Relay-enabled backscatter communication (BC) is an intriguing paradigm to alleviate energy shortage and improve throughput of Internet-of-Things (IoT) devices. Most of the existing works focus on the resource allocation that considered the unequal and continuous time allocation for both source-relay and relay-destination links. However, the continuous time allocation may be infeasible since in practice, the time allocation shall be carried out in integral multiple of the subframe duration unit. In this article, we study a discrete time scheme from the perspective of frame structure, where one transmission block is divided into two phases and the linear mapping is employed as a re-encoding method to determine the number of subframes for both phases and the power allocation for each subframe in a relay-enabled BC system. Based on this, we derive an accurate system-throughput expression and formulate a mixed-integral non-convex optimization problem to maximize the system throughput by jointly optimizing the power reflection coefficient (PRC) of the IoT node, the power allocation of the hybrid access point (HAP) and the linear mapping matrix, and solve it via a three-step approach. Accordingly, we propose a low complexity iterative algorithm to obtain the throughput maximization-based resource allocation solution. Numerical results analyze the performance of our proposed algorithm, verify the superiority of our proposed scheme, and evaluate the impacts of network parameters on the system throughput.