Forms of Fourier Series

 

Introduction

Forms of Fourier Series

1. Trigonometric Form 

2. Harmonic Form

3. Complex Exponential Form

Relationship Between The Fourier Series Forms

Odd and Even Functions

Power Content

 

Introduction

Any function f(t), which is periodic with period T, can be approached with the sum of an infinite number of sinusoidal and cosinusoidal terms, together with a constant term. This form of representation of a periodic function with trigonometric series is called Fourier Series:

Where the coefficients are:

Every multiple of the basic frequency f (or angular velocity ω) is called harmonic. So the first harmonic is f, the second harmonic is 2f, the third harmonic is 3f, etc. Each harmonic has a specific value. For example the first harmonic (or basic frequency) f=1.5 kHz, the second harmonic 2f=3 KHz, etc.

At the following figure we can see the approximation of a periodic 50% square pulse, adding sines (see Example 1).

 Figure 1: Approximation of a Periodic Square Pulse with Fourier Series

As we can see from the above diagrams, at the points of discontinuities of the original function, the Fourier Series converge at the average value at the point of the discontinuity.

Also we can see from the above diagrams, that the approximation becomes better and better as the number of harmonics increases. Theoretically for an infinite number of harmonics, the initial function and the Fourier Series will be identical. 

 

Forms of Fourier Series

Using the Trigonometric Identities, the Fourier Series of a function f(t), can be expressed by the following three forms:

 

1. Trigonometric Form

The coefficients are given by the formulas:

This is the basic form of Fourier Series, from which the other two forms are derived from.

 

2. Harmonic Form

The coefficients are given by the formulas:

Here is how we can derive the Harmonic Form from the Trigonometric Form just using some of the trigonometric identities:

 

3. Complex Exponential Form

The complex coefficients are given by the formula:

Here is how we can derive the Complex Exponential Form from the Trigonometric Form just using some of the trigonometric identities.

For the Fourier Series:

For the Fourier Series Coefficients:

Also for the complex coefficients, from the above equations, it is:

 

Relationship Between The Fourier Series Forms

Trigonometric-Harmonic

 

Trigonometric-Complex Exponential

 

Harmonic-Complex Exponential

 

Odd and Even Functions

The calculations of the integrations for the Fourier Series coefficients can sometimes become very tedious. There are some “tricks” however, which simplify these calculations. These tricks are based on the feature that a function might be odd or even.

A periodic function is odd when:

Figure 2: Example of Odd Function

These functions are symmetric towards the t axis. The Fourier Series of these functions contain only sine components. They have no constant term and no cosine components.

 

 A periodic function is even when:

Figure 3: Example of Even Function

These functions are symmetric towards the y axis. The Fourier Series of these functions contain only cosine components and they might also contain constant term.

 

Power Content

The average power P of a periodic function f(t) with period T can be calculated from the following formula:

Parseval’s theorem states that the average power of a periodic function can also be calculated using the coefficients of the Complex Exponential form of the Fourier Series: