The signal's spectrum is determined by the modulation scheme and data rate used by the transmitter, but can be modified with a pulse shaping filter. This pulse shaping will make the spectrum smooth, leading to a time limited signal again. Usually the transmitted symbols are represented as a time sequence of dirac delta pulses multiplied with the symbol. This is the formal transition from the digital to the analog domain. At this point, the bandwidth of the signal is unlimited. This theoretical signal is then filtered with the pulse shaping filter, producing the transmitted signal. If the pulse shaping filter is a rectangular in the time domain (like this is usually done when drawing it), this would lead to an unlimited spectrum.
