What are the applications of fourier transform in communication?

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I have learned about Fourier transform .But,I have not deep understanding of it.  I heard about it is using in radios . But,I don’t know how and why it is using in radios. Can anyone explain it in very detailed manner ? What are other applications of Fourier transform in communication ?communicationfourierShareCiteEditFollowFlagasked yesterdayJanith11133 bronze badges New contributor

• 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. – jonk yesterday
• #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. – tlfong01 yesterday   
• 1I 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 . – Janith yesterday
• 2This 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) – The Photon yesterday
• #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, … – tlfong01 yesterday   

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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 yesterdayTony Stewart Sunnyskyguy EE75105k22 gold badges3737 silver badges146146 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 20 hours agoMatt48011 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 3 hours ago   

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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 3 hours agoanswered yesterdayuser28700117.7k22 gold badges99 silver badges3131 bronze badgesAdd a comment1

Question

What are the applications of Fourier transform in communication?

Answer

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

(1) Series (mathematics) – Wikipedia

(2) Function (mathematics) – Wikipedia

(3) Using Rpi python and MCP3008 ADC to read flame (RGB) sensor data for FFT analysis – RpiSE, 2021mar09

(4) Chat record of the above RGB sensor FFT analysis Q&A

(5) Spectrum analyzer (Section 4.2 ~ 4.4 FFT Based) – Wikipedia

(6) Laplace transform – Wikipedia

(7) Napier’s bones – Wikipedia

(8) How we used log tables before electronic calculators

(9) Slide rule – Wikipedia

(10) Euler’s number, e, (45 min YT)- BBC In Our Time

(11) Imagimery Number j (45 min YT) – BBC in Our Time

(12) But what is the Fourier Transform? A visual introduction (20 min YT) – 3Blue1Brown, 2018jan27, 5,493,780 views

/ 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, …

ShareCiteEditDeleteFlagedited 4 hours agoanswered yesterdaytlfong011,58411 gold badge66 silver badges1111 bronze badgesAdd a comment1

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.ShareCiteEditFollowFlaganswered 1 hour agoLvW19.5k22 gold badges1818 silver badges4343 bronze badgesAdd a comment