Fourier Transformations

In summary, the conversation is about the difficulty understanding Fourier transforms and the suggestion to check Wikipedia for more information. The person asking about Fourier transforms has read about it but still doesn't understand, and the other person asks for their level of mathematical knowledge. The conversation then shifts to discussing vectors and the basics of Fourier transforms, with a mention of its application in spectral analyzers and its use in solving differential equations.
  • #1
skaboy607
108
0
Can anyone explain the above-i've read about in books, internet sites and still do not understand what its doing or the maths.

Thanks
 
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  • #2
Try wikipedia. It can't be explained in a few sentences.
 
  • #3
hmm, I've read that and although i understand the basic, i.e complicated function and representing with smaller functions with sin and cosine waves, the rest doesn't make sense no matter how many times I read it.
 
  • #4
how old are you? what level mathematical maturity do you have?
 
  • #5
Not really sure why it matters but I am 21. hmmm not that much i guess?
 
  • #6
you know anything about vectors? how you can express any vector as a sum of basis vectors?
 
  • #7
hmmm can't say that I do, is that where I should start then?
 
  • #8
what do you know then? why do you want to know about Fourier transforms?
 
  • #9
An intuitive explanation:
I am sure you have seen one of these http://en.wikipedia.org/wiki/Spectral_analyzer" [Broken] found on Hi-Fi's or digital audio players, that plot frequency vs. amplitude. They take the audio signal (amplitude/time), apply the (discrete) Fourier transform, and display the resulting function (amplitude/frequency).

To illustrate, take the function [tex]\cos(2\pi at)[/tex], which is a wave with frequency [tex]a[/tex]. Its Fourier transform is zero except for two "spikes" at [tex]-a[/tex] and [tex]a[/tex].


A more mathematical reason why the Fourier transform is important, is that it turns differentiation into multiplication, see http://en.wikipedia.org/wiki/Fourier_Transform#Analysis_of_differential_equations", which is quite useful for solving some differential equations.
 
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1. What is a Fourier Transformation?

A Fourier Transformation is a mathematical tool used to break down a complex signal into its individual frequency components. It converts a signal from its original time domain to a frequency domain, making it easier to analyze and process.

2. What are the applications of Fourier Transformations?

Fourier Transformations have numerous applications in various fields, including signal processing, image processing, data compression, and solving differential equations. They are also used in music and audio analysis, medical imaging, and telecommunications.

3. How does a Fourier Transformation work?

A Fourier Transformation works by breaking down a complex signal into simpler sinusoidal components of different frequencies. It uses a mathematical formula to calculate the amplitude and phase of each component, which are then used to reconstruct the original signal.

4. What is the difference between a Fourier Transformation and a Fourier Series?

While both Fourier Transformation and Fourier Series are used to analyze signals, they differ in the type of signals they can handle. Fourier Series is used for periodic signals, while Fourier Transformations can handle both periodic and non-periodic signals. Additionally, Fourier Series decomposes a signal into a sum of discrete frequency components, while Fourier Transformations decompose a signal into a continuous spectrum of frequencies.

5. What are some common misconceptions about Fourier Transformations?

One common misconception about Fourier Transformations is that they can only be applied to signals with a finite duration. In reality, they can be applied to both finite and infinite signals. Another misconception is that Fourier Transformations can only be calculated using complex numbers, while in fact, they can also be calculated using real numbers.

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