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April 15th, 2016, 09:42 PM   #1
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Explain Fourier transform please

I bumped into Fourier transform and from its applications I found it was very important for scientists. But I don't know why. I tried going through Wikipedia, but it didn't help. I really want to understand this.

Please help. In layman's terms as far as possible.

Last edited by skipjack; April 16th, 2016 at 05:10 AM.
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April 16th, 2016, 12:52 AM   #2
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Quote:
I bumped into Fourier transform and from its applications I found it was very important for scientists.
There are two different mathematical processes that go under this name.
You need to be careful which one you mean.

There are many repetitive phenomena in Nature and in Man's experience.

The tides in the oceans
Soundwaves in the air
Music
Electrical waves in electrical apparatus
Mechanical vibrations in buildings and machinery.

The list goes on and on.

The Fast Fourier Transform (often abbreviated to FFT or sometimes FT) is a numerical mathematical method for extracting Fourier Series coefficients from tables of measurements on the phenomena.
These coefficients are used by engineers and scientists for many design, repair and inspection processes.

The second mathematical process is an analytical mathematics technique for solving some difficult mathematical equations by replacing the variables with different ones that offer easier equations.
This is what is meant by 'transform'.

There are many examples of transform methods in mathematics. Do you know the equation of a circle and of a parabola?

Last edited by skipjack; April 16th, 2016 at 05:13 AM.
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April 16th, 2016, 08:03 AM   #3
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Yes i do know the equation of circle and a parabola.I want to know about the general Fourier transform.
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April 16th, 2016, 08:51 AM   #4
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Quote:
Yes i do know the equation of circle and a parabola.I want to know about the general Fourier transform.
So if we take a circle

$\displaystyle {x^2} + {y^2} + 2gx + = {R^2}$

which has two variables, x and y

we can transform it into a pair of equations in one variable

$\displaystyle x = R\cos t$

$\displaystyle y = R\sin t$

Similarly with the parabola

$\displaystyle {y^2} = 4ax$

we can transform it into a pair of equations in one variable.
$\displaystyle x = a{t^2}$

$\displaystyle y = 2at$


First some more introduction.

M = 5.73204 x 1597.235676

Find M

In the past this was a difficult multiplication.
It was made easier by the logarithmic transformation into an addition.
So a multiplication equation is transformed into and addition

logM = log(5.73204) + log(1597.235676)

M = antilog (logM)

The General Fourier Transform includes the Laplace transform, amongst others.
These types of transformation do the same thing for differential equations,

That is they transform a difficult differential equation into an addition.
Once you have perfome the addition you have to perform the inverse transformation, just as taking the antilog above.

Do you understand differential equations?

Last edited by studiot; April 16th, 2016 at 08:53 AM.
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April 16th, 2016, 03:55 PM   #5
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Yes I took the course of differential equation of first degree.Please go ahead.
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