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This question has been bothering me for a very long time.

Imagine a wire carrying electric current. It carries two alternating current (AC) signals of different frequencies (say $50$ Hz and $60$ Hz). Now, since both the signals are on the same medium (wire), if I see the resultant signal on an oscilloscope or other device, I would actually see a new resultant signal that is an algebraic sum of individual waves at every point, but not the original signals anymore.

How can the individual signals (waves) be separated from the wire after they combined to form a new wave? I know the telephone lines and TV cables carry multiple frequency signals on the same medium, and we successfully separate them at the receiver, but was thinking how is it possible to separate out individual signals when they combine and result in a different wave altogether? How can receiver figure out the original two waves or hundreds of waves that resulted in the combined wave?

I read about Fourier series analysis and signal filters but it just does not make sense fundamentally to me.

It is like adding two numbers 2 and 3 which results in 5 and being able to separate them again in to 2 and 3 and not 1 and 4 or some other combination. The medium (wire) has only the combined superimposed wave. How does the receiver know that 5 should be separated as 2 + 3 but not 1 + 4 or 2.5 + 2.5 I hope I'm making sense. I tried googling this but I think I'm not framing the right search terms.

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  • $\begingroup$ You are on the right track about Fourier series. Since you are adding two functions with the same period, if you expand the resulting function in a Fourier series, it will give you back what the two original frequencies were, and what their relative amplitudes were. $\endgroup$
    – Sayan Mandal
    Commented Sep 2, 2019 at 13:17
  • $\begingroup$ Maybe the thing you don't understand is that the receiver doesn't split just one signal value into 5 = 2+3 or 5 = 1+4 or whatever. It takes a block of values, and then works out that the complete graph of that bit of signal looks like two sine waves superimposed, plus some random noise that can be ignored. For just two sine waves, the number of values you need to do this is small - the minimum number would be just 4 is there was no noise on the signal - but you can't do it with fewer than 4 values. $\endgroup$
    – alephzero
    Commented Sep 2, 2019 at 13:56
  • $\begingroup$ … in the real application, this also means that there is a delay when the receiver processes the signal. It might have to wait to read say 1024 samples of the signal at different times, then process all of them, and then deliver the two separated output signals. You can demonstrate that yourself if you can receive a radio station that generates time signals (like the BBC "pips" in the UK). If you receive the radio broadcasts on a digital and an analog radio, you will hear the delay caused by the digital radio's signal processing. $\endgroup$
    – alephzero
    Commented Sep 2, 2019 at 14:02
  • $\begingroup$ @alephzero: I was using 5 = 3 + 2 or 5 = 4 + 1 as a simple example. Let's say the inputs to a black box are 2 and 3 and the black box just adds the numbers and spits out the sum as the output. Isn't it amazing the receiver connected to the output which just sees the sum 5 knows that the inputs were 2 and 3 and not 1 and 4 or some other combination. It is like we're storing the additional information without actually storing it. Sorry if I don't make sense $\endgroup$
    – Pavan
    Commented Sep 3, 2019 at 4:55
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    $\begingroup$ Let $z=3+2i$. Now solve $x+iy$ for real $x$ & $y$. The solution is unique. This is a simplistic example, but Fourier analysis (both the continuous form and the discrete form) is ultimately an expansion of this simple idea. $\endgroup$
    – PM 2Ring
    Commented Sep 6, 2019 at 19:33

2 Answers 2

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Your analogy between adding waves and adding numbers is not accurate, not a faithful representation of the same idea. There are many ways to get 5 from smaller numbers. The decomposition of a function into sine and cosine waves is unique. This is because these functions form an orthonormal basis for a function space. The proper analogy would be getting the components of a vector in a set of coordinates. Once the coordinates are chosen a given vector has a unique vertical component for example. If you make a new vector by adding orthogonal pieces, say a little in the x-direction and a little in the z-direction, then when you look for the x-component of your vector sum you will get back exactly what you put in! In the FFT the sin and cos terms at different frequencies are "orthogonal" in the abstract sense. You cannot make a 50Hz pure sine by adding together other frequencies no matter how hard you try. So, adding a 50Hz and 60Hz sine together is like adding some x and some z together. Another analogy is the Taylor series in calculus, this uses polynomials as a basis and they are linearly independent. No amount of a parabola will give you a line. Same deal.

The FFT is similar to projecting your complex wave onto the infinite number of bases in the function space (a continuously infinite dimensional space), each f is unique and orthogonal to each other. Now remember that to get the separation you need to sum over all points in space (or time in this case). So you need to capture a very large time window of data to perform the FFT accurately. You cannot just take a single data point and say wow, there is some 50Hz and some 60Hz in this. That will never work. If you had some book or web resource on FFT go back and reread it, there should be some equations that tell you that each mode is independent of the other modes (mode is equivalent to basis in my jargon). If your resource does not state this start looking deeper into it.

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  • $\begingroup$ Thanks @ggcg, really appreciate you taking time to explain it. $\endgroup$
    – Pavan
    Commented Oct 8, 2020 at 16:24
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You aren't just separating out one sum. You have thousands of sums, and you find patterns among them.

Each different frequency can carry a little bit of other information with it, information that correlates with the carrier. If the sine wave is amplified for a thousand cycles, but the background noise in the midpoints is not correlated with the peaks, then that's a signal.

I should note this is not the only way to do it. If you have light that's mixed wavelengths, you can put it through a prism to separate them. Different frequencies get refracted different amounts, so they come out separate.

Maybe if you studied refraction carefully you might find that's really an application of the same principle, I don't know. I haven't studied it that carefully.

Similarly, if you can design an antenna that mostly picks up 50 hertz and not 60 hertz, you've separated them. You'll have set it up so that the 50 hertz carriers from different sources add, while the 60 hertz carriers tend to cancel.

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  • $\begingroup$ Thanks for the reply. Refraction makes some sense to me. I wonder, in the medium (wire, air etc.), when all the waves combine, we just see a combined wave, but it is amazing that individual waves can still be separated out from this combination. $\endgroup$
    – Pavan
    Commented Sep 3, 2019 at 4:49
  • $\begingroup$ We aren't perfectly separating them out. We are getting averages over many cycles. $\endgroup$
    – J Thomas
    Commented Sep 4, 2019 at 4:56

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