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Ultra-wideband (UWB) systems having a bandwidth on the order of gigahertz have received wide attention both in the US and in Europe. The family of UWB systems may communicate either, by generating ultra-wideband signals or with the aid of innovatively combining conventional narrowband, wideband, or broadband signals. At the time of writing, UWB systems have only been implemented using ultra-wideband signals, such as those known from impulse radio systems. Hence, in this paper, UWB systems using narrowband signals are explored as a design alternative, which are based on the well-known family of frequency-hopping (FH) spread-spectrum multiple-access techniques. In the proposed UWB system, FH is implemented using multistage frequency-hopping multiple access (MS/UWB FHMA). We highlight the principles of the synchronous MS/UWB FHMA communication system, investigate the associated spectrum assignment, and the residue number system (RNS) based FH strategy: detection of the received signal can be achieved with the aid of existing fast FH signal detection schemes. Our study shows that the RNS assisted FH strategy is capable of efficiently dividing the huge number of users supported by the synchronous MS/UWB FHMA system into a number of reduced-size user groups, where the multiuser interference only affects the users within the same group. Since the number of users in each group is only a small fraction of the total number of users supported by the synchronous MS/UWB FHMA system, advanced multiuser detection algorithms can be employed for achieving near-single-user performance at an acceptable complexity. Our results show that MS/UWB FHMA is capable of supporting an extremely high number of users, while employing relatively simple receivers.