Can one generate a signal with an asymmetric power spectral density?
@user129412, there are many ways to generate both continuous signals and discrete sequence having asymmetric PSDs. One way, for discrete sequences, is shown in the following diagram.



$x_r(n)$ is a narrow bandpass signal having spectral energy in the vicinity of $±f_c$ Hz. The complex-valued $x_c(n)$ output sequence is your desired asymmetrical sequence having spectral energy only in the vicinity of $+f_c$ Hz. (If your Hilbert transformer has M taps, where M is odd, then the upper path delay must be (M-1)/2 samples to achieve time synchronization between the two signal paths.)  If you desire sequences whose asymmetrical spectra are centered at zero Hz, many ways to obtain such sequences can be found at:

https://www.dsprelated.com/showarticle/153.php