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The Flexible Wireless System (FWS) has been proposed as a networked system for a User-Centric Wireless Networks (UCWN). UCWNs allow users to make network connections easily at all times without being conscious of any upgrades or differences in wireless systems. The FWS is a unified wireless platform that simultaneously deals with various types of wireless signals. It consists of flexible access points and a wireless signal processing platform. Various types of wireless signals are received at a distributed flexible access point and transferred to a server in the wireless signal processing platform through the wired access line. Transferred signals are separated and demodulated at the server. To achieve highly flexible and efficient radio wave data transfer between the access point and the server, this paper proposes a sparse-matrix-based compressed sensing method under the framework of the belief propagation and cavity method. Low cost implementation using interlevers and adders is also proposed. An empirical study with real data shows the proposed method achieves greater efficiency and reduced calculation cost compared to the conventional compressed sensing method.