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# What are the applications of the Fourier transform in communications?

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 little bit more understanding about Fourier series and Fourier transformation by reading answer section/comments and googling things so many times.But,I don’t know I am correct or wrong. If I am wrong, please comment below.

The general form of sinusoid .

That means sinusoids can be defined using amplitude ,frequency and phase . Sinusoids can be represented in complex plain using Euler’s formula.

Fourier series is a method to express an arbitrary periodic function as a sum of cosine terms.

C – a complex constant That is complex form of Fourier series.

That is general form of Fourier series.

But,I don’t know where is the phase information in general form of Fourier series.

I think both amplitude spectrum and phase spectrum can be plotted using complex form of Fourier series.But,in general form of Fourier series I can’t find the phase in that.communicationfourierShareCiteEditFollowFlagedited 5 hours agoasked Mar 20 at 2:02Janith12144 bronze badges New contributor

Physics state some facts which stay. One of them is that devices which generate or detect radio waves in a predictable and controlled way are simplest if they work in certain frequency. That made sinusoidal function sin(2Pift) especially important.

In 1800’s mathematicians gave to us tools to analyze circuits which contain sinusoidal voltages and currents. In the first half of 1900’s radio signals became more complex than pure sine – that’s because pure sinusoidal signal carry no information except “it exists and has a certain frequency”.

Presenting signals as sums of sinusoidal ones – Fourier transform makes it – made possible to use design methods which told how circuit works in different frequencies, no matter the actual signals were complex – for ex. speech or TV-image which was modulated onto a sinusoidal carrier.

Most practical communication signals need an infinite number of sinusoidal components. Fourier series cover it if the signal repeats. Fourier transform gives how the needed sinusoidals distribute (as relative amplitudes and phase angles) over continuous frequency range when the signal is non-repeating.

If you take a book of communication theory you will find Fourier transform is used nearly continuously. The text probably talks about signal spectrums, but in the start of the book or as an appendix there’s a chapter how the spectrums are calculated as Fourier transforms.

Numerical calculation method (=FFT) of Fourier transforms of stored digitized signals works so fast that in radios it’s effective to detect modulations with it. Transmitters code the actual signal to frequency, amplitude and phase angle variations. FFT in the receiver founds them. Especially useful FFT is for compression to reduce needed bitrate, target detection in radars and generally making signal filtering fast.ShareCiteEditFollowFlagedited 2 days agoanswered Mar 20 at 8:56user28700117.7k22 gold badges99 silver badges3131 bronze badgesAdd a comment3

This could fill up a book. So rather than be redundant, study the characteristics that change between the time and frequency domain with amplitude and phase with a visual approach.

Go here http://www.falstad.com/fourier/ and enable mag/phase and log view boxes.

Then use your mouse to select any standard signal and then modify any of the 3 views. The top is an arbitrary repetitive waveform that you can hear and its Fourier Spectrum below.

An impulse is extreme broad-spectrum and shown as repetitive. Square wave is the derivative of a triangle wave and thus the spectrum and phase changes accordingly.

f-3dB = 0.35 / Trisetime = half-power point when measured from 10 to 90% of the peak amplitude in time domain. Harmonics of any fundamental will have an envelope in the harmonics with nulls which are equivalent to the frequency ( and its harmonics) of that partial pulse width at 50% amplitude.

Then imagine this for any spectrum from ULF geomagnetic or vibration earth tremors used to find oil, gas & minerals from a shock impulse to a radar pulse to galactic noise.

Draw any time waveform and see the Fourier response, extend the bandwidth by the number of harmonics displayed. Remember that pattern and repeat. Then delete and try to create the time signal from the harmonic amplitudes and phases using a small n value. If you can do this then you understand how Fourier works.ShareCiteEditFollowFlaganswered Mar 20 at 2:25Tony Stewart Sunnyskyguy EE75106k22 gold badges3737 silver badges146146 bronze badgesAdd a comment2

Here is my very short answer:

The output spectrum of a frequency-dependent system is nothing else than the signal input spectrum multiplied by the transfer function A(jw) of the system. Both spectrums are the FOURIER transforms of the correspondiung functions in the time domain. In this context, the transfer function A(jw) can be derived from the system function H(s) for s=jw.

