This paper presents a novel audio indoor localization system. In the proposed system, four speakers placed at known positions transmit chirp signals according to the time-division multiple access (TDMA) plus frequency-division multiple access (FDMA) transmission scheme. A smartphone receives the signal via a built-in microphone and calculates the time differences of arrival (TDOAs). Using TDOA measurements, the position is estimated by the shrinking-circle method. In particular, to reduce the positioning error in moving conditions, a TDOA correction method based on Doppler shifts is proposed. The performance of the proposed system was evaluated in real-world experiments using a 10.971 m × 5.684 m positioning area. The results of the static-target positioning experiment showed that the TDMA+FDMA transmission scheme has more advantages in improving the update rate of the positioning system than the TDMA-only transmission scheme. The results of the moving-target positioning experiment under three different speeds demonstrated that the positioning errors were reduced by about 10 cm when the Doppler-shift-based TDOA correction method was adopted. This research provides a possible framework for the realization of a TDOA-chirp-based acoustic indoor positioning system with high positioning accuracy and update rate.