  In this paper, a compressive sensing (CS) approach is proposed for sparse
binary signals' compression and reconstruction based on analog fountain codes
(AFCs). In the proposed scheme, referred to as the analog fountain compressive
sensing (AFCS), each measurement is generated from a linear combination of L
randomly selected binary signal elements with real weight coefficients. The
weight coefficients are chosen from a finite weight set and L, called
measurement degree, is obtained based on a predefined degree distribution
function. We propose a simple verification based reconstruction algorithm for
the AFCS in the noiseless case. The proposed verification based decoder is
analyzed through SUM-OR tree analytical approach and an optimization problem is
formulated to find the optimum measurement degree to minimize the number of
measurements required for the reconstruction of binary sparse signals. We show
that in the AFCS, the number of required measurements is of O(-n log(1-k/n)),
where n is the signal length and k is the signal sparsity level. We then
consider the signal reconstruction of AFCS in the presence of additive white
Gaussian noise (AWGN) and the standard message passing decoder is then used for
the signal recovery. Simulation results show that the AFCS can perfectly
recover all non-zero elements of the sparse binary signal with a significantly
reduced number of measurements, compared to the conventional binary CS and
L1-minimization approaches in a wide range of signal to noise ratios (SNRs).
Finally, we show a practical application of the AFCS for the sparse event
detection in wireless sensor networks (WSNs), where the sensors' readings can
be treated as measurements from the CS point of view.