That means: The transfer function A(jw) connects the FOURIER transforms of the input as well as output signal functions. In contrast to the LAPLACE transform, the FOURIER transform has a clear physical meaning.ShareCiteEditFollowFlagedited 2 days agoanswered 2 days agoLvW19.6k22 gold badges1818 silver badges4343 bronze badgesAdd a comment1

Lots of good, very complicated answers. Here’s what I hope is a simple one…

A Fourier transform tells you the frequency content of a signal. That’s all it does. If I record myself playing a single note on a flute, and plot it as a function of time, it will look similar to a sine wave. If I take the Fourier transform of that wave, it will tell me what note I’m playing. I’ll get a spectrum of the signal, and if I’m playing an A at 440 Hz, it will have a large peak at 440, and smaller peaks at 880, 1320, etc., all the multiples of 440, called the harmonics. You can use this to build a simple instrument tuner.

That’s not really communications, at least as EEs would define it. So let’s say you run an FM radio station that broadcasts at 100 MHz. You want to play music that has a bandwidth of 20 kHz. Your neighbor runs an FM station at 100.1 MHz. Will you interfere with each other? You need to find the Fourier transform of your 100 MHz carrier modulated with the 20 kHz signal to find out if the other station is too close to yours.ShareCiteEditFollowFlaganswered Mar 20 at 16:12Matt50011 gold badge33 silver badges66 bronze badges

• Yes, I agree that what FFT does is digital signal processing (DSP), which can have nothing to do with communication. Similarly, AM and FM radio are analog signal processing, (modulating and demodulating 20kHz range audio signals) amplitude or frequency. Of course you need to transmit and receive AM/FM signals carriers, around 500kHz to 1MHz. And transmitting modulated carrier signals is of course communication engineering. – tlfong01 2 days ago

Question

What are the applications of Fourier transform in communication?

Part 1 – Getting a rough idea by starting with Falstad’s FFT app.

(1.0) Introduction

So I am using the square wave example to start messing around. The app seems newbie friendy, I just clicked, clicked, clicked, and after 3 clicks, I now “more or less understand” what is a Fourier Series.

If you have a cheapy US$300 scope like my Rigol 50MHz, you can play with realtime FFT to get a deeper understanding of how a function in time domain (1 kHz square wave) is transformed to frequency domain. (See Ref 3 for a real example of FFT application in detecting flames.) (1.1) What actually is a “series”? Series – Wikipedia says the following about “Trigonometric series” A series of functions in which the terms are trigonometric functions is called a trigonometric series. The most important example of a trigonometric series is the Fourier series of a function. (1.2) What is Laplace Transform? In mathematics, the Laplace transform, named after its inventor Pierre-Simon Laplace, is an integral transform that converts a function of a real variable t (often time) to a function of a complex variable s (complex frequency). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms differential equations into algebraic equations and convolution into multiplication. Part 2 – An example of application of FFT like a spectrum analyzer of a range of frequencies of audio/sound or video/colour signals. Introduction 2.1 If three sound components of frequencies Freq1, Freq2, and Freq3 are superimposed/added together to form a combined wave/time dependent signal, a spectrum analyser or FFT can be used to extract the original/composing frequencies Freq 1, 2, 3. 2.2 Now a burning flame can be considered as similarly sending out video/light signals which can be considered as a composite signal of different Red, Green, Blue colour signals. Each of the R, G, B signal has a fixed wave length/frequency. For a particular burning flame the amount/intensity of R, G, B components would be different according to the chemical property of the stuff being burnt. So if we use a spectrum analyzer or FFT to find out the relative intensities of the R, G, B components, and we can determine which chemical substance is being burnt. One big important difference between FFT and the traditional spectrum analyzer is the FFT can do its job real time, say in milliseonds, so it can be used for fire alarm in a chemical or nuclear plant. / to continue, … References / to continue, … Appendices Appendix A – Napier’s bones Napier’s bones – Wikipedia How we used log tables before electronic calculators Napier’s bones is a manually-operated calculating device created by John Napier Scotland for the calculation of products and quotients of numbers. Using the multiplication tables embedded in the rods, multiplication can be reduced to addition operations and division to subtractions. Napier’s bones are not the same as logarithms, with which Napier’s name is also associated, but are based on dissected multiplication tables. / to continue, … ShareCiteEditFlagedited 13 hours agoanswered Mar 20 at 3:49tlfong011,58411 gold badge66 silver badges1111 bronze badgesComments disabled on deleted / locked posts / reviewsAdd Another Answer Chat Record 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…communicationfourier jonkJanith, 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. tlfong01#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. JanithI 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 . The PhotonThis 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) tlfong01#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).. Janith7:18 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. Marcus MüllerI’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. tlfong01#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. jonk@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. Janith@jonk are you talking about complex form of fourier series ? jonk@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. tlfong017:18 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. KD9PDPI 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.

Janith@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] .

tlfong01@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.

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

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