i need to understand fourier transform.

elmolla wrote

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


hello elmolla

i want to know about orthogonal functions and dirchlet conditions and its significance in fourier series
(i know orthogonal signals that are right angle to each other)

Hi Wolfheart and kumar,

Wolfhart, I'm really embarassed for not uploading the files uptill now, I've already made an electronic copy of it, but it was corrupt!!! So I'm managing to make another one again. It'll be online by thursday.

Kumar, well, orthogonality of functions is different from the orthogonality of two lines, or as you studied about how to get the orthogonal set of functions on another function at a certain point.

By Definition:
Two functions A(x) and B(x) are orthogonal wrt a weight function W(x) on an interval (p,q) if the integral ∫A(x)B(x)W(x)dx over (p,q) =0 and such that ∫A(x)²W(x)dx≠0 over (p,q) where A(x) B(x) and W(x) are non zero over (p,q). But this concept is used for a set or families of functions. Using this definition, we can define what is known as the Generalized Fourier Series which is defined for any set of orthogonal functions of which the Fourier Series used in Signal Processing is just one of them known as the Fourier Trigonometric Series. There are also the Forier-Bessel and the Fourier-Legendre Series.

I see that Kumar and Wolfheart are both interested in this topic, so I'll post one for this for you guys in another post just after I upload the files for Wolfheart because I'm really busy these days, but I'll post it as soon as I'm ready.

Sorry for taking so long folks. I apologize again Wolfheart.

i am going to try to explain from a different point of view. Lets say you have a signal which is a function of time. Whatever form it is. What the FOURIER transform does is: it multiplies the signal to a cosine(or sine) and then integrates this multiplication. What you obtain is a number. This number represent how much the signal is "close" to the sine or cosine. In other words, when you multiplies two signals,(and then integrates ) you are obtaining a relationship number( close definition to correlation coeficient). If the signal is close in form and time(phase) to the cosine(or sine) the number is high. For other hand if the signal is not close to cosine then the number is small. So,the Fourier coefficeint you obtain is just an indication of how much of that frequency(of cosine) the signal contains. When you do this to the rest of frequencies the whole signal can be represented. You need to this with multiplying signals that dont have any relationship to each other. Two cosine wich one is harmonic of the other one are orthogonals. Then comes things like this, if the signal is even you will need only cosine to represent it. If the signal is odd you will need sine to represent it. If the signal is not even or odd(but a mix) you will need both. That is why signals can be represented by two set of other signal(cosine and sine) which they dont have any relationship to each other. If you try to do a multiplication of a pure cosine and a pure sine the coeficient is zero.( this is said they are orthogonal).You can do that with any type of multiplying signal as long they are not related(the relationship is zero = correltion is zero), that is the principle of Wavelet.Evey cosine is orhogonal to the next and to the sine of the same frequeccy or multiples. However, wavelet introduce one more element, "locality". In the case of sine/cosine the multiplying signals are present all the time, form -infinite to +infinite. In wavelet you can change that..That is the concept of scale in Wavelet. The closeness(look alikeness) becomes localized in time(scale).

Is like saying, when you do a multiplication of two numbers, the results of the multiplication of these two numbers is an indication of how much one contain the other one.Tha t is the principle of Fourier.
I hope this will help

Hello Mr. wolfheart

I am not sure if it is too late, but this was and to a certain extent still is a question that bugs me, but after a couple of years I have come to get a reasonably good feeling for what the Fourier transform is doing.

