## Example 3.1

#### A four-level baseband data stream has a symbol period of 100 ms.

1. What is the minimum bandwidth required for transmission, assuming a root raised cosine filter is used with a = 0.3?
2. What is the time taken to transmit 1 million bits?
3. If it is required to transmit the information in half the time, how many symbol states would be required using the same transmission bandwidth?

### Solution

1. The minimum bandwidth required for transmission is half the symbol period for baseband signalling, assuming brick-wall filtering (a = 0). For a = 0.3, this bandwidth must be increased by a factor of (1 + a). Thus for a symbol period of 100 ms, the symbol rate is 10 000 symbols/second, and the bandwidth required is 5000(1 + a) = 65 000 Hz.
2. With a four-level signalling scheme, there can be two bits encoded in each symbol, hence the bit rate is twice the symbol rate, i.e. 20 kbps. The time taken to send 1 million bits is thus 1 000 000/20 000 = 50 seconds.
3. To halve the transmission time, we need to encode double the number of bits in each symbol. Thus 4 bits/symbol requires 16 symbol states.