Suppose I want to measure a signal $x \in \mathbb{R}^n$ subject to i.i.d. noise $\epsilon$. In traditional Nyquist Sampling, I can increase my signal-to-noise ratio by measuring $x + \epsilon$ for $k$ times and averaging over the measurements.
$$\overline{x} = \frac{1}{K}\sum_{k=1}^{K}{(x +\epsilon_k)}$$
Instead of $x$ i now have $k$ compressed measurements $y \in \mathbb{R}^m$ and their corresponding k differing measurement matrices $\Psi \in \mathbb{R}^{m*n}$ with $m<n$, that contain e.g. gaussian random entries so that $y = \Psi*(x+\epsilon_k)$. Assuming $x$ is sufficiently sparse so that reconstruction via $l_1$-Norm minimization is possible.
Can I also improve my SNR in the second case?
What are the noise implications if I reconstruct each of the $k$ measurement pairs $(y,\Psi)$ and take the average of the reconstructed signals?
What will be the noise implications of taking the average of all measurements $\overline{y}$ and $\overline{\Psi}$ and recover the signal from the single pair?