  In this paper, we propose a method of structured construction of the optimal
measurement matrix for noiseless compressed sensing (CS), which achieves the
minimum number of measurements which only needs to be as large as the sparsity
of the signal itself to be recovered to guarantee almost error-free recovery,
for sufficiently large dimension. To arrive at the results, we employ a duality
between noiseless CS and analog coding across sparse additive noisy channel
(SANC). Extending Renyi Information Dimension to Mutual Information Dimension
(MID), we show the operational meaning of MID to be the fundamental limit of
asymptotically error-free analog transmission across SANC under linear analog
encoding constraint. We prove that MID polarizes after analog polar
transformation and obeys the same recursive relationship as BEC. We further
prove that analog polar encoding can achieve the fundamental limit of
achievable dimension rate with vanishing Pe across SANC. From the duality, a
structured construction scheme is proposed for the linear measurement matrix
which achieves the minimum measurement requirement for noiseless CS.
