# Fourier Series of Even Square Wave

• sandy.bridge
In summary, the task at hand is to determine the Fourier Series of a given wave. The formula for the Fourier Series is f(t)=a_0+{\sum}_{n=1}^{\infty}a_ncos(n\omega_0t)+{\sum}_{n=1}^{\infty}b_nsin(n\omega_0t). In this case, a_0=0 and b_n=0 due to the given wave being an even function. The remaining term, a_n, is found by taking the definite integrals of the cosine function over two intervals and then substituting in the values for the endpoints of the given wave. Simplifying further, the final equation for f(t) is
sandy.bridge

## Homework Statement

$-0.5\leq{t}\leq{1.5}, T=2$
The wave is the attached picture.

I need to determine the Fourier Series of the wave in the picture.

I know that $$f(t)=a_0+{\sum}_{n=1}^{\infty}a_ncos(n\omega_0t)+{\sum}_{n=1}^{\infty}b_nsin(n\omega_0t)$$

where $a_0=\bar{f}=0$ due to being an even function. Furthermore, $b_n=0$ due to being an even function also.

That leaves,
$$a_n=\int_{-0.5}^{0.5}cos(n\omega_0t)dt-\int_{0.5}^{1.5}cos(n{\omega}_0t)dt=0-\frac{1}{n\omega_0}(sin(1.5n{\omega}_0t)-sin(0.5n{\omega}_0t))=\frac{1}{n{\omega}_0}(sin(0.5n{\omega}_0t)-sin(1.5n{\omega}_0t))$$

therefore,
$$f(t)=\frac{1}{{\omega}_0}\sum_{n=1}^{\infty}cos(n{\omega}_0t)(sin(0.5n{\omega}_0t)-sin(1.5n{\omega}_0t))$$

Is this suffice as an answer, or am I missing something? My textbook is lacking examples so I just would like to know if I am doing it right. Thanks!

#### Attachments

• wave.jpg
7.3 KB · Views: 515
I also had encountered another equation deeper in the chapter that states
$$f(t)=A/2+(2A/\pi)\sum_{n=1}^{\infty}\frac{sin((2n-1)\omega{_0}t)}{2n-1}$$

## What is a Fourier Series?

A Fourier Series is a mathematical tool used to represent a periodic function as a sum of sine and cosine functions. It allows us to break down complex signals into simpler components, making it easier to analyze and understand.

## What is an Even Square Wave?

An Even Square Wave is a periodic function that alternates between two values, typically +1 and -1, with an equal period of time between each switch. It is considered an even function because it is symmetric about the y-axis.

## How is the Fourier Series of an Even Square Wave calculated?

The Fourier Series of an Even Square Wave can be calculated using the following formula:
f(x) = (4/pi) * (sin(x) + (1/3)sin(3x) + (1/5)sin(5x) + (1/7)sin(7x) + ...).
This formula takes into account the odd harmonics of the square wave, which are the frequencies that contribute to its unique shape.

## Why is the Fourier Series of an Even Square Wave important?

The Fourier Series of an Even Square Wave is important because it allows us to analyze and understand complex signals in a simpler way. It also has practical applications in fields such as signal processing, electrical engineering, and physics.

## What are the key properties of the Fourier Series of an Even Square Wave?

Some key properties of the Fourier Series of an Even Square Wave include:
1. It is an infinite series, meaning it can be represented by an infinite number of sine and cosine functions.
2. It is an even function, meaning it is symmetric about the y-axis.
3. It has a period of 2 pi.
4. It converges to the original Even Square Wave as the number of terms in the series increases.
5. It has a constant DC component of 0, meaning the average value of the square wave is 0.

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