# Fourier Series Help: Isolated Pulses, Width "w", +-D

• AntonVrba
In summary, the conversation discusses the Fourier series of two isolated pulses with a pulse width of "w" and a spacing of "2D" apart or at +-D. The function is symmetrical with coefficients B_n equal to 0 and period of 2D. The equation for A_n is given and the final equation for y(x) is shown to be correct through a plot and correction of a previous error.
AntonVrba
I am 52, not yet senile and would be greatfulif someone can give me the Fourier series of two isolated pulses, pulswidth "w" spaced "2D" apart or at +- D.

#### Attachments

• Pulses.gif
2.6 KB · Views: 464
I will try it, but don't have a program to check my calculations.

This function is symetrical. So the coeficients $B_n$ are 0. The period is 2D.

$$A_n = \frac{2}{D}\int_{0}^{D} cos\left( \frac{n\pi x}{D}\right)y(x)dx = \frac{2h}{D}\int_{D-W/2}^{D} cos\left( \frac{n\pi x}{D}\right)dx = \frac{2h}{D}\left[\frac{D}{n\pi}sin\left( \frac{n\pi x}{D}\right)\right]_{D-W/2}^{D}$$

$$A_n = \frac{2h}{n\pi}sin\left( \frac{n\pi W}{2D} - n\pi \right) & n=1,2,...$$

$$A_0 = \frac{Wh}{D}$$

$$\Rightarrow y(x) = \frac{Wh}{2D} + \frac{2h}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}sin\left( \frac{n\pi W}{2D} - n\pi \right) cos\left( \frac{n\pi x}{D}\right)$$

This makes sense, because the bigest W can be is 2D wide. In this case, the function is y(x) = h, which is correct. (h is the height of the pulse)

Last edited:
I get:

$$y(x)=\frac{Wh}{2D}+\frac{2h}{\pi}\sum_{n=1}^{\infty}\frac{1}{n}Sin[\frac{n\pi W}{2D}]Cos[\frac{n\pi}{D}(x-c-\frac{W}{2})]$$

with:

$$c=\frac{2D-W}{2}$$

I've included a plot for:

W=5
D=10
h=1

for the first 25 terms of the series

#### Attachments

• pulse.JPG
6.3 KB · Views: 371
Our expressions are equivalent, in case you haven't noticed.

$$-c-\frac{W}{2} = \frac{-2D+W-W}{2} = D$$

and

$$\cos\left(\frac{n\pi}{D}(x+D)\right) = \cos\left(\frac{n\pi x}{D}+n\pi\right)=-\cos\left(\frac{n\pi x}{D} \right)$$

So it's cool.

Edit: Wait, that's not true

Last edited:
Ok I corrected an error in my original post due to this false identity that I had used.

Now our expressions are equivalent.

## 1. What is a Fourier series?

A Fourier series is a mathematical representation of a periodic function as a sum of sinusoidal functions. It is used to analyze and describe the behavior of periodic signals in various scientific fields, including physics, engineering, and mathematics.

## 2. How is a Fourier series used to represent isolated pulses?

A Fourier series can be used to represent isolated pulses by considering them as a periodic function with a very large period. The amplitude and width of the pulses can then be adjusted to create a series of sinusoidal functions that, when combined, approximate the shape of the pulse.

## 3. What is the significance of the width "w" in a Fourier series?

The width "w" in a Fourier series represents the duration of the isolated pulse. It is an important parameter as it determines the shape and frequency components of the pulse in the Fourier series representation.

## 4. What is the meaning of "+-D" in Fourier series?

The "+-D" in a Fourier series represents the symmetry of the isolated pulse. The "+" indicates an even symmetry, meaning the pulse is symmetric about the y-axis, while the "-" indicates an odd symmetry, meaning the pulse is symmetric about the origin.

## 5. How can Fourier series help in signal processing and analysis?

Fourier series can help in signal processing and analysis by decomposing a complex signal into simpler sinusoidal components, making it easier to understand and manipulate. It also allows for the identification of specific frequency components within a signal, which can be useful in filtering and noise reduction.

• Introductory Physics Homework Help
Replies
10
Views
250
• Introductory Physics Homework Help
Replies
8
Views
496
• Biology and Medical
Replies
6
Views
772
• Calculus
Replies
139
Views
6K
• Introductory Physics Homework Help
Replies
6
Views
2K
• Introductory Physics Homework Help
Replies
4
Views
1K
• Introductory Physics Homework Help
Replies
6
Views
6K
• General Math
Replies
12
Views
2K
• Introductory Physics Homework Help
Replies
2
Views
1K
• Introductory Physics Homework Help
Replies
1
Views
3K