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I have a discrete stochastic function $f(t_k)$ (in my case, it's a time signal related to atmospheric parameters). I know this function to have a certain smoothness. One way I can specify its smoothness is using its auto-correlation function, or equivalently, its power spectral density which can be estimated by $S_f(omega_k) = mathcalF(f(t_k)) overlinemathcalF(f(t_k))$, where here $mathcalF$ is the discrete Fourier transform over some arbitrary window of data and $bar$ means complex conjugate. Let's assume that the SDF is the a-priori information I have available for defining the signal smoothness.

I want to be able to define sequential or batch estimation algorithms to estimate this signal given noisy measurements. For simplicity, let's say:

$$
y(t_k) = f(t_k) + w(t_k)
$$

is our measurement, where $E[w(t_k)w(t_l)] = sigma^2delta(k-l)$, where $sigma^2$ is the noise variance and $delta$ is the Dirac delta function.

If I want to predict/estimate $hatf(t_k)$ given past estimates $hatf(t_k-1), hatf(t_k-2), dots $, one way to do this is using an auto-regressive (AR) model:

$$
hatf(t_k) = c + sum_i=1^P a_i hatf(t_k-i)
$$

where $c$ and $a_i$ are model coefficients. (There is a direct relationship between an AR model and its corresponding power spectral density, correct?)

Another way is by estimating and using derivatives of $f(t_k-1)$:

$$
hatf(t_k) = hatf(t_k-1) + Delta t hatdotf(t_k-1) + fracDelta t2 hatddotf(t_k-1) + cdots
$$

where $Delta t = t_k - t_k-1$, $dotf$ denotes the first time derivative (i.e. rate) of $f$, and $ddotf$ denotes the second time derivative (i.e. acceleration) of $f$, etc. (I don't at all understand how this method of predicting $f$ relates to its power spectral density.)

Intuitively, I feel there must be a relationship between these two ways of predicting/estimating a smooth function, but I can't quite pin it down, nor find any references that do. So...

What is the relationship between using an auto-regressive model and using function derivatives to predict a smooth signal $f(t_k)$, particularly in regard to knowing a-priori its power spectral density $S_f(omega_k)$.

Thank you for any help/guidance/direction. If this question belongs in another community, please let me know.

From the point of view of radiolocation, the general task can be divided into two parts: the consideration of jamming trends and the allocation of signal information. As far as I understand the essence of the issue, in meteorology the same information can be perceived as an interfering trend (with a short-term forecast), and as signaling information (with a long-term one).

Time trends (including polynomial one) can be used to equalize the signal level at the edges of the samples, which eliminates the use of weight windows that reduce the resolution of the DFT.

The same task can be performed by methods based on an autoregressive smoothing model, for which fast algorithms also exist.

Trends computed in the spectral region are used to normalize the density of the power spectrum. The alignment of the interference spectrum brings the consistent signal processing to the optimum.

The signal spectrum is chaotic and poorly predictable. Non-parametric processing methods are better suited for it — for example, median filtering in a sliding window.

Waiting for detalization in comments.