On Time Delay Estimation

An important signal parameter estimation problem is time-delay estimation. Here the unknown is the time origin of the signal: s (l, θ) = s (l − θ). The duration of the signal (the domain over which the signal is de ned) is assumed brief compared with the observation interval L. Although in continuous time the signal delay is a continuous-valued variable, in discrete time it is not. Consequently, the maximum likelihood estimate cannot be found by di erentiation, and we must determine the maximum likelihood estimate of signal delay by the most fundamental expression of the maximization procedure. Assuming Gaussian noise, the maximum likelihood estimate of delay is the solution of