An online approach to system identification based on the least-mean squares (LMS) algorithm is presented in this paper. This recursive method is actually an extended version of the LMS-like identification method based on binary observations (LIMBO), whose practical requirement is a simple comparator (1-bit quantizer). This method can be applied in the case of finite impulse response (FIR) systems in the presence of noise and offset at the comparator input. Moreover, contrary to classical LIMBO approach, the unknown parameters are rigorously identified, and not up to a positive multiplicative constant. The idea consists in introducing a known dithering signal at the input of the quantizer, which acts as reference amplitude and allows us to identify the gain of the system. Some simulation results are given in order to compare the performances of this extended version of LIMBO with the usual one, in terms of convergence speed and estimation quality.