I am a beginner in probability, estimation and statistics. I can understand the physics behind the problem really well, but when it comes to estimation using some statistical algorithms I am very confused.
My problem:
I want to estimate the velocity distribution of an ensemble of particles. The signal I receive is a combination of signals received from all the particles. Let's consider there are $N$ particles and it is not directly the velocity that is there in the signal. It looks like the following.
$$ x_{t_i} = \sum_{n = 1}^{N} A_n e^{j (2\pi/\lambda) u_n t_i} + \mathcal{N}(0, \sigma_a^2), \quad i = 1, 2, ..., N_t $$
Here, $x$ is the model of the received signal, $N$ is the number of particles, $t_i$ are the time samples and $\lambda$ is the wavelength of the sensor. The $u_n$ are the velocities of each particle. $A_n$ is the amplitude from each particle and we can assume it to be $1$ for now. $\mathcal{N}(0, \sigma_a^2)$ is an added zero-mean Gaussian noise.
Conventionally to see the velocity spectrum of such a model, a lot of time samples can be gathered and a Fourier transform of the time samples would give us the spectrum of the velocities. The velocities in the spectrum vary from $[-v_{a}, v_{a}]$ where $v_a = \lambda/(4 . dT) $ with $N_t$ samples between $[-v_{a}, v_{a}]$.
I want to address the problem when a few samples are available. In this case, a Fourier transform wouldn't give an accurate estimate of the spectrum because of low resolution, because the number of steps in the velocity axis would be much lower than $N_t$. The figures of the velocity spectrum are shown for a $N_t = 128$ (ground truth) and measurement with $10$ samples.
The measurement model looks like this,
$$ z_{t_k} = x_{t_k}, \quad k = B . dT + \sum_{i = 1}^{C} i . dT $$
Where $dT$ is the time step. Basically, $t_k$ is uniformly distributed time samples with a gap. The time sample indices are shown in the following figure.
At each blue line, there are 10 samples.
With these measurements, I want to estimate the velocity distribution $u$. I thought of using MCMC to do this. I don't know how to proceed through. I understand that MCMC samples are from a posterior distribution. I don't know how to proceed. Should I linearize the model? When people talk about the distribution of the unknown variables, is it similar to the spectrums in the figure I attached?
A solution would be to generate a spectrum for $u$ with 128 samples from a measurement of 20 measurement samples [10 consecutive samples and a huge gap and then again 10 consecutive samples].