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In this paper, we study the target tracking problem in wireless sensor networks (WSNs) using quantized sensor measurements where the total number of bits that can be transmitted from sensors to the fusion center is limited. At each time step of tracking, a total of R available bits need to be distributed among the N sensors in the WSN for the next time step. The optimal solution for the bit allocation problem can be obtained by using a combinatorial search which may become computationally prohibitive for large N and R. Therefore, we develop two new suboptimal bit allocation algorithms which are based on convex optimization and approximate dynamic programming (A-DP). We compare the mean squared error (MSE) and computational complexity performances of convex optimization and A-DP with other existing suboptimal bit allocation schemes based on generalized Breiman, Friedman, Olshen, and Stone (GBFOS) algorithm and greedy search. Simulation results show that, A-DP, convex optimization and GBFOS yield similar MSE performance, which is very close to that based on the optimal exhaustive search approach and they outperform greedy search and nearest neighbor based bit allocation approaches significantly. Computationally, A-DP is more efficient than the bit allocation schemes based on convex optimization and GBFOS, especially for a large sensor network.