1. A method performed by a compressed sensing system comprising a sparse signal transmission device and a sparse signal recovery device, for recovering a sparse signal of a finite field from a measurement signal, the method comprising: measuring, by the sparse signal transmission device, a target signal by applying a measurement matrix of the finite field to the target signal; generating, by the sparse signal transmission device, the measurement signal of the finite field and transmitting the generated measurement signal via a wireless channel; receiving, by the sparse signal recovery device, the measurement signal from the sparse signal transmission device, the received measurement signal comprising a noise signal added via the wireless channel; updating, by the sparse signal recovery device, discrete probability information of a target signal element of the transmitted measurement signal and discrete probability information of a measurement signal element of the finite field by exchanging the discrete probability information of the target signal element with the discrete probability information of the measurement signal element a predetermined number of times, wherein the target signal element and the measurement signal element are related to each other; calculating, by the sparse signal recovery device, a final posteriori probability of the target signal element based on a priori probability of the target signal element and the updated discrete probability information of the measurement signal element; and outputting, by the sparse signal recovery device, a recovered target signal by performing maximum posteriori estimation to maximize the calculated final posteriori probability.