This paper introduces identification algorithms for finite impulse response systems under quantized output observations and general quantized inputs. While asymptotically efficient algorithms for quantized identification under periodic inputs are available, their counterpart under general inputs has encountered technical difficulties and evaded satisfactory resolutions. Under quantized inputs, this paper resolves this issue with constructive solutions. A two-step algorithm is developed, which demonstrates desired convergence properties including strong convergence, mean-square convergence, convergence rates, asymptotic normality, and asymptotical efficiency in terms of the Cramer-Rao lower bound. Some essential conditions on input excitation are derived that ensure identifiability and convergence. It is shown that by a suitable selection of the algorithm's weighting matrix, the estimates become asymptotically efficient. The strong and mean-square convergence rates are obtained. Optimal input design is given. Also the joint identification of noise distribution functions and system parameters is investigated. Numerical examples are included to illustrate the main results of this paper. (C) 2015 Elsevier Ltd. All rights reserved.