I want to sample a signal whose derivative I know to be bounded by physical constraints. The sampling is disturbed by gaussian noise, hence I need to filter the sample with a lowpass filter.


Since I know precisely the bound on the derivative magnitude, I was wondering if there is a way to translate this bound in a frequency bound, in order to determine the cutoff-frequency of the filter.


My intuitive idea is that the low frequencies make up the "smooth" part of the sample (i.e. the part with a derivative that is compatible with the bound, the signal), while the higher frequencies are responsible for the sudden changes in the sample (i.e. the part with a slope that exceeds the bound, the noise); so, I think there should be a relation between the derivative and the frequency components of the sample. I'm looking for something that formalizes this concept. Thanks!



Relation between signal derivative and frequency spectrum
A lot depends on how you want to formalize your question. Here is one possible approach. Let's say that the signal can be any function on $\mathbb Z$ with the derivative bounded by $1$ and the noise has the standard deviation $\sigma$ and its values at different samples are independent. You apply a linear filter and you want to minimize the expectation of the $L^2$ norm of the error over a long period of time *in the worst case scenario*. What should this filter be?


Taking the Fourier transform of everything, as usual, we see that the question reduces to finding a function $\varphi$ on the circle with unit measure that minimizes $\sigma^2\int|\varphi|^2+\sup\_{a}\frac 1N\int{|1-\varphi|^2\left|\sum\_{k=1}^N a\_k z^k\right|^2}$ where $a\_k$ is an arbitrary sequence of real numbers with $|a\_{k+1}-a\_k|\le 1$ and $a\_0=a\_{N+1}=0$ ($N$ is the duration of the signal). Note that any such sum is just $\frac{1}{1-z}\sum\_{k=0}^N b\_kz^k$ where $|b\_k|\le 1$ and $\sum\_k b\_k=0$.
This brings us to the problem of finding the supremum of $\frac 1N\int\psi^2\left|\sum\_{k=1}^N b\_k z^k\right|^2$ for a given $\psi=\frac{1-\varphi}{|1-z|}$.


This supremum is, of course, not greater than $\sup\psi^2$, but it is also not much less than that since if we allow complex coefficients instead of real ones, we can approximate the delta-measure at any point we want. Thus, if we do not care too much about factors like $2$, we can restate our problem as follows:


Minimize $\sigma^2\int(1-M|1-z|)\_+^2+M^2$. If we pass to the continuous case of the line (which makes a decent approximation if you sample frequently enough, so in this normalization $\sigma\gg 1$) and assume that our Fourier transform is given by $\widehat f(\omega)=\int f(t)e^{-2\pi i \omega t}$ (so that the $L^2$ norm is preserved, which corresponds to $z=e^{2\pi i \omega}$), we see that we are to minimize $\sigma^2\int(1-2\pi M|\omega|)\_+^2+M^2=\frac{\sigma^2}{3\pi M}+M^2$, which results in the minimum at $M=\sqrt[3]{\frac{\sigma^2}{6\pi}}$. Thus, from this point of view, the optimal filter should pass $e^{2\pi i\omega t}$ for $|\omega|\le \omega\_0=\sqrt[3]{\frac{3}{4\pi^2\sigma^2}}$ with linear decline in amplification from frequency $0$ (amplification $1$) to frequences $\pm\omega\_0$ (amplification $0$).


Now about scaling. Suppose that you sample at time intervals $\tau$, your time derivative is bounded by $D$ and the noise standard deviation at each sample is $\Sigma$. Then $\sigma=\frac{\Sigma}{D\tau}$ and the final answer should become $\Omega\_0=\omega\_0/\tau=\sqrt[3]{\frac{3 D^2}{4\pi^2 \Sigma^2\tau}}$.


Note once more that it is *the worst case scenario* optimization under *the only restriction concerning the derivative* with the objective *to minimize the average square error*. If you have more restrictions on your signal (say, some amplitude bound in addition to the derivative bound), or wish to optimize for a "typical signal" (which has then to be defined) and do not care much about outliers, or prefer a different objective, the answer may change. Also I believe that my logic is correct but I'm notoriously bad in algebra after midnight, so check the numbers involved before applying the final answer.


