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What is the basic idea behind (discrete and continuous) Fourier transform (FT)? In short, what is the difference between discrete and continuous FT?

I have read multiple answers on the web related to this topic, but they are all long, provide long examples or cryptic equations. I would prefer not to read another long answer or an answer with the example of the circles or ellipses. I do not care about that. Ideally, I would like to read short answers which give the idea behind this concept. Your target audience should be high-school students (or, definitely, not math or physics college students), or, in general, people that do not have a very solid mathematical background and do not want to read equations. Hence, I hope it is clear that equations should be avoided as much as possible, unless they can be understood easily by high-school students.

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closed as unclear what you're asking by Stanley Pawlukiewicz, Matt L., lennon310, MBaz, A_A Jan 21 at 10:19

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  • $\begingroup$ Here is a blog article of mine on one conceptual meaning of the DFT: dsprelated.com/showarticle/768.php It's a lot of math, but you should be able to ignore if you understand the concept. There are pretty pictures that illustrate all the points. If you get those, you get what the DFT does. The article is based on real valued signals. Complex signal work the same way, with appropriate rotations involved. If you don't "get it", sleep on it, maybe you will wake up with an "aha". $\endgroup$ – Cedron Dawg Jan 19 at 17:01
  • $\begingroup$ Also, check out my answer here: dsp.stackexchange.com/questions/53706/… Feel free to upvote it ;-) $\endgroup$ – Cedron Dawg Jan 19 at 17:10
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    $\begingroup$ I suggest you watch this amazing video :) $\endgroup$ – mateC Jan 21 at 10:08
  • $\begingroup$ because your question is unclear. people often answer questions that are unclear or vague or subjective. the way that the answers evolved would seem to support my assessment. the answers are not the problem, the question is. you can edit your question to improve it $\endgroup$ – Stanley Pawlukiewicz Jan 21 at 16:15
  • $\begingroup$ what is a layman? are all high school students the same. why would this generic high school student have trouble? can you demontrate why any answer is successful? $\endgroup$ – Stanley Pawlukiewicz Jan 21 at 17:33
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I think the discrete Fourier transform (DFT) is typically the easiest to use as a base of explanation:

A DFT is a linear transform that operates on a time domain signal. After the transform, the signal is in “the frequency domain”, i.e the signal has now been broken down into a vector/signal which contains information about what sinusoids are contained in the signal, and how strong they are. That’s pretty much it, but I’ll continue on to be more thorough.

So let’s say I have a time domain signal that has a 200Hz size wave and some noise: assuming my sampling rate is sufficient, after I do my DFT, I’ll easy be able to see the sine wave as a peak in the frequency domain, whereas the noise won’t really “add up” to much in the frequency domain.

Of course it’s important to mention that once you do a DFT, there is also an inverse DFT operation that “unapplies” the transform; doing this returns you to the time domain and gives you your original signal back.

For a continuous Fourier transform, the idea is essentially the same, except it is just expanded to calculus rather than discrete mathematics/linear algebra

EDIT:

Okay, we'll try to make this even simpler using some pictures via MATLAB. I've added a good amount of words to explain in detail, but there is almost zero math in the text; hopefully that will be sufficient for you.

We'll start off by showing two signals: one is a sinusoid (with a frequency of 50Hz, that is it oscillates 50 times per second. If you need more information than that, please consult a high school text book), the other is that same sinusoid, but with noise added.

Since you asked what noise was, you can think of noise as random numbers added into the signal which "corrupts" the signal, i.e. it is no longer a pure sinusoid. A plot of such signals in the time domain looks like this: Time Domain Signals

On the top plot, we can see that there is clearly a sinusoid present. In the bottom plot, it pretty much just looks like a bunch of random numbers. A DFT essentially breaks a signal down into sinusoidal components. It does this by "looking" for the each sinusoid in the signal (it's a bit more complicated than that, but since you wanted limited to no math we'll skip it).

So now that we know that a DFT can "look" for signals, we take the DFT of our two signals, and plot the results to find the following figure:

DFT of Signals

In the top plot, we have the DFT of our pure sinusoid. Recalling that I assigned it a frequency of 50Hz, we would expect there to be a peak at 50Hz, and indeed there is. You may notice there's a negative frequency component. There is more mathematics required to explain that which involves unit circles, which you said in your original question that you "don't care about", so we'll just leave that be.

Now, on the bottom plot, we have the DFT of the sinusoid plus our random noise. We can see that those two peaks are still there at -50Hz and +50Hz. When we just looked at the time domain signal, it looked like "nothing" was there, just a bunch of random numbers. Through using the DFT, we were able to show that there is actually a sinusoid buried in that noise, and we were able to identify it easily in the frequency domain. This is a very basic motivation for how/why one might want to use a DFT.

Naturally, the inverse DFT (IDFT) does this operation in reverse, i.e. it maps a signal in the frequency domain to the time domain. Knowing that, it's easy to see that if we take the DFT of a signal, perform no other operations, and then eventually take the IDFT of the output, we are given back the signal we started with. This is because the DFT is what is called a linear transform. If you'd like to learn more about that, you'll have to learn more about linear algebra.

This is just one application of the DFT, there is a myriad of other applications, and it is a VERY important subject/tool in signal processing. The Wikipedia article for the DFT has a list of a wide range of applications. If you're interested in learning more, I'd start there.

