This is a really nice problem.
Problem Formulation
I will formulate it as following:
Let $ x \in \mathbb{R}^{n} $ be a signal. Given $ y \in \mathbb{R}^{n} $ which is a noisy measurement of $ x $ such that $ y = x + v $ and $ z $ be a noisy measurement of the derivative of $ x $ such that $ z = F x + w $ where $ F $ is the finite differences operator. It is known that both $ v $ and $ w $ are White Noise which is independent form each other. Find a reasonable estimator of $ x $.
OK, actually, I did the hard part by formulating the problem the way I did (Formulating a connection between $ x $ and its derivative).
I find this formulation to be reasonable. It doesn't assume too much but gives us enough to work with.
So my first of choice would be something like:
$$ \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| F x - z \right\|}_{2}^{2} $$
This is really easy to solve.
The tricky part is $ \lambda $. Well, I put it there to allow playing with the ratio between the noise level of the measurements. If the $ \sigma $ of both noises is the same, set $ \lambda = 1 $. Else, set it as $ \lambda = \frac{ {\sigma}_{v}^{2} }{ {\sigma}_{w}^{2} } $. The intuition is straight forward, if the noise level of $ w $ is smaller then $ v $ we want $ \lambda $ to be bigger than $ 1 $ in order to maximize the knowledge coming from $ z $.
Remark
The model above is actually the Maximum Likelihood Estimator of the model above if you assume the noises are AWGN.
I don't want to get into the Math of deriving it, but Least Squares is almost (Always) the ML for the AWGN model.
Solution to the Model
As mentioned above:
$$ \hat{x} = \arg \min_{x} \frac{1}{2} {\left\| x - y \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| F x - z \right\|}_{2}^{2} $$
This is a strictly convex problem hence the solution is given in the stationary point.
$$\begin{align*}
\frac{\mathrm{d}}{\mathrm{d} \hat{x}} \frac{1}{2} {\left\| \hat{x} - y \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| F \hat{x} - z \right\|}_{2}^{2} & = 0 && \text{Definition of Stationary Point} \\
& = \hat{x} - y + \lambda {F}^{T} \left( F x - z \right) && \text{} \\
& \Leftrightarrow \left( I + \lambda {F}^{T} F \right) = y + \lambda {F}^{T} z && \text{} \\
& \Leftrightarrow \hat{x} = {\left( I + \lambda {F}^{T} F \right)}^{-1} \left( y + \lambda {F}^{T} z \right)
\end{align*}$$
MATLAB Simulation
The model I chose is using an harmonic signal - sine.
We use Finite Differences as the Derivative Model.
Here are the signals and the estimation using $ \lambda = \frac{ {\sigma}_{v}^{2} }{ {\sigma}_{w}^{2} } $:
It can be seen that the estimation is much better then any of the given signals.
Let's verify the optimality of $ \lambda $:
It is hard to see 2 circles at the bottom but they are close to verify the optimality of $ \lambda $.
The code is available on my StackExchange Signal Processing Q52150 GitHub Repository (Look at the SignalProcessing\Q52150 folder).
