multiply with e^(-j) in Fourier transform?

Discussion in 'Math' started by screen1988, Mar 20, 2013.

Mar 7, 2013
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I am curious why we have to multiply with $e^{-j\omega t}$ in Fourier transform? What is the purpose of this?

Oct 2, 2009
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3. WBahn Moderator

Mar 31, 2012
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You mean when we are performing the actual transform itself?

There are a number of ways of explaining it depending on how rigorous you want to be. Here is a pretty simple and intuitive way of looking at it.

If so, keep in mind that a transform is merely a change of variables from one set of coordinates to another. Think of the hoops you have to jump through to convert an equation from spherical to cylindrical coordinates! But it is possible that it might be easiest to understand a particular set of equations in spherical but easiest to solve them in cylindrical. So we would jump through the hoops, solve them, and then jump through the hoops again to go back to spherical.

For the Fourier Transform, multiplying by the complex expenential and integrating is merely the process (the hoops) of transforming between the two sets of coodinates that are used. We jump through them because, for systems operating in AC steady state, it is much easier to solve the set algebraic equations that result in the transformed coordinate system than the set of differential equations we have to deal with in the original time-domain coordinate system.

screen1988 likes this.

Mar 7, 2013
310
4
yes, I meant that.
Can you give me an example?
For example, there is a signal$x(t)= \sin \left( \omega _{0}t\right)$ after fourier transform we have$X\left( \omega \right)= \dfrac {\pi } {j}\left( \delta \left( w-\omega _{0}\right) -\delta \left( \omega +w_{0}\right) \right)
$
that is a complex function. What does this result mean? What is the meaning of j in the result?

Last edited: Mar 21, 2013
5. MrChips Moderator

Oct 2, 2009
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i and j are the same thing.

Think of j as a mathematical operator that puts a signal 90° out of phase.

In mathematics we write i. Because this causes confusion with the symbol for current, we use j instead of i.

screen1988 likes this.
6. WBahn Moderator

Mar 31, 2012
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The first thing that you need to understand is that there is no fundamental reason why any part of the transformed equation has to have any physical meaning. It could just be a purely mathematical transform to a set of coordinated in which the math is easier to carry out.

In the case of the Fourier transform, the coordinates are known as the "complex frequency" plane and we can tie pieces parts of it to physical quantities. The imaginary part carries steady state frequency information while the real part (which you see in the more general Laplace transform) carries information about the transient response. The j is merely a mathematical artifact of the transform.

screen1988 likes this.

Mar 7, 2013
310
4
Could you give me the link explain in detail how to change of variables from one set of coordinates to another in fourier transform?
I can't actually figure out how is it performed. I think first I need to understand the conversion coordinates in fourier transform and its meaning.

8. WBahn Moderator

Mar 31, 2012
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For heaven's sake, just Google Fourier Transform.

$
F( \omega ) = \int_{- \infty }^{+\infty} f( t ) e^{-j \omega t} dt
$

Your original function was in the t domain. Since you are integrating over t, the t goes away and you are left with a function in the f domain/ That's the transform!

9. mikedb New Member

May 16, 2013
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If you want to bake a cake -google it--FOR EXAMPLE

1.I found lots of info on eggs

2.I now know were milk comes from.

3.Icing is very fine suger.

4.A stove heats up food and can also bake a cake.

5.Butter is made from milk-how do i make butter?

I still do not know how to bake a cake,after searching 100's of sites for months,i do not know how to impliment FFT'S and what i saw on the NET
the people who know does not tell.

Just a simple samplel -64 samples will do

1.example--SIN of all 64 samples call it S1 to S64

2.example--COS of all 64 samples call it C1 to C64

3.AND THEN WHAT DO I DO NEXT????????????

10. mikeleeson New Member

Aug 22, 2012
26
4
This is how to bake your cake:-

You have 64 samples of a waveform f(t) that has a duration of T. If I define $\omega =\frac{2\pi}{T}$ then

$f(t)=\frac{a_0}{2} + \sum ^N_{r=1} \left(a_r cos(r\omega t) + b_r sin(r\omega t) )\right$

This is a Fourier series, if you can find the values of $a_0$, $a_r$ and $b_r$ (for r=1 to 64) then you have done a Fourier analysis of your sampled signal.

$a_0 = \frac1T \int ^T_0 f(t)dt$

$a_r = \frac1T \int ^T_0 f(t)cos(r\omega t)dt$ r=1, 2, 3, ...

$b_r = \frac1T \int ^T_0 f(t)sin(r\omega t)dt$ r=1, 2, 3, ...

Evaluate the integrals for every value of r and there you have it, your cake.

This is not the most efficient way to find the values: you can combine the cos and sin series using complex exponentials; you can speed up the calculation process by using certain properties of exponential functions and integration (FFT). These are clever tricks that you learn along the way...

But you didn't come here to learn or to contribute. You walked into a room full of busy, professional chefs and shouted "TELL ME HOW TO MAKE A CAKE".

11. WBahn Moderator

Mar 31, 2012
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So? There are lots of threads on this and any other forum that I've read and not found anything useful. Does that give you the right to hijack someone else's thread complaining that it didn't answer a question you didn't ask?

It's particularly ironic that you hijack a thread asking about a particular aspect of the Fourier transform and you appear to be whining about implementing an FFT, which is a related, but very different critter.

If this is all you could find after searching 100's of sites for months, then you might as well give up any notion of ever doing anything technical because this involves problem solving and you clearly have neither any aptitude for it nor any willingness or ability to gain any.

Using your example of how to bake a cake. Okay, let's Google for "How to bake a cake":

Just the first few hits on the first page put that lie to rest.

So let's turn our attention to "How to take an FFT":

Are you REALLY claiming that there isn't any information there that would help you figure out what to do with the 64 samples of your waveform?

Why on Earth are you taking the sin and cosine of your samples? Are they angles?

How about reading, say, the first paragraph of the first hit from the link above. Notice how it says that the FFT (as discussed there) is an optimization of the more general DFT (discrete Fourier transform)? So how about learning how to take a DFT before trying to tackle the mathematical trickery involved in optimizing it into the FFT?

Can you at least track down the defining summation equations for the DFT, or is that too much for you?

mikeleeson likes this.
12. t_n_k AAC Fanatic!

Mar 6, 2009
5,448
790
I can fully understand the response ....

"For heaven's sake, just Google Fourier Transform."

Member mikedb undoubtedly has their reason(s) for lampooning the suggestion.

For mukedb's benefit I suggest you consider the extraordinary contribution that WBahn makes to the AAC forums. His generally patient attention to often inane questions is exemplary. His consistently detailed contribution is usually well regarded and many members have him to thank for sorting out their problems.

Personally I have nowhere near the patience of WBahn - although I rarely express it in the imperative to " ... just Google it". But I am often tempted to do so.

13. WBahn Moderator

Mar 31, 2012
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6,982
Thank you for the kind words, t_n_k. I seldom tell people to just Google it, either. This was a special case. Also, notice that even here I did not only say to Google it. I also provided the key part of the answer.

14. MrChips Moderator

Oct 2, 2009
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I would bet that there are not too many people in this world who know how the FFT algorithm works. And that would make even fewer people here on AAC that can explain it to you in one post.

So if you want to learn how to code it yourself you better start learning about butterflies and twiddle factors. Go get a good text book on DFT and FFT or at least try to look it up in the library or on the internet.