n pulse amplitude modulation (PAM) where an analog signal is sampled and represented by varying-height pulses, a minimum bandwidth on the order of 1/(2T) is required to transmit a pulse train. In pulse code modulation (PCM) the varying amplitude of each pulse is further represented by a code consisting of several pulses (the amplitude of these pulses is constant in binary coding which uses simple on-off pulses). This increases the required bandwidth in proportion to the number of pulses in the code. For example, if 8 quantizing levels are used in a binary PCM system, 3 pulse positions (that is, 3 bits) are used, and thus the PCM signal requires 3 times more bandwidth than if 1 pulse were transmitted (the reason is that 3 pulses are now sent in the same time that it took to send 1 pulse). Similarly, the bandwidth is increased by a factor of 4 for 16 levels in a binary system. Extending this to a binary system with n quantization levels, we see that the required bandwidth is increased by a factor of at least log2n.
(a) In binary PCM coding, find the increase of the transmission bandwidth if 4, 64, and 256 quantization levels are used.
(b) Find the increase in bandwidth required for transmission of a PCM signal that has been quantized to 64 levels if each pulse is allowed to take on the following number of levels: 2, 3, and y.