0}.

Note that $T_0$ is the fundamental period, meaning it can be at any time instant, as long as it’s a period.

\int_{T_0} \cos n\omega_0 t \cos m\omega_0 t \, dt = \begin{cases} 0 & n \neq m \newline \dfrac{T_0}{2} & m = n \neq 0 \end{cases}

\int_{T_0} \sin n\omega_0 t \sin m\omega_0 t \, dt = \begin{cases} 0 & n \neq m \newline \dfrac{T_0}{2} & m = n \neq 0 \end{cases}

\int_{T_0} \sin n\omega_0 t \cos m\omega_0 t \, dt = 0 \quad \text{for all } n \text{ and } m.

So:

f(t) = a_0 + a_1 \cos \omega_0 t + a_2 \cos 2\omega_0 t + b_1 \sin \omega_0 t + b_2 \sin 2\omega_0 t + \ldots

So:

f(t) = a_0 + \sum_{n = 1}^{\infty} (a_n \cos n\omega_0 t + b_n \sin n\omega_0 t).

a_n = \dfrac{\int_{t_1}^{t_1 + T_0} f(t) \cos n\omega_0 t \, dt}{\int_{t_1}^{t_1 + T_0} \cos^2 n\omega_0 t \, dt},

a_0 = \dfrac{1}{T_0} \int_{t_1}^{t_1 + T_0} f(t) \, dt,

a_n = \dfrac{2}{T_0} \int_{t_1}^{t_1 + T_0} f(t) \cos n\omega_0 t \, dt,

b_n = \dfrac{2}{T_0} \int_{t_1}^{t_1 + T_0} f(t) \sin n\omega_0 t \, dt.

We can write this more compactly.

a_n \cos n\omega_0 t + b_n \sin n\omega_0 t = C_n \cos(n\omega_0 t + \theta_n).

Where:

C_n = \sqrt{a_n^2 + b_n^2},

\theta_n = \tan^{-1} \left(\dfrac{-b_n}{a_n}\right).

So:

f(t) = C_0 + \sum_{n = 1}^{\infty} C_n \cos(n\omega_0 t + \theta_n).

Exponential Fourier series

Given the set: $\{e^{jn\omega_0 t}\}$ , with, $n = 0, \pm 1, \pm 2, \ldots$ , this is a complete, orthogonal set.

Note that, $j$ here is complex.

\int_{T_0} e^{jm\omega_0 t} (e^{jn\omega_0 t})^{*} \, dt = \int_{T_0} e^{j(m - n)\omega_0 t} \, dt = \begin{cases} 0 & m \neq n \newline T_0 & m = n \end{cases}

f(t) = \sum_{n = -\infty}^{\infty} D_n e^{jn\omega_0 t},

D_n = \dfrac{1}{T_0} \int_{T_0} f(t) e^{-jn\omega_0 t} \, dt.

Using Euler’s formula, we can switch between trigonometric and exponential series.

\begin{align*} C_n \cos(n\omega_0 t + \theta_n) & = \dfrac{C_n}{2} \left(e^{j(n\omega_0 t + \theta_n)} + e^{-j(n\omega_0 t + \theta_n)}\right) \newline & = \left(\dfrac{C_n}{2} e^{j\theta_n}\right) e^{jn\omega_0 t} + \left(\dfrac{C_n}{2} e^{-j\theta_n}\right) e^{-jn\omega_0 t}. \end{align*}

Let’s introduce, $D_n$ :

D_n = \dfrac{1}{2} C_n e^{j\theta_n},

D_{-n} = \dfrac{1}{2} C_n e^{-j\theta_n}.

f(t) = D_0 + \sum_{n = 1}^{\infty} (D_n e^{jn\omega_0 t} + D_{-n} e^{-jn\omega_0 t}).

In an even more compact form:

f(t) = \sum_{n = -\infty}^{\infty} D_n e^{jn\omega_0 t}.