Forward error correction (FEC) is an effective means of reliable communications in wireless networks. Among all error-correcting codes, the recently developed fountain codes are known for their low complexity and rateless features. In the literature, fountain codes are mostly adopted in point-to-point communications. In this paper, we will investigate decomposed fountain codes for distributed dual-hop systems. In this type of codes, two layers of random XOR encoding are performed, but only a single layer of decoding is needed. By implementing each layer of encoding at one hop, the dual-hop systems can ensure end-to-end communication reliability with significantly reduced computation cost. Since Luby Transform (LT) codes are the first class of practical fountain codes and the core of more recent fountain codes, we will focus our study on decomposed LT (DLT) codes. To construct the DLT codes, we first analyze general LT code decomposition, and then propose a unique decomposition algorithm tailored for the LT code with robust Soliton distribution (RSD). The performance of the resultant DLT code will be evaluated in terms of the decoding probability and computation cost.