Compare FM synthesis sound signal to equivalent additive synthesis signal

I currently try to familiarize myself with the theories behind FM synthesis. I found this page with a lot of useful information about FM synthesis: https://ccrma.stanford.edu/software/snd/snd/fm.html

One can find an example on the linked page where the author compares FM to additive synthesis. He states that the FM synthesis sounds louder and richer. He also provided an image of both signals. I'm interested in recreating this example on my own. I would like to use an example like that for a term paper in my university.

The example is in the upper third of the page. Right after the 3D Bessel plot and the three frequency spectrums but before the sum of multiple sinus waves. The author wrote:

One hidden aspect of the FM expansion is that it produces a time domain waveform that is not "spikey". If we add cosines at the amplitudes given by the Bessel functions (using additive synthesis to produce the same magnitude spectrum as FM produces), we get a very different waveform. Doesn't the FM version sound richer and, far more importantly, louder? FM waveform (index: 3.0) vs sum of cosines with the same (relative) component amplitudes

I'm aware that the additive synthesis is basically a Fourier Series which sums the different sideband frequencies up. I'm also aware how to create a frequency spectrum for the FM synthesis, following Chownings paper or Wikipedias formula.

So how can I reproduce the shown signals from the page? The FM synthesis seems to have a quite high carrier frequency, a relative low modulation frequency and a modulation index of 3. But after I calculate the amplitudes of the sideband frequencies and but them together to a additive synthesis, I basically recreate the original FM synthesis. Which makes sense because I just used the sum form of the FM synthesis formula.

That's why I'm confused how the author of the webpage could create the aforementioned example?

Thank you for any help!

• I suspect that the difference is due to the phases (+/- signs) of the component signals in the additive synthesis -- maybe try summing |a_n|*cos(w_n t), i.e. take the absolute value of the amplitudes. – Dave Jun 9 '15 at 20:06
• You're right. That would be actually the answer – MBulli Jun 25 '15 at 8:42