Radio-On-Demand Networks (ROD) has been proposed in order to improve energy-efficiency of WiFi device, which offers wireless access in an on-demand manner. Each WiFi device in ROD is equipped with a low-power wake-up receiver, which detects the length of wireless LAN frames through envelope detection and on-off-keying (OOK) processing. A STA, upon communications demands, transmits a wake-up signal consisting of WiFi frames with their length corresponding to a target wakeup ID. The receiver detects a wake-up request by finding the predefined length of WiFi frames. However, when WiFi devices transmitting non-wake-up data frames are present around the wake-up receiver, it is possible that the length of these frames match with the predefined length. This causes an adverse event of False Positive (FP), where WiFi device is turned on without actual wake-up requests. In this paper, in order to reduce the number of FPs, we propose a wake-up frame detection exploiting the correlation among the received signal strength of WiFi frames received from a single STA in a fixed position. By computer simulations using the experimentally measured values of received signal strength, we show that the proposed scheme can significantly reduce the number of FPs without degrading the wake-up success probability.