If someone else would like to provide details for the continuous time Fourier Transform, please do! I think you'll find its more or less going to be the same, but a bit more mathematical in its explanation.

For completeness, here's the MATLAB code for the figures I've shown:

f = 50;               % sinusoid frequency
fs = 500;             % sample rate
t = linspace(0,1,fs); % create a time vector 
% create a sinusoid
x = sin(2*pi*f*t);
% create a noise signal
n = 1.5*randn(size(x));
% add in the noise to the signal x
xn = x+n;
% create a frequency vector that we'll use to label the plot
f = linspace(-fs/2, fs/2, numel(t));

% create a some plots
figure(1);
subplot(2,1,1);
plot(t,x);
xlabel('Time (s)'); ylabel('Magnitude');
title('Pure Sinusoid (No Noise)');


subplot(2,1,2);
plot(t,xn);
xlabel('Time (s)'); ylabel('Magnitude');
title('Sinusoid Buried in Noise');

figure(2);
subplot(2,1,1);
plot(f,20*log10(abs(fftshift(fft(x)))))
xlabel('Frequency'); ylabel('Magnitude (dB)');
xlim([-250,250]); ylim([0 60]);
title('DFT of Pure Sinusoid');

subplot(2,1,2);
plot(f,20*log10(abs(fftshift(fft(xn)))))
xlabel('Frequency'); ylabel('Magnitude (dB)');
xlim([-250,250]); ylim([0 60]);
title('DFT of Sinusoid Buried in Noise');
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  • $\begingroup$ 1. What is a linear transform? 2. What is an example of a signal on a time (and frequency) domain? 3. How can a time domain signal have 200Hz, if Hz are the units of frequency and not time? 4. What is an example of "noise"? What do you mean by that? 5. What do you mean by "assuming my sampling rate is sufficient"? Please, just edit your answer to add these, IMHO, important details. I know that these may sound trivial questions, but just to be explicit and clearer. $\endgroup$ – nbro Jan 19 at 14:42
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    $\begingroup$ Sure, I'll attempt to help you out, but I will say eventually we're going to hit a mathematical barrier, and I'll certainly exceed that ten to fifteen line limit you imposed earlier. The reason you keep finding explanations of the Fourier transform which are heavy on mathematics is just simply the fact that to understand and utilize the transform properly, one has to know all/most of the underlying mathematics, which for the DFT is linear algebra (i.e. it's somewhat of a ore-requisite to know what a transform is). I'll edit my original answer in a bit to see if I can make it even simpler. $\endgroup$ – matthewjpollard Jan 19 at 15:19
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    $\begingroup$ Your new explanations definitely helped me a lot. One more question: why do we care about sinusoids? Of course, sine and cosine waves are commonly studied in mathematics, but, in the real world, why would they be interesting? $\endgroup$ – nbro Jan 19 at 17:19
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    $\begingroup$ Great question! The answer to that comes from an area of mathematics called Fourier Analysis. In simple terms, you can think of it as basically every signal/recording can be represented as a series of summations of sine waves; this is exactly what a DFT does. If you'd like to learn more about it, the internet is full of great resources on Fourier Series and Fourier Analysis in general, though I will caution you that the math heavy "stuff" you saw in the other DFT explanations is all Fourier Analysis! So if you're going to research more, be prepared to learn a great deal of math along the way $\endgroup$ – matthewjpollard Jan 19 at 17:25
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    $\begingroup$ @nbro: The answer to your question about why sinusoids are of interest is that sinusoids are eigenfunctions of linear systems. That is, if you put a sinusoid into a linear system, you get a sinusoid out, with possibly a change in its amplitude and phase. See this old question for a discussion of why impulse and frequency responses are useful. $\endgroup$ – Jason R Jan 19 at 18:35
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I usually try by explaining a matched filter over a fixed length. Then I might graph how a sinewave matched filter of the same number of cycles as the signal adds up, and when using a filter of a different integer number of cycles, the positives and opposites cancel out, and nothing gets through. If the students know trigonometry, then I might show how a pair of filters, cosine and sine, can detect any phase of sinusoid at their same frequency. Or show how any waveform can get split into odd and even, and thus get picked up by some combination of cosines and sines as even and odd filters at the same frequency.

A short DFT is just a bunch of fixed length matched filters, with each filter a different number of sinewave cycles. If they know about matrices, you can show the set of filters in matrix form as a vector space transform. You can then talk about about orthogonality and completeness of the vector space. Or hand-wave about the same if they don't know any linear algebra.

If the students know about the sqrt(-1), then you might mention Euler's theorem, and how the sine and cosine filters are commonly combined into one complex exponential filter in many common DFT toolboxes.

Then mention the historical controversy when Fourier suggested that this works to represent any (reasonable?) signal or function, even in the continuous case (with an infinite number of infinitely long filters), before other mathematicians cleaned it all up (with concepts beyond typical HS level).

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  • $\begingroup$ This works because everything (except the last step) can be demonstrated by a short Python/Swift/Basic program with arrays and graphical output. $\endgroup$ – hotpaw2 Jan 19 at 17:52
  • $\begingroup$ An audio equalizer visualization might also be helpful for step #2 if linear algebra isn't available. $\endgroup$ – hotpaw2 Jan 19 at 17:55

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