| In the domain of Signal Processing, Direction of Arrival (DOA) is always to be one of the most famous problems. So far, there has been plenty of theory and method serving for DOA. Among them, one method has been paid much attention for worldwide researchers who have spent more than 30 years to develop and perfect it. This is the namable "Time Delay Estimation (TDE)".In this paper, the TDE problem will be studied deeply and comprehensively. At the beginning, a summary and comparison of the main TDE methods will be made, at the same time their property will be illustrated. Then the paper will discuss the usability of these methods in real place. After this, this paper will mainly research the "Least Mean Square Error TDE Method" because of its wide applicability, high reliability, less computation and good ability for real-time. Therefore, Least mean square error TDE method will be applied widely in practice.The Least mean square error TDE method can be divided into two kinds in theory. One is the traditional method and the other is the constrained method. In the paper, one classical algorithm named LMSTDE from traditional method and two algorithms named ETDE and ETDGE from constrained method will be selected to make a deep research. The research work will be in two aspects: in the background of uncorrelated noise and background of correlated noise. Moreover, in this paper, we assume the delay to be static.From the analysis in theory, in the background of uncorrelated noise, the algorithms of LMSTDE, ETDE and ETDGE have good property in estimating delay. In the background of correlated noise, the LMSTDE is failed, but the ETDE and ETDGE can still estimate delay in sometime. Their effect of estimation is strongly affected by the real time delay. If the real time delay can be set properly, we find from the simulating results that even at the time of signal to noise ratio being -15db, the ETDE and ETDGE algorithm can estimate the time delay properly.
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