Power Spectral Density From Variance

For an exercise, we made a noise signal by generating a normally-distributed random vector in MATLAB. The mean is defined to be equal to zero and the variance is set to 0.5.

I know that the power spectral density is just the fourier transform of the variance in frequency domain. In this case, does the PSD just equals to 0.5 since the variance is constant with time?

The power spectral density is NOT the Fourier transform of the variance. For wide-sense-stationary (WSS) signals, the power spectral density is the Fourier transform of the Autocorrelation function, and the variance is the autocorrelation at time offset $$\tau = 0$$ (when the WSS process is zero-mean).
Update: Dilip stated this all much more elegantly and concisely in the comments, paraphrased here: The area under the power spectral density function of a WSS process equals the value of the autocorrelation function at offset $$\tau=0$$. For a zero-mean WSS process, this value is the variance. Therefore deducing the power spectral density from knowledge of the variance is equivalent to asking for a curve given that we only know the area, which we know is not possible.
• +1 You might want to add that the area under the power spectral density function of a WSS process equals the value of the autocorrelation function at offset $\tau=0$ and for a zero-mean WSS process, this value is indeed the variance whose value the OP knows. That is, deducing the power spectral density from knowledge of the variance is equivalent to asking for a curve given only that the area under the curve equals some number; what is wanted is just not possible. Nov 4 '18 at 16:35