How to create a synthetic time series where power spectral density estimation is achieves better results than a direct Fourier transform?

I am trying to create a synthetic time series where PSD estimation is necessary and useful to recover the correct spectral information of the time series. But so far I can only create a time series where either a) Fourier transform is superior and recovers sharper peaks; or b) Neither Fourier transform or PSD recover useful spectral information.

I have been trying to show this by creating a time series in MATLAB which has a peak frequency of 40 Hz but is also a wide-sense stationary random process. I then want to use power spectral density estimation to recover the spectral content (with a peak at 40 Hz).

I have tried doing a variety of things including:

1) Create a time series and add random noise. Both the Fourier transform and the PSD recover the true spectrum and the Fourier transform recovers sharper peak. So why use PSD?

2) Create a time series and add a random phase. Fourier transform recovers sharper peak at 40 Hz. Why use PSD?

3) Create a time series with random noise and then randomly sample the time series by pulling data samples from the time series with replacement. This results in random spectra with no peaks using either Fourier transform or PSD.

4) Create a time series with random noise and then re-sample the time series at a lower sample rate. This results in both Fourier transform and PSD recovering the peak at 40 Hz but, once again, the Fourier transform has sharper peaks.

What am I doing wrong? How can I create a synthetic wide-sense stationary time series in MATLAB?

• Maybe you are expecting something that the PSD cannot do? If you know that there is a deterministic 40 Hz sine wave in the signal, the DFT will find it as well as or better than anything else. Also, consider that all numerical PSD estimates are based on the DFT and do not have access to the actual random process.
– MBaz
Feb 3, 2020 at 18:45
• Here's an example where the PSD performs better: if you take time series from Gaussian noise and find the DFT, it will be quite noisy. You know from theory that the actual spectrum is flat (within the Nyquist range). And the PSD in fact is able to produce a flatter, less noisy spectrum.
– MBaz
Feb 3, 2020 at 18:47