A Multi-Constellation Software-Defined Receiver (MC-SDR) is designed and implemented to extract the Doppler measurements of Low Earth Orbit (LEO) satellite’s downlink signals, such as Orbcomm, Iridium-Next, Globalstar, Starlink, OneWeb, SpaceX, etc. The Doppler positioning methods, as one of the main localization algorithms, need a highly accurate receiver design to track the Doppler as a measurement for Extended Kalman Filter (EKF)-based positioning. In this paper, the designed receiver has been used to acquire and track the Doppler shifts of two different kinds of LEO constellations. The extracted Doppler shifts of one Iridium-Next satellite as a burst-based simplex downlink signal and two Orbcomm satellites as continuous signals are considered. Also, with having the Two-Line Element (TLE) for each satellite, the position, and orbital elements of each satellite are known. Finally, the accuracy of the designed receiver is validated using an EKF-based stationary positioning algorithm with an adaptive measurement matrix. Satellite detection and Doppler tracking results are analyzed for each satellite. The positioning results for a stationary receiver showed an accuracy of about 132 m, which means 72% accuracy advancements compared to single constellation positioning.