7:18 PM
2

I have learned about the Fourier transform, but I do not have a deep understanding of it. I heard that it is used in radios, butI don't know how and why. Can anyone explain it in a very detailed manner? What are other applications of the Fourier transform in communications? EDIT 1: I got a littl...

Janith, There is much depth to it. It's even used in filter analysis instead of Laplace, at times. In any case, I'd recommend googling up "3blue1brown" on youtube. He has some very excellent videos that will help, immensely. They are unique -- you won't find an equivalent to them. And he focuses on getting at the intuition and less so on math details, which is kind of important when you are just getting into something. Try him out.

#Janith, your question is interesting, but as #Tony says, it is a big subject, but we can eat the big elephant bite by bite, perhaps in 10 bites. I have started the first bite/step, playing with falstad's FFT app. I played with the app using a square wave sample and got a very, very rough picture what FFT is about. I am now showing as my answer what I am messing around with the app. I found that you need to know some basic middle school maths to understand better. Perhaps I can start writing up an short answer and you can ask me questions to expand the short answer into a long one. Cheers.

I watched 3blue1brown video related to Fourier transform . It has a nice example of decomposing signals .But,I need more details .After googling several times .I found it is using in spectrum analyzers .

This is like asking "What are the applications of wheels in motor vehicles?". The F.T. is so fundamentally important to communications technology that you can hardly discuss communications without the F.T. being involved. (Even if you want to start with Shannon's theorem, you have to define "bandwidth", and now you're back to Fourier transforms)

#Janith, (1) I am glad that you mentioned the spectrum analyzer. Actually FFT is sort of spectrum, and spectrum analyzer can use FFT to do its job. (2) Just now I watched the 3blue1brown video and found the first little part using sound components with different frequencies vary good. But the remaining part is too mathematical and too abstract, not to mention that the complex variable and 2D complex plane is used to illustrate the idea. Now let me try to explain the use of FFT to do flame sensing. / to continue, ...
1B3B's sound example is about composing and decomposing sound components of different frequencies. For flame sensor, it is about decomposing/extracting different colours of different frequencies (Ref 3, 4 of my answer)..

7:18 PM
#The Photon , I know there are many applications of impulse transform in different field of engineering .But,I just want to know several applications of Fourier transformation in communication and working principle in depth of at least one application for understanding.

I've written down a couple of examples: electronics.stackexchange.com/questions/425008/… but, honestly, your question is waaaaaaay to broad. When you start learning about communications technology systematically (instead of from limited-scope youtube videos!), you'll see that.

#Janith, another FFT app example you can read in depth/detail is Convolution, which is FFT's favourite app. You can read Wikipedia for a detailed description of the process of convolution. Then Wiki says that to do convolution of one function into another, you simply do FFT on each sequence, multiple point by point, then do inverse FFT. In short, FFT transforms complication convolution to multiplication.
@jonk, (1) the first time I watched the 3B1B FFT Intro YT non stop, I found it confusing. Then I watched it a second time. This time I paused a couple of times during the first 10 minutes or so. I needed to pause for about one or two minutes at the terms new to me, eg. "winding frequency", "centre of mass" etc. I was glad that at the end of the first 10 minutes, I pretty much understand what is going on in using animation to decompose the frequencies of eg. 2Hz, and 3Hz, from the composite sound signal. So the first 10 minutes shows the operation of FFT. / to continue, ...
(2) Then the short "sound editing" part that immediately follows is a very good example of one application of FFT. The application of using FFT to find the "noise frequency" and then edit the original noisy sound track by filtering out the noise frequency signal is indeed educational! Now I agree every word of your praise of the 3B1B video. No wonder it has millions of views. So I think the first 10 minutes of the 3B1B video is already a very good answer to the OP's question.

@Janith Hmm. Something I didn't notice earlier. Your equations shown are not Fourier, which is based upon \$j\,\omega\$, but Laplace which is based over the entire complex plane and is based upon \$s\$. Just FYI.
@tlfong01 There are two parts to a complex number. The Laplace transform includes a real number part, which represents either decay over time or else expansion without limit over time, and the imaginary part which is just that winding frequency thing. Multiplication in the complex domain normally includes both spiraling in or out as well as rotation. The Fourier transform only has the imaginary (winding frequency) part and sets the real part (spiraling part) to 0.

@jonk are you talking about complex form of fourier series ?

@tlfong01 When you hold \$\omega\$ constant then the function is rotated (wound around the circle) at that "speed," so to speak. If \$\omega\$ happens by coincidence to have a repeating period at that frequency then it will "add up" over and over again and show up as a "signal" over the entire integral. It's kind of like a pulse-height analyzer, of sorts.
@Janith Eq. 1 shown here, Fourier Transform, shows it. Yours doesn't match that one. It's more like the usual Laplace definition except that you got the signs wrong in the power of e. See the Laplace section later on in the above link.

7:18 PM
@jonk, Many thanks for your clarification on Laplace Transform and its relation to Fourier Transform. I must confess I have too little knowledge and skills in phasor diagrams, complex number analysis, and relation between Laplace Transform and Fourier Transform. So I am googling more stuff to study (Refs 14, 15, and 16 of my answer). I guess I need to spend at least 10 more hours to appreciate your comments, because I am very weak in complex numbers and their application in circuit analysis.

I think it's a Fourier transform; it's just using \$s=-\omega\$. Not conventional, but technically still outputs a Fourier transform with a variable substitution.

@jonk It is a form of Fourier transform .I think it can be derived by getting limits (T->infinity) of Complex form of Fourier series.There are some information related to it in this book(drspmaths.files.wordpress.com/2019/01/…) [page number 933] .

@jonk, Your comment "When you hold ω constant then the function is rotated, ..." lets me guess that there is something I don't know that I don't know, and this something is actually a stumbling block that makes me hesitate to move on. So I google 1B3B and found there are some 10 episodes including one in "complex numbers". I watched its first 10 minutes and the handsome guy said something that is eureka to me: He was asked for a better adjective to "imaginary" ? His answer is "rotation ...". Now I have a deeper meaning of winding and rotation. Thanks for your inspiration. Cheers.

I edited my question and wrote down what I understood by answer section/comment and googling facts .If I am wrong ,Please comment below .