First off, I assume that you know that the complex exponential is just a fancy way of writting cos(x)+i*sin(x). This is from euler's formula. This is easy to look into on google. Anyways, when people say functions are orthogonal to other functions, they are esentially saying something very similiar to when you say one vector is orthogonal to another vector. I am going to assume that you have some kind of engineering background based on the fact that you even care about this topic so I assume you know what the dot product is for vectors. Basically when we dot product two vectors together you are multiplying the x components together, the y components together, and the z components together, and then adding together these products.

dot product = x1*x2 + y1*y2 + z1*z2

You are essentially doing the same thing with the fourier transform. You can think of the complex exponential, and the signal under study as the two vectors. You can think of every instant in time as the x,y, z, components (except it is no longer discrete in the same way the x,y,z used to be) The integration is just a fancy sumation of all the products of the two functions (vectors). This is how I generally think about it. You are essentially dot producting tw vectors. Remeber when you do a dot prooduct with vectors you are esentially trying to see how well the two vectors line up. When they are perpendicular to each other they don't line up at all and the result is zero. when they line up perfectly, the result is simply the product of the magnitudes of the vectors. Everything else is somewhere in between.

There is another way to look at this too. If you are comfortable with the concept of correlation the fourier transform is essentially just trying to find out how well a signal is correlated with signs and cosines.

For more informaiton, I think one of the easiest websites to understand is wikipedia. lookup correlation cross correlation, fourier transform, signal processing, etc.

Personally I think oppenhiem's book is very hard for begineers to understand. Try looking in a signal and systems book such as "signals and systems" by Chi-Tsong Chen.

I hope this helps and I would be curious to know if helps or not. I have spent years trying to get a feel for this thing and alot of the theory generally is hard to get a feel for. I think most people are more comfortable thinking in terms of vectors.

Hey guys,
Sorry for the bad handwritting, and if you want anything else tell me!!

Let me cut to the chase!

Have you heard something from Claude Debussy famous french composer? Fourier is french so I chose a french composer. Bummer!

Orchestra is composed by 3 violinists,each violinists play violin different than the other, one piano and one cello **for example.

Music is so beautiful because is made of many harmonics.

Separate them and you will not hear the same music because the beautiful Debussy music of Clair de Lune sounds great if played by all people in orchestra.

If you are able to separate in a music sound (**or any signal) each component then what you have done is the famous fourier transform.

Mathematics behind it is just doing by means to extract component you like. Things shall get complicated if signal is a corrupt,encrypted etc.


Vive La France!

Hi Wolfheart

I'm a newcomer.
I hope the following link can help you understand about equation Fourier and how to compute coefficients.
**broken link removed** (Tutorial 4,5,6)

Hi there,

First of all, you have to admitt, why did they choose sinusoids ?

One reason for that is what's called sinusoidal fidelity, which in simple words means that if you apply a sinusoid to a linear system then the output will be sinusoid of the same frequency but with some delay and attnuation factor.

Cool, now we can say: if we apply a signal (which can be broken up to a bunch of sinusoids, as has been illustrated in previous posts) then the output can be characterized by at most the same sinusoids.

To think of frequency domain, you have to think of time as another dummy domain, and not THE PROMINENT DOMAIN, in this manner you can pose the question :
how many dimensions do we live in?
That's a good question which can be answered by a thing called m-theory.......

In teaching DSP lab, i found a tutorial in the appendix of the book : Digital communications , by sklar

Hope this helps ia m saying the same things that the others said but in a slightly different manner

Fourier representation of a signal is basically unique resolution of the signal into its components
In physics when dealing with vectors , u might have learnt resolution of vector into components along x,y and z axis.The fourier transform does the same to signals i.eit resolves the signals into its components. When dealing with vectors U resolve it into ortogonal components.

U may ask Why represent a signal as a sum of sinusoids?
This is because is a smooth function (easier to deal with infinitely differentiable) and certain natural phenomena are also sinusoidal in nature ( displacement , velocity and acceleration for a simple pendulum)

Why orthogonal functions?
Given a vector F ,its x component can be easily obtained by Fx=F.i .
similarly component of signal with frequency "F" can also be computed easily (finding fourier coefficients An,Bn or Cn).

Difference between Fourier series and Fourier transform
Fourier series generally is an infinite series of discrete elements placed uniformly.Fourier series is used for periodic signals. A signal with fundamental frequency "Fo" will have spectral components at Fo 2Fo 3Fo .......
For noperiodic signals the period can be thought of as infinity,Infinity period is not practical assuming a very large period we get fundamental frequency = df. therefore the frequency domain representation will have components at df 2df 3df ... .This gives us a continuous frequency spectrum i.e Fourier Transform.

