## Examples 2.2

#### The specification for a Class 1 telephone link is a guaranteed flat bandwidth of 300 Hz to 3400 Hz and a minimum signal to noise ratio of 40 dB. The specification for a Class 2 telephone link is a guaranteed flat bandwidth from 600 Hz to 2800 Hz and a minimum signal to noise ratio of 30 dB. A company has a requirement to send data over a telephone link at a bit rate of 20 kbps without error. Would you advise the company to rent the more expensive Class 1 service or the cheaper Class 2 service? Justify your decision.

### Solution

#### The Shannon–Hartley equation gives us the required relationship between channel capacity in bps, bandwidth and signal to noise ratio as follows:
Channel capacity C = B.log2(S/N + 1) bps
For the Class 1 line, B = 3400 – 300 = 3100 Hz and S/N = 40 dB, thus:
C = 3100.log2(10000 + 1) = 41.2 kbps
Note, it is essential to convert the S/N value from dB to a ratio for use in the Shannon–Hartley expression.
For the Class 2 line, B = 2800 – 600 = 2200 Hz and S/N = 30 dB, thus
C = 2200.log2(1000 + 1) bps = 21.9 kbps
These calculations show that both the Class 1 and Class 2 line will meet the specification of 20 kbps error-free transmission however, the performance of the Class 2 line is very close to the Shannon bound, and allows little margin for error. In practice, it is unlikely that a modem could be realized that would give the desired result on the Class 2 line.