I know linear regression, $l_1$ regularization etc in statistics. How is the problem being studied in compressed sensing/signal processing different from that in statistics/linear regression (estimating coefficients of linear model)?
From the book A mathematical introduction to compressive sensing http://www.springer.com/us/book/9780817649470, "observed $y$ is connected to signal via $Ax=y$, where $A$ is 'measurement matrix'. My question will be, for example, is this matrix $A$ customary? If your answer can include, for example how cellphone works and how signal processing works in this process etc. that'd be cool.
$\mathbf{A}$ is an arbitrary linear operator. It can be composed of FFTs, resamplings, any linear filter, subbanding, data corruption/loss, change of basis, decimation, etc. Basically anything that you can represent with a matrix (which is just a linear system by definition). For example, suppose that your input signal has 5 samples but the second and third ones got corrupted/lost somehow, then
$$ \mathbf{y} = \mathbf{A} \mathbf{x} = \left[ \begin{array}{ccccc} 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} x_0 \\ x_1 \\ x_2 \\ x_3 \\x_4 \end{array} \right] = \left[ \begin{array}{c} x_0 \\ x_3 \\x_4 \end{array} \right] $$
Keep in mind that the product of any two linear operators is also a linear operator, so basic individual operators can be chained together to form more complex ones.