Combining compressed measurements from the same source

Question

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?

1. What are the noise implications if I reconstruct each of the $k$ measurement pairs $(y,\Psi)$ and take the average of the reconstructed signals?

2. 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?

Answer 1

If you have $k$ individual measured vectors $\mathbf y$, each taken with an individual measurement matrix $\mathbf \Psi$, you actually have an interesting, third option: $$\mathbf y = \begin{bmatrix} \mathbf y_1 \newline \vdots \newline \mathbf y_k \newline \vdots \newline \mathbf{y}_K \end{bmatrix} = \begin{bmatrix} \mathbf \Psi_1 \newline \vdots \newline \mathbf \Psi_k \newline \vdots \newline \mathbf \Psi_K \end{bmatrix} \mathbf x$$ So now you effectively have a (much) taller measurement matrix, so you are effectively "under-sampling less" and that will improve your estimate $\mathbf x$ based on the above equation. How much, I cannot recall an expression for off the top of my head.

Note also that you may have a tall measurement matrix now due to the stacking of the $\mathbf \Psi_k$'s. That is, you do not necessarily have an under-determined system anymore. This gives you the option of for example using least-squares estimation instead of $\ell_1$-norm optimisation. If you know that $\mathbf x$ is sparse, then it still makes sense to use sparse estimation ($\ell_1$-norm optimisation etc.), but if $\mathbf x$ is only approximately sparse, compressible then I would try least squares estimation as well.