# Connexions

You are here: Home » Content » Meet the Fourier Coefficients

### Recently Viewed

This feature requires Javascript to be enabled.

# Meet the Fourier Coefficients

Module by: Richard Baraniuk. E-mail the author

Summary: An introduction to the Fourier coefficients.

ft=nZ c n ei ω 0 nt f t n n c n ω 0 n t
(1)
c n =1T0Tfte(i ω 0 nt)d t =ft,ei ω 0 nt c n 1 T t T 0 f t ω 0 n t f t ω 0 n t
(2)
Equation 2 is equal to "the strength of ft f t along the ei ω 0 nt ω 0 n t direction" as well as "the similarity between ft f t and ei ω 0 nt ω 0 n t ."

Multiply and integrate (Reference) and (Reference) (real and imaginary parts) to get c 2 c 2 , which is a complex number.

Signal ft f t : Figure 2.

## Fourier Coefficients

Also known as c n c n . In Figure 3, c 0 =1T0Tftd t c 0 1 T t 0 T f t and is the average value of ft f t and c 3 =1T0Tfte(i ω 0 3t)d t c 3 1 T t 0 T f t ω 0 3 t , where e(i ω 0 3t) ω 0 3 t is the real part (Figure 4).

### Note:

We can plot c n c n and c n c n as well.

## Example 1: FS of Square Pulse I

c n =1T0Tfte(i ω 0 nt)d t =1T0T4e(i ω 0 nt)d t +1T3TTTe(i ω 0 nt)d t =1(i ω 0 Tn)(e(i ω 0 T4n)1)+1(i ω 0 Tn)(e(i ω 0 Tn)e(i ω 0 T34n))=1(i2πn)(e(i2π4n)1)+1(i2πn)(e(i2πn)e(i2π34n))=1(i2πn)(e(iπ2n)1)+1(i2πn)(1e(i3π2n))=1i2πn(e(i3π2n)e(iπ2n))=1i2πn(eiπ2ne(iπ2n))=1πneiπ2ne(iπ2n)2i=1πnsinπ2n=12sinπ2nπ2n=sincπ2n2 c n 1 T t 0 T f t ω 0 n t 1 T t 0 T 4 ω 0 n t 1 T t 3 T T T ω 0 n t 1 ω 0 T n ω 0 T 4 n 1 1 ω 0 T n ω 0 T n ω 0 T 3 4 n 1 2 n 2 4 n 1 1 2 n 2 n 2 3 4 n 1 2 n 2 n 1 1 2 n 1 3 2 n 1 2 n 3 2 n 2 n 1 2 n 2 n 2 n 1 n 2 n 2 n 2 1 n 2 n 1 2 2 n 2 n sinc 2 n 2
(3)
where e(i3π2n)=eiπ2n 3 2 n 2 n by the periodicity of complex exponentials. To check Equation 3, we see that c 0 =12 c 0 1 2 . This is correct since it is the average value of ft f t . Figure 6 is a plot of c n c n .

### Square Pulse

1. Only odd harmonics (average value of c 0 c 0 doesn't count as even).
2. sinc sinc envelope on harmonics.

### Notice

sinπ2n={0  if   n even 1  if  n159-1  if  n3711 2 n 0 n even 1 n 1 5 9 -1 n 3 7 11
(4)
Therefore,
c n ={12  if  n=01πn  if  n±1±5±91πn  if  n±3±7±110  if   n even c n 1 2 n 0 1 n n ± 1 ± 5 ± 9 1 n n ± 3 ± 7 ± 11 0 n even
(5)

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks