In digital communication systems, transmission of digital data is encoded using bandlimited pulses in order to utilize the channel bandwidth efficiently. Unfortunately, bandlimited pulses are nontimelimited; they have infinite duration, which causes pulses representing successive digits to interfere and possibly cause errors. This difficulty can be resolved by shaping a pulse p(t) in such a way that it is bandlimited yet causes zero interference at the sampling instants. To transmit R pulses/second, we require a minimum bandwidth R/2 Hz (see Prob. 3.1-11). The bandwidth of p(t) should be R/2 Hz, and its samples, in order to cause no interference at all other sampling instants, must satisfy the condition
where T = 1/R . Because the pulse rate is R pulses per second, the sampling instants are located at intervals of 1/R s. Hence, the preceding condition ensures that a pulse at any instant will not interfere with the amplitude of any other pulse at its center. Find p(t). Is p(t) unique in the sense that no other pulse satisfies the given requirements?