## Example 2.1

#### A modem claims to operate with a bandwidth efficiency of 5 bits/second/Hz when using 1024 symbol states in the transmission constellation.

1. How many bits are being encoded in to each symbol, and what is the modem capacity if the baud rate is 4000 symbols/second?
2. How many symbol states must be employed if the user wishes to send his information in half the time?

### Solution

1. For 1024 symbol states, the number of bits represented by each symbol is log21024 = 10 bits/symbol.
   For a baud rate of 4000, this means that the information transfer rate is 4000 x 10 = 40 kbps.
2. In order to send the information in half the time, it would be necessary to send data at 80 kbps and hence to encode 20 bits in each symbol. The number of symbol states is thus a massive 220 = 1048576.