i need to understand fourier transform.

[wolfheart_2001]

how to understand fourier transformation

i cant understand fourier transform at all, i can solve the problems in the books but i cant gain an imagination for it so i dont know when to use it.

i searched in many books but i couldnt find anyone who really understand it also they all also bypass the reason of the theory which will declare the idea behind it and they also discuss things in a very abstract mathematical manner which is useless and have no meanings for the users who wants to use this theory in real applications.

i dont think mr fourier put this theory from nothing, he must needed somthing so the theory came out.

i wonder how a scientist from that era put that very intellegent and complex theory!!!

well do anyone really understand that theory will give me help???

 

[echo47]

how to understand fourier transform

Any signal waveform can be constructed by adding up a bunch of sinewaves having suitable frequency, amplitude, and phase. The fourier transform converts your waveform into that bunch of sinewaves.

 

[mayyan]

Transforms are made for your convenience so its means you should use it whenever you think it will make your life easyer. here are some examples:
1) you want to work in the frequacy domain.
2) there are thing which are much easyer to do mathematically like convloluation becoms multiplication, differentiate , time shifting and more


mayyan

 

[armsgroup]

try this it may help
www.cs.otago.ac.nz/cosc453/ student_tutorials/fourier_analysis.pdf

 

[ishtiaq]

The real meaning of fourier transform is that to convert a signal which is in time domain into frequency domain and vise versa. The advantage to use fourier transform is there when a signal is in time domain of infinite length and you want to analyze its response for any given channel. Since the signal is of infinite time length so you convert it into frequency domain where it will be periodic for a frequency range (so you take its first part) and you also convert channel in freq domain and there you just convolute/ multiply (as required) with each other and easily gets the answer.
In nut shell Signal in time domain of infinite length is finite in frequency domain.

I hope now it will clear you the basic idea behind Fourier Transform.

Thanks

 

[mayyan]

The advantage to use fourier transform is there when a signal is in time domain of infinite length and you want to analyze its response for any given channel. Since the signal is of infinite time length so you convert it into frequency domain where it will be periodic for a frequency range (so you take its first part)

As far as i know what you say is not correct. It will become true only if you sample an infinite signal and that is DTFT not FT.

mayyan

 

[steve10]

The Fourier transform itself does not restrict you to apply to any problems. As long as the veriable(s) range in the infinite space or are extendable to the infinite space, you can always find help from the Fourier transform. Dont' restrict yourself to the problems that have to have a special variable "time". For example, you can solve the elliptic equation (d^2/dx^2)u + (d^2/dy^2)u = 0, -INF<x<INF, 0<y<INF by Fourier transforming "x". Keep in mind that the purpose of the Fourier transform is converting the differential operations to algebraic operations, which mean to simplify the problems.

 

[suvendu]

u can understand it clearly when u read wavelet transform etc.where fourier transform is very much useful.otherwise u read communication thory wwwhich is totally based on fourier transform.
olz refer the book "signal processing and linear system" by lathi.

 

[wolfheart_2001]

sorry guys........but we usually need it in "finite" time problems not "infinite"...........i know that fourier transform things between time domain and frequency domain but i dont understand the equation itself . its bizzare, i need only to know how this equation really works, this is my mistry??

 

[Elenzo]

yes i understand what do you mean...
the equations of fourier isnt very easy to understand... like a:2/pi ∫f(t). sen(t)
I have the same question about ALL the equations that the fourier analisys use...

 

[amarnath]

read the initial portion of communication systems by lathi as to why we need a transform in the first place and how these equations came to place.

regards
amarnath

 

[wolfheart_2001]

hi amarnath,

isnt there any free source to this chapter, i asked many libraries about this book and they adviced me to by it through the web but this is costly since i need only that first chapter

 

[elmolla]

Hey guys, I know this might be late, but I'm a new user and this'll be my second post.
Hope it helps.

About the Physical meaning of Fourier Transform and how did J.B. Fourier got the idea:

A very comprehensive approach dealing with this matter is given in the third chapter of "Signals and Systems" by Alan V. Oppenheim and Alan Wilsky with S Hamid Nawab . Prentice-Hall . 2001.

Anyway, here is a summary of what I understood of it from that book and other sources.


First of all, lets take a look at the waves of the sea, they all look a bit like sinusoids moving through the sea or the ocean. When we usually cause a disturbance causing that wave as by throwing a pebble into the water or the periodic attraction force caused by the moon orbiting earth, we find sinusoidal looking waveforms.

So what does this have to do with anything?????

Well, Fourier was a wise guy (8) <--Fourier) !!!! If all the waves were sinusoids, they why do seamen talk about HUGE waves that have a large amplitude and look more like a wall than a slopy sinusoid???? How do they even build up to this huge no sinusoids?? [If you have seen a movie about the Tsunami or The Perfect storm, you can imagine what I'm talking about easly]..

