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I am working with PWM signals. These signals are generated by comparing a modulating (at frequency $f_m$), and a carrier (at frequency $f_c$), as shown in the following image:

Time-domain plots for PWM

In the resulting spectrum, there are baseband harmonics (at frequencies $n \cdot f_m$; $n \in \mathbb{N}$), carrier harmonics (at frequencies $m \cdot f_c$; $m \in \mathbb{N}$), and sideband harmonics (at frequencies $m \cdot f_c \pm n \cdot f_m$). In the following image, $m_f$ is the frequency modulation index, defined as $m_f = f_c/f_m$.

Frequency-domain plot for PWM

The thing is, when the frequency modulation index gets very low (for example, $m_f = 2$), some sideband harmonics overlap with each other, and some end up at frequencies below zero.

My question is, what happens to those harmonics going below zero? It seems obvious that they "bounce" back to positive frequencies, but, how does it affect the phase?.

I am calculating the harmonics in their sine-cosine form using the double Fourier integral. My first guess was that the cosines remain the same and the sines change sign, as the following statement holds for negative frequencies:

$ A\cos\left(2\pi f t\right)\ +\ B \sin\left(2 \pi f t\right) = A \cos\left(2 \pi |f| x\right)\ -\ B \sin\left(2 \pi |f| t\right)$

But this does not seem to give me exact results. Here is an image of the first three harmonics for $m_f = 2$, calculated with FFT and by summing-up sidebands: enter image description here

In one case I get an amplitude of $0.979$ for the first harmonic and in the other case I get $0.962$. I know that these are not rounding errors for two reasons:

  1. When I calculate the same thing with a high frequency modulation index (when there are no sidebands going at negative frequencies), the results are very accurate.
  2. The error obtained is of the same order as the harmonics at those negative frequencies. Witch makes me conclude that I am not adding them correctly.

In this page https://cmtext.indiana.edu/synthesis/chapter4_fm3.php, it says that they bounce and dephase 180 degrees, but it doesn´t seem to give me exact results either.

This question is more or less similar of Calculating amplitudes of FM Sidebands (unanswered).

How do I add the contribution of the sideband harmonics when $m\cdot f_c - n f_m < 0 $?.

Olayo
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    More than you probably want to know about the spectra of PWM signals can be found in the paper Z. Song and D.V.Sarwate "The Frequency Spectrum of Pulse Width Modulated Signals," Signal Processing, vol. 83, pp. 2227-2258, October 2003. It is not an easy read but the main point is that with natural sampling PWM (which is what you seem to be doing), there are no harmonics of the modulating signal in the baseband, and the passbands centered around the $k$-th harmonic of the carrier frequency are identical to the spectrum of phase modulation of the harmonic with modulation index $k\pi$. – Dilip Sarwate Dec 06 '22 at 03:05
  • Thanks for the contribution. I will check out the paper. It seems very usefull for my research. – Olayo Dec 07 '22 at 07:43

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