n the problem of time delay estimation (TDE), the desired source signals of interest are correlated and with a specific spectral distribution. In such cases, the convergence speed using the conventional approaches, viz., time domain adaptive constrained and unconstrained LMS TDE algorithms, becomes slowly and the performance of TDE will be degraded, dramatically. In fact, the convergence rate depends highly on the distribution of spectral density of the desired signal sources. Also, the performance of TDE is affected by the background noises, accordingly. To circumvent the problem described above, in this thesis, a transformed domain adaptive constrained filtering scheme, refers to the constrained adaptive DCT-LMS algorithm, for TDE is devised. We show that this new proposed constrained algorithm, with the so-called direct delay estimation formula, for non-integer TDE does perform better than the conventional time domain adaptive constrained and unconstrained LMS TDE algorithms and the unconstrained adaptive DCT-LMS TDE algorithm. Finally, to further reduce the spread of eigenvalue in the unconstrained adaptive DCT-LMS algorithm, the Gram-Schmidt orthogonalizer approach realizing by the adaptive Escalator is investigated. It indicates that bias of TDE will occur without using the constraint of weight vector. That is, it could not be used to alleviate the effect due to background noises.