The above paragraph isn't how Forier thought of it, its just an example to make you think about how can WE think of Fourier's ideas. In the real life, Fourier made his theory from the study of the conductive heat transfer through metals. But here is a method to think of it using Fourier's own study but with a clear example. When you solve the exact PDE of the heat transfer by conduction through a body, for example, a metallic bar, this well lead you to the wave equation solution describing the temperature distribution across the bar. Now, if you expose this bar to static thermal conditions, example 100°C at one side and 0°C at the other side, do you get anything that looks like a the sinusoidal wave given by the solution??? Well, then it must be a SUPERPOSITION of many waves, that gave this result. [Remember the Complementary function thing in the solution of PDE's]



That was about the physics, now how did he formulate that in mathematics? Lets see, but this discussion will be made for continuous time signals because they are more easier to imagine:

The last three lines means that generally:
f(x)=∑a1n.sin(bn.x+c1n) [n is the index of the summation]

so to get rid of the phase shift we can say more generally
f(x)=∑(a2n.sin(bn.x)+b2n.cos(bn.x)) [sincesin(a+b)=sin(a)cos(b)+cos(a)sin(b)]

Now, if f(x) was periodic with a period from -\pi to +\pi we could use the orthognality of the series. [Thats a very big subject which I'll assume you know, if not, tell me. It is very important and explains many of the equations] To overcome this condition, a guy called Dirchlet made some modifications to make it applicable for any periodic signal with any general period, but thats not out subject. SO the above form is complete and is known as the Fourier Trigonometric Series. This was the one that was used by Fourier.

A more general form of f(x)=∑an.exp(jwn.x) is prefered, because it reduces the calculations.



Now to get more insight into the transform:

We usually need the transform when we need to deal with non-periodic signals. Now, how to get on with that??

Just use the above form of Fourier Serier as the period tends to infinity (use a limit and make use of Reimann's Sum method to convert it to an integral) and you'll end up with the Forier Transform form.



How to apply the Fourier transform for periodic signals:

The Fourier Transform as well as the Laplace transform are integral transforms, ie they are linear operations. So we can easly just convert any periodic signals to their Fourier series and find their transform so that we can easly find the transform of any periodic signal and they we can concentrate on finding the transform of sin(tx) or cos(tx) or exp(jtx).

To solve that, we make use of complex analysis, their transform can be easly found using the Euler formula and making use of what is known as Cauchy's theory. These integrals are so much encountered that they are called the Fourier Integrals. 8O :x


Hope I helped. If you don't understand anything of the above or you're confused in anything in it. Just tell me. All your comments are welcome. Hope my tone of speech wasn't bad. I just wanted to be informal so that this post doesn't look like that FORMAL thing which we usually find in books. I didn't write the equations because I'm sure you know them well.

 

[wolfheart_2001]

hi elmolla,

thnx for ur long reply but u know it still sound uncomfortable, these equations are very complex in my imagination and u didnt mention also clearly the reason of the equation . i think it will be very helpful if u placed a problem and solve it with fourier(simple one of course). i mean this way we can know how to analyze a problem and when to use fourier

 

[steve10]

I don't think anyone else in this world can help you except one person, which is yourself. Apparently, you need some exercises.

 

[elmolla]

hi,
Wolfheart, I've made for you the derivation for the Fourier Transform from the Fourier Series. I'll scan it and upload it ASAP. If are thinking for a proof of the Fourier Series, consult any book about the Orthogonality of functions, or tell me to explain it as well, or just think of it in this way:

If x =5, I can say x=Σ(0,2,3)=Σ(-3,-2,-1,0,1,2,3,5), or more generally
x=Σ(.........,(a-2)*2,(a-1)*-1,(a0)*0,(a1)*1,(a2)*2,...........) and I'll choose the a's to suit the value I need.


The Fourier Transform is used with aperiodic functions, the Fourier Series is used with periodic functions.

 

[wolfheart_2001]

hi elmollah,

what is the book u nominate for me which describe the subject in full details , thanx for ur interest, and please send me the derivation for the Fourier Transform from the Fourier Series.

i will be thx also if u explained the fourier series proof, but the most important for me is fourier transform not series.

 

[raghav]

The thing of beauty that one should appreciate in the fourier transforms is the way it helps in analyzing the non-periodic functions as periodic ones. for the application part you can refer to the book :

Advanced Communication Methods --- Simon Haykin

 

[elmolla]

hi all,
Haykin is a very good book and it will help you much. But I nominate you to read the chapters 3, 4 and 5 of Signals and Systems by Alan Oppenheim, Alan Wilsky and Hamid Nawab.

I'm sorry I didn't upload it up till now but I'll do as soon as I can because I don't have a scanner .

The proof of the Fourier series can be easly understood but after learning about the Orthogonality of functions, you can search for it online and you'll find much helpful information.

 

[samcheetah]

here are the first five chapters of the book by Oppenheim (not a very nice scan)

and here is its solution manual