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In this paper, a design and simulation for Asymmetric Digital Subscriber Line (ADSL) modem is presented which can be applied to different telephone networks. The ADSL modem is modeled and simulated under the MATLAB version 7.0 environment by which the simulation is achieved for both the downstream and upstream directions of the modem. The ADSL modem is divided into the modulation system that uses a Multicarrier Modulation (MCM) technique called Discrete Multitone modulation (DMT) and an error control system that consists of error detection and correction codes to achieve a desirable low Bit Error Rate (BER). The designed system can be applied to a channel with a relatively low Signal to Noise Ratio (SNR). As a result, asymmetric transmission rates were achieved, that is 6.16 Mbps is achieved with BER = 3.61x10-6 at SNR=33.5 dB in the downstream direction, and 687.5 kbps is achieved with BER= 7.673 x10-6 at SNR=23.5 dB in the upstream direction.