Code acquisition in CDMA systems is conventionally conducted with a matched-filter based structure. However, the performance of this method degrades greatly while multiple access interference presents. Recently, an adaptive filtering scheme was proposed to solve this problem. It has been shown that the computational complexity of this approach is proportional to the delay uncertainty and inversely proportional to the required acquisition time. When propagation delay is large and the required acquisition time is short, the computational complexity of the adaptive filtering approach will become high. In this paper, we propose a multirate adaptive code acquisition approach to alleviate this problem. The proposed scheme is comprised of several acquisition units operating in different processing rates. Thanks to the decimation property in multirate processing, the overall computational complexity can be greatly reduced. Theoretical analysis of adaptive filters and mean acquisition time is also provided. Experimental results show that while the proposed scheme can have comparable performance with respect to the original adaptive acquisition scheme, its computational complexity is much lower.