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I try to get my sound program right, and hoped to find some help here with the understanding of frequency modulation by a square wave.

My modulation looks like this: (please excuse if my formula naming and writing is not standard, glad to learn it)

$$y_m=A \cdot sin( 2 \pi f_c x + \\I \cdot \color{green}{( \frac4 \pi \frac 1 1 sin(1 \pi f_mx) + \frac 4 \pi \frac 1 3 sin(3 \pi f_mx) + \frac 4 \pi \frac 1 5 sin(5 \pi f_mx ) + \frac 4 \pi \frac 1 7 sin( 7 \pi f_mx)} ) $$

$f_c :$ carrier frequency
$f_m :$ frequency of the modulation
$A :$ amplitude
$I :$ impact of the modulation
$x :$ time

I tried to set up the formulas (on desmos) for a pulse modification and a graph, to validate what I should hear. But the graph does not look the way I expected. Is this from the acoustic perspective correct? Because I expected, that the low frequency area from the carrier signal would appear in the valley of the pulse.

enter image description here

Appendix, 09.06.2020: The formula for the frequency modulation, I got from an example, that modulates a sinus wave. And my intention was to replaced the sinus with a pulse wave. It was from an article from the American Mathematical Society webpage, looking like this:

$$y_m=A \cdot sin( 2 \pi f_c x + I \cdot \color{purple}{( sin( 2 \pi f_m x)})$$

The original pulse function I got from a desmos example and it was looking like this:

$$y_{s} =\color{green}{\frac4 \pi \frac 1 1 sin(1 \pi x) + \frac 4 \pi \frac 1 3 sin(3 \pi x) + \frac 4 \pi \frac 1 5 sin(5 \pi x ) + \frac 4 \pi \frac 1 7 sin( 7 \pi x)}$$

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    $\begingroup$ That's not what frequency modulation looks like. (You're modulating the phase, not the frequency; you'd need a factor of $x$ on the series for the former.) Where did you get that formula from? $\endgroup$ – Emilio Pisanty Jun 9 at 9:34
  • $\begingroup$ @EmilioPisanty Thank you for your feedback. I added the original formulas to my post. Thank you for any further hints. $\endgroup$ – spikey Jun 9 at 14:55
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    $\begingroup$ Welcome to this site. The preferred way here to present formulas is MathJax instead of images. $\endgroup$ – Thomas Fritsch Jun 9 at 19:03
  • $\begingroup$ @ThomasFritsch Its the first time I use MathJax, looks nice. Glad I looked into it. $\endgroup$ – spikey Jun 9 at 20:59
  • $\begingroup$ @EmilioPisanty I do not understand exactly what you mean with "I need a factor x in the series". You mean $x$ should not take place inside the $\sin()$ functions? $\endgroup$ – spikey Jun 10 at 19:44

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