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I have learned about the Fourier transform, but I do not have a deep understanding of it.

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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 .

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

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

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C – a complex constant That is complex form of Fourier series. enter image description here

That is general form of Fourier series.

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EDIT 2:

General form of Fourier series is a simplification of Fourier series . enter image description here

Fourier transformation is a mathematical procedure for converting a signal in time domain to a complex number in frequency domain.

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From http://www.falstad.com/fourier/ site

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Amplitude spectrum can be plotted by getting absolute value of complex valued function against frequency and Phase spectrum can be plotted by getting angle of complex valued function against frequency.

EDIT 3:

Continuous Signal and Discrete signals

A continuous signal is a function of the time which is continuous (there are no breaks in signal).

A discrete signals is a signal whose value is taken at discrete measurements.

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FFT-Fast Fourier Transformation

FFT is an algorithm which computes the Discrete Fourier Transformation of a sequence .

I coded two python scripts for Continuous Fourier Transformation and Discrete Fourier Transformation .But, there is a significant difference between phase spectrums .I don’t know that is because of discrete values or a problem of the script.

For Continuous Fourier Transformation

import matplotlib.pyplot as plt
import numpy
from scipy.integrate import quad

def comp_quad(func, a, b, **kwargs):
    def real(x,g,f):
        return numpy.real(func(x,g,f))
    def imag(x,g,f):
        return numpy.imag(func(x,g,f))
    real_p = quad(real, a, b, **kwargs)
    imag_p = quad(imag, a, b, **kwargs)
    return (real_p[0] + 1j*imag_p[0], real_p[1:], imag_p[1:])

def con_four(a,func,g):
    return  (1/2)*numpy.exp((-1)*(1j)*g*a)*func(a)

def square_wave(a):
    A=2
    T=2
    return (-1)*A*((a%T)<T/2)+(A)*((a%T)>=T/2)


fig,(ax1,ax2)=plt.subplots(nrows=2,ncols=1)
fig1,(ax3,ax4)=plt.subplots(nrows=2,ncols=1)


#Wave
r=[x/100 for x in range(200)]
ax1.plot(r,[square_wave(w) for w in r],color="darkblue")
ax1.set_xlabel("Time(s)")
ax1.set_ylabel("Amplitude")
ax1.set_title("Signal")
ax1.grid()


#Fourier Transform
freq=[e/2 for e in range(0,100)]
ang=[x*2*numpy.pi for x in freq]
y=[comp_quad(con_four,0,2,args=(square_wave,c,))[0] for c in ang]
#Amplitude
l=[abs(t)*2 for t in y]  #Normalize amplitude
l[0]=l[0]/2 #For DC component(frequency 0)
ax2.plot(freq,l,color="red")
ax2.set_xlabel("Frequency(Hz)")
ax2.set_ylabel("Amplitude")
ax2.set_title("Amplitude Spectrum")
ax2.grid()
#Phase
k=[0 if (abs(j)<0.001) else (numpy.pi/2)+numpy.angle(j) for j in y] 
ax3.plot(freq,k,color="red")
ax3.set_xlabel("Frequency(Hz)")
ax3.set_ylabel("Phase")
ax3.set_title("Phase Spectrum")
ax3.grid()



#Reconstruction
d=[sum([l[i]*numpy.sin(2*numpy.pi*freq[i]*t+(k[i])) for i in range(len(freq))]) for t in r]
ax4.plot(r,d,color="green")
ax4.set_xlabel("Time(s)")
ax4.set_ylabel("Amplitude")
ax4.grid()


plt.show()

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For Discrete Fourier Transformation

import numpy as np
from scipy import fftpack
import matplotlib.pyplot as plt
import cmath


def square_wave(a):
    A=2
    T=2
    return (-1)*A*((a%T)<T/2)+(A)*((a%T)>=T/2)
    
fig,(ax1,ax2)=plt.subplots(nrows=2,ncols=1)
fig1,(ax3,ax4)=plt.subplots(nrows=2,ncols=1)
fig2,ax5=plt.subplots(nrows=1,ncols=1)

#wave
p=[c/100 for c in range(200)]
sig=[square_wave(x) for x in p]
ax1.plot(p,sig,color="darkblue")
ax1.set_xlabel("Time(s)")
ax1.set_ylabel("Amplitude")
ax1.set_title("Signal")
ax1.grid()

#Discrete Fourier Transform
sig_fft=fftpack.fft(sig)
#Amplitude Spectrum
Amplitude=2*((np.abs(sig_fft)/len(sig))[0:int(len(sig)/2)]) #Normalize amplitude
Amplitude[0]=Amplitude[0]/2  #For DC component
sample_freq=fftpack.fftfreq(len(sig),d=0.01)[0:int(len(sig)/2)]
ax2.plot(sample_freq,Amplitude,color="red")
ax2.set_xlabel("Frequency(Hz)")
ax2.set_ylabel("Amplitude")
ax2.set_title("FFT-Amplitude Spectrum")
ax2.grid()
#Phase Spectrum
k=[0 if (abs(j)<0.01) else (np.pi/2)+np.angle(j) for j in sig_fft[0:int(len(sig)/2)]] 
ax3.plot(sample_freq,k,color="red")
ax3.set_xlabel("Frequency(Hz)")
ax3.set_ylabel("Phase")
ax3.set_title("FFT-Phase Spectrum")
ax3.grid()


#Reconstruction
d=[sum([Amplitude[i]*np.sin(2*np.pi*sample_freq[i]*t+(k[i])) for i in range(len(sample_freq))]) for t in p]
ax4.plot(p,d,color="green")
ax4.set_xlabel("Time(s)")
ax4.set_ylabel("Amplitude")
ax4.grid()

#IFFT
rf=fftpack.ifft(sig_fft)
ax5.plot(p,rf,color="brown")
ax5.set_title("IFFT")
ax5.set_xlabel("Time(s)")
ax5.set_ylabel("Amplitude")
plt.show()

enter image description here enter image description here

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  • \$\begingroup\$ Comments are not for extended discussion; this conversation has been moved to chat. \$\endgroup\$ – Voltage Spike Mar 23 at 19:18
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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.

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  • \$\begingroup\$ After researching a little bit ,I understood it is a big field .There are so many applications like AM(envelop detectors),FM receivers,detecting dtmf tones ,decoding morse codes,band pass filters , radars and more and some are implemented in hardware (for continuous signals) and other are based on software (for discrete signals-ADC analog to digital converters) and got a little bit interesting way to play with signals(basically rf) that is SDR(software defined radios) using spectrum(FFT) and spectrogram(SFFT) . So, I decided to accept your answer . \$\endgroup\$ – Janith May 5 at 14:10
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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.

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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.

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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.

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  • \$\begingroup\$ 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. \$\endgroup\$ – tlfong01 Mar 21 at 9:52
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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.


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.)


scope fft


(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.


fourier series


(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

(13) Fourier transform -Wikipedia

(14) Phasor Diagrams and Phasor Algebra - Electronis Tutorials

(15) Complex Numbers and Phasors - Electronics Tutorials

(16) Laplace Transform (And relation between Fourier Transform) - TutorialsPoint

(17) Lockdown math announcement - 2020apr17

(18) Lockdown math's 11 videos 2020may21

(19) Complex number fundamentals | Lockdown math ep. 3 - 2020apr25

(20) Abacus - Wikipedia


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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.


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