Why is PSD estimated, and not simply computed?

Question

I apologize if this comes as a basic question, but I am struggling to understand why the PSD can only be estimated and not directly computed. For example, this thread discusses several such PSD estimation methods.

If I have a deterministic signal with a fixed number of samples, shouldn't I be able to directly determine its spectral information? Also, it is well known that PSD can be defined as the Fourier transform of the auto-correlation function. Isn't this calculation deterministic?

Finally, in this paper the authors compute the PSI directly from the FFT coefficients as shown below. There doesn't seem to be anything stochastic about the computation, and the computation of FFT coefficients is not really an estimation, to the best of my understanding. What am I missing?

                   

Answer 1

Well, because in a lot of real world problems this

If I have a deterministic signal with a fixed number of samples, shouldn't I be able to directly determine its spectral information?

is just not the case. Very often, measured signals are more of a random process. A simple and common case would be to have the desired signal and some additive noise, very often Gaussian in nature.

While you can capture the signal for a while and then deterministically calculate the PSD of those samples, what you get is the instantaneous PSD of these samples which may or may not be close to the actual PSD of the whole random process that you are sampling. It all boils down to what you want to do but in a lot of cases you want the PSD of the random process that you are sampling and also very often, just capturing some samples and calculating the instantaneous PSD just does not come close to it.

There is a lot of different methods for PSD estimation. Some may work better in some cases than other, maybe some are more computationally expensive than others. Or possibly some reach a better estimate with fewer samples as input data. In the end it boils down to select the most appropriate method for your given scenario.

Answer 2

If you are performing a finite computation, then what you are missing is the assumption that the PSD of the longer or infinite signal outside of the samples you are using in the computation is or us not identical to that of your finite sample set, as well as assumptions about the possibility that there was or was not some non-zero probability distribution of sampling errors.