# Compressed Sensing: Missing Fourier Coefficients?

This question is regarding the problem of reconstructing a signal given only a subset of the Fourier coefficients are observed:

$$\min_x \|x\|_1 \text{ subject to } y = Ax$$

where $x = (x_1,x_2,\dots,x_t)$ is a time-domain representation of our signal of interest, $y$ are $m << t$ observed or known Fourier coefficients of $x$, and $A$ is the measurement matrix.

My question. In practice, how are certain Fourier coefficients randomly sampled? For example, how does one collect these $m$ coefficients from the spectrum of $x$ in a random uniform (unbiased) fashion?

Some Additional Elaboration. I imagine that, in an MRI scan for example (which is one of the popular applications of compressed sensing thanks to Candés), shortening the amount of time used to collect Fourier coefficients means that we can only accurately capture the higher frequency components of $x$. Contrarily, shortening the number of samples taken from the underlying signal (but keeping the time fixed) results in a lower-resolution signal that may only capture the low-frequency components of $x$. Both of these methods mean that the observed Fourier coefficients are not randomly distributed on the spectrum of $x$, but are in fact biased to one end or another. However, a key assumption of the $\ell_1$ reconstruction used in compressed sensing is that (in the domain for which the $x$ is sparse, in this case some over-complete dictionary), the sampling operator must procure elements in a random fashion and the subspace generating $x$ must be incoherent (uncorrelated with canonical axes).