The problem of analytically calculating the value of a pure discrete time delay from input-output measurements is considered for the case of square-summable input. This is a class of input signals that are often encountered in applications but do not possess persistent excitation property as they vanish at infinity. By expressing the input and output signals in terms of Laguerre series, i.e. being considered in Laguerre domain, it can be turned into a parameter estimation problem that has a closed-form solution. Several delay value estimates, both for a general and pre-defined input sequence, are proposed and evaluated in simulation.