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 perodic function, into a sum of trigonometric functions.

So the power is really, given a complex perodic 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 \ | \ \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_{t1}^{t2} 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 function.

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

\int_{t1}^{t2} 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 + \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 = \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 = \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 = 0 \text{ for all n 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 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_{-\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.

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)

\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}

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_{-\infty}^{\infty} D_n e^{jn\omega_0 t}