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I'm trying to understand the difference between the sums and products of two sine signals. I also wonder where and how these two situations are used in practice.

Fs = 100e6;
t = 0:1/Fs:0.1;

x = sin(2*pi*10*t);
y = sin(2*pi*1e3*t);

mult_sig = x.*y;
add_sig = x+y;

The sum and fft of the two signals are as follows: enter image description hereenter image description here

The product and fft of the two signals are as follows:

enter image description hereenter image description here

Could you give the mathematical background and examples of its practical use? Isn't it strange that different frequencies appear especially in fft graphs?

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Could you give the mathematical background [...] Isn't it strange that different frequencies appear especially in fft graphs??

  • For addition, it's trivial: $$\mathcal{F}\big\{\sin(\omega_1t) + \sin(\omega_2t)\big\} = \mathcal{F}\big\{\sin(\omega_1t)\big\} + \mathcal{F}\big\{\sin(\omega_2t)\big\}\tag{1}$$ where $\mathcal{F}$ is the Fourier Transform operator.

    With $\omega_1 = 2\pi\cdot10$ and $\omega_2 = 2\pi\cdot1000$, you correctly see a spike at $10\tt{Hz}$ and a spike at $1000\tt{Hz}$ in your first FFT plot.

  • For multiplication, recall from your algebra classes that: $$\sin(\omega_1t)\times\sin(\omega_2t) = \frac{1}{2}\bigg(\cos(\omega_1-\omega_2)t - \cos(\omega_1+\omega_2)t\bigg)$$ So multiplication becomes an addition, you can use $(1)$, and verify that you do get the expected result in your second FFT plot.

examples of its practical use?

There are too many to list here, but I'll name a few:

Addition: Sound synthesis, Signal Mixing, Vibration analysis, etc.
Multiplication: Modulation (AM and FM), Frequency Translation, Phase Locked Loops, etc.

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  • $\begingroup$ Thanks for the descriptive answer. When we multiply, we now change the frequency of the signal. When we add it up, those two frequencies are still preserved. Is the thing in multiplication that a certain carrier frequency is already known, as in modulations? $\endgroup$
    – bb0667
    Commented Jul 6 at 10:06
  • $\begingroup$ That’s outside the scope of this question, but yes, kind of. If you’re interested in modulation in particular, I’d advise you to either look for questions already asked here, or ask a new one! $\endgroup$
    – Jdip
    Commented Jul 6 at 10:21

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