# Generating a phase modulation frequency spectrum with jitter

I would like to produce a frequency spectrum plot like this:

I know that one can take the modulation frequency and multiply it by a given order and scale by the modulation depth to get a similar graph with infinitely sharp peaks, but I am interested in the fact that this spectrum shows some spreading of the frequency around each modulation order. Is there an equation that produces something like this plot, with a variable that influences the frequency spread (I guess caused by phase noise or jitter)?

I'm not interested in the time domain signal, just a means of plotting a signal in the frequency domain with some assumption about frequency spread and modulation depth.

• What is the duration of your signal in the time domain that was used for the plot above? I would like to understand how much of that spread is due to the window of the time domain data itself (rectangular window or otherwise). – Dan Boschen May 11 '19 at 0:55
• @DanBoschen, I am not sure. The picture there is just an example and there seems to be no data (see the link I posted). – Sean May 11 '19 at 15:05
• For clarity, I would just like to produce something similar to the plot above, but without the asymmetries. For phase modulation the modulation depth will give the different amplitudes for each order, but the spread must be governed by some other property which I guess is phase noise. I hope there is an equation that can predict the frequency spectrum of such a phase modulation - and that's basically what I'm looking for. – Sean May 11 '19 at 15:07
• You can do the spreading by windowing the data and the formula is the kernel (DTFT) of the window—- as that would convolve in frequency with the impulses representing your tones. For example a rectangular window is a Sinc function with first nulls at 1/T where T is the time duration. Shorten your time duration and you widen the spreading. The rectangular window has strong side-lobes so you can choose other windows (such as Kaiser) to minimize that. – Dan Boschen May 11 '19 at 15:09