I am attempting to add phase noise to a modulated signal.
- I am doing this by taking a phase noise mask (single sided 0 Hz to 1 MHz), which shows dBc values away from central value or reference value. Phase noise is decreasing as we move in frequency away from the reference point which is 0 Hz. [Top left image in the 4-plot image below]
- I then make this double sided.
- I convolve this phase noise plot with a spectrum of a clean complex exponential at 1 MHz, this is my oscillator for converting frequency. I think this has imparted the phase noise onto the complex exponential. [Top right image in the 4-plot image below]
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The spectrums are convolved to make a oscillator with phase noise:
$\mathcal{F}(e^{j \omega t}) * \mathcal{F}( e^{j \phi(t) }) = \mathcal{F}(e^{j \omega t})* \mathcal{F}(e^{j B}) = \mathcal{F}(e^{j( \omega t + \phi(t))}) $
where $\mathcal{F}$ is fourier transform, $ \omega$ = 1 MHz, $\mathcal{F}( e^{j \phi(t) }) $ is the double sided phase noise mask on 0 Hz, then also this can be interpreted as a bandwidth $B$ of the upper and lower sidebands of the phase noise centered on 0 Hz (the double sided spectrum in the top left plot below when not in logarithmic x-axis scale). The convolution result is the top right image, symmetric phase noise translated from 0 Hz onto my a oscillator or CW or sine wave at 1 MHz.
- I take the Inverse Fourier Transform of this convolution to produce an oscillator in the time domain, my thought is that the phase noise is on it.
- I then multiply my complex modulated signal (16-APSK) by this oscillator (complex exponential with phase noise) to produce an up conversion and then downconvert with a clean complex exponential with no phase noise. This should leave the phase noise on the modulated signal.
- The math now being done in time domain:
Up-conversion using the inverse transform of the result above
$A(m) e^{ j( \phi (m))} e^{j( \omega t + \phi(t))} $
Down-conversion of the upconversion
$ A(m) e^{ j( \phi (m))} e^{j( \omega t + \phi(t))} e^{-i \omega t} $
where $A(m) e^{ j \phi (m)} $is a modulated signal
- However I am showing results that are not like classical phase noise, which would be rotation acrcs around constellation points but it appears as an amplitude modulation? [Final constellation image]
- I know this is wrong, but I stuck why. If I subtract in the time domain the clean sinusoid without phase noise and the oscillator with phase noise, I get this variation across the full time domain simulation. [Central images are the real and imag values of the oscillators in time and the bottom images are the difference]
Why is this producing amplitude variation and not phase variation? Is there another way to do this?
--Edit Updated constellation based on replies below--