DC component
This component is a constant component of signal (note it is orthogonal to sinusoids over a period of 2*pi )and can be never resolved into sinusoids.

Fourier transform is nothing but the expansion of an arbitrary function in terms of a complete basis. you can find more information on this issue in the book by Arfken, Mathematical Methods for Physicists.

fourier transform is used to transfrom signals from time domain to frequency domain. sometimes, signal in time domain is hard to analyse. hence, we need to transform it into frequency domain to 'see things' we can't in time domain.

by fourier transform you can see the all harmonic that exist in the time domain signal and easily can be filtered them.

Just a side note:

I remember reading or hearing from a Proff. that Mr Fourier was working
on problems that the French Army was having while shooting cannons.
The Pulse vibrations.

But his theory applies to any repeating signal that has a period. And if you
make the period infinite it'll still work!

Cheers

Hello,

I am a grad student at Berkeley. I need to learn Convolution and Fourier Transformation very quickly for my new research project and am trying to teach myself by using Oppenheim's Signals and Systems. (2nd edition) Is there a way to get a hold of the Solutions Manual for this book?

Thank you !!

The first thing you need to know is how to do a search.
The solutions of the book is in this forum in some place.

Fourier transform is the mathematical bridge connecting the TD and FD worlds. It just allows you to see the problem in different angles. Also, somtimes it is easier to solve the problem in one domain than the other.

there are good tutorial in **broken link removed**

A real good discussion ..

But ..

everything was about using Fourier Transform to get a spectrum of a signal so as to ease the dealing with it .. On ther hand .. something like Discrete Fourier Transform (DCT) and it's fast implementation (FFT) is basically a Fourier Transform .. but they are implemented in Hardware .. a question here comes .. what's the use of implementing an illusion ? .. (if it's really an illusion , as it's only used to transform an equation from a diffecult form to another easier one) ..

And then .. someone comes and says : DFT/FFT is used in many applications like data compression .. modulation .. etc.

So, how come an illusion used in an application ?? .. if X(F) is the FT of x(t) .. then still both are 'x' .. nothing happened to the signal itself .. so, would someone tell about how can a conversion from one shape to the other help in Modulation for example (like in OFDM) ?

very nice question. i will try to answer it because im still a student and i dont claim to be a master in DSP.

when you take the fourier transform of a signal the result you get is a combination of different frequencies. a signal can have one fundamental cosine at 50Hz and harmonics like 20kHz 100kHz. in the time domain you just had one signal x(t) but in the frequency domain you now have a number of signals. lets say you want to filter out the noise from the signal. you know that the noise is the high frequency components, say above 50kHz. now you will design a system such that when these signals are given at its input it attenuates the signals above 50kHz and does nothing to the others. so in the frequency domain you can see the different components of a signal and discard the components that you dont need and keep the components that you need.

i hope that helps

samcheetah said:

very nice question. i will try to answer it because im still a student and i dont claim to be a master in DSP.

when you take the fourier transform of a signal the result you get is a combination of different frequencies. a signal can have one fundamental cosine at 50Hz and harmonics like 20kHz 100kHz. in the time domain you just had one signal x(t) but in the frequency domain you now have a number of signals. lets say you want to filter out the noise from the signal. you know that the noise is the high frequency components, say above 50kHz. now you will design a system such that when these signals are given at its input it attenuates the signals above 50kHz and does nothing to the others. so in the frequency domain you can see the different components of a signal and discard the components that you dont need and keep the components that you need.

i hope that helps

Click to expand...

this can be done at the simulation level .. to know the characteristics of the filter that will remove the unwanted frequencies .. yet, u won't need to design any other piece of hardware to convert the signal entering the filter from time domain to frequency domain .. simply because signal is signal .. and domains are only existing in simulations ..