[Okta Webinar] Learn how to a build a cloud-first strategyRegister Now

x
  • Status: Solved
  • Priority: Medium
  • Security: Public
  • Views: 410
  • Last Modified:

What does this newly arranged equation for Fourier Transform mean?

Here's an equation of the Fourier transform: Fourier Equation

In the same equation image above, in the second line, is an equation I read from a paper. It seems that the a_n and b_n are turned into functions of A(u) and B(u) respectively, which depend on the variable u, which is the frequency. There is also an additional 2*pi in the sin and cos functions.

In other words, the eventual equation can get to become: New Equation

I don't understand several things in this new equation.
First, what are the A(u) and B(u) functions? Why do they depend on u?
Second, why is there a 2*pi in the sin and cos functions? What do the 2*pi mean?
Third, how does the new equation differ from the original equation?
0
xenonn
Asked:
xenonn
  • 6
  • 5
  • 2
1 Solution
 
aburrCommented:
A(u) and B(u) are just functions rather than constants. It depends on the problem
The 2 pi  is just 2 x 3.14159... and are used to change the frequency variable from radians to cycles per second.
0
 
xenonnAuthor Commented:
@aburr, Thanks. Just to clarify a little, since u is the frequency, is there any special reason/occasions when one would choose to have the functions A(u) and B(u), which are replacing the constants, to depend on u?
0
 
ozoCommented:
Where did the summation index  n go?
Could they have meant A(n) where they wrote A(u)?
0
Concerto Cloud for Software Providers & ISVs

Can Concerto Cloud Services help you focus on evolving your application offerings, while delivering the best cloud experience to your customers? From DevOps to revenue models and customer support, the answer is yes!

Learn how Concerto can help you.

 
aburrCommented:
I thnk ozo has hit upon a fundamental difficulty
The second line is  missing more than just the missing n. The last line it should have a cos or a phase angle.
Also if u is a frequency, what is x?  It cannot be a distance.
0
 
xenonnAuthor Commented:
Sorry about the variable n in the summation. It should be u. I had a typo.

I believe x refers to the time in the time domain.
0
 
aburrCommented:
"Sorry about the variable n in the summation. It should be u. I had a typo. "
I am still confused.
which n's should be u's? Or which u's should be n's.    If  you sum over u it cannot be a frequency. n is not a variable, it is an index.
Saying x is a time clears up a number of points
0
 
ozoCommented:
Can you restate the equations with the typos corrected?
0
 
xenonnAuthor Commented:
There shouldn't be n in the equation. It was a typo.

This is the correct equation: Corrected Equation

The variable x is referring to the time domain, and the variable u is referring to the frequency domain.

Sorry about the confusion. Hope this clarifies and helps to answer my question better.
0
 
aburrCommented:
There is still something seriously wrong with the equation.
If u is a frequency, what is the difference in cycles per second between u = 5 and u = 7?
-
The indices of a summation are usually integers and dimensionless, in which case the dimensionality of the arguments of sin are incorrect
0
 
xenonnAuthor Commented:
I'm not quite sure what is wrong. In this case, how should a correct Fourier transform probably look like if it has to be in this "sort of arrangement"?
0
 
aburrCommented:
The last three lines of my post told exactly some of the things wrong
0
 
aburrCommented:
a correct formulation (which might have nothing to do with your problem) would be

    \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos(nx) + b_n \sin(nx)
above line not rendered very well. Try
f(x) = a_0/2 + sum (from n = 0 to infinity)(a_n cos(nx) + b_n sin(nx)
0
 
xenonnAuthor Commented:
Thanks. The latex doesn't render on EE, but works fine when I paste it on a latex editor. The latex form actual reads better. Thanks! :)
0

Featured Post

Free Tool: SSL Checker

Scans your site and returns information about your SSL implementation and certificate. Helpful for debugging and validating your SSL configuration.

One of a set of tools we are providing to everyone as a way of saying thank you for being a part of the community.

  • 6
  • 5
  • 2
Tackle projects and never again get stuck behind a technical roadblock.
Join Now