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This paper focuses on the problem of estimating and tracking time-varying direction-of-arrivals (DoAs) with an antenna array. A sequential DoA estimation method is proposed by extending the capon and sparse iterative covariance-based estimation (C-SPICE) method, which is an iterative off-grid method for estimating constant DoAs. Then, a moving average initialization technique is introduced such that the spatial spectrum information estimated in this snapshot can be utilized in the next one. In uniform linear arrays (ULAs), we replace the uniform grid in direction domain with that in a "frequency" domain, to improve estimation accuracy without additional complexity in practical applications. The validity and efficiency of the proposed methods are demonstrated through numerical experiments. [ABSTRACT FROM AUTHOR]