Telephone channels have a bandwidth of about 31 kHz Do the f

Telephone channels have a bandwidth of about 3.1 kHz. Do the following in Excel.

1.1 If a telephone channel’s signal-to-noise ratio is 1,000 (the signal strength is 1,000 times larger than the noise strength), how fast can a telephone channel carry data? (Check figure: Telephone modems operate at about 30 kbps, so your answer should be roughly this speed.)

1.2 How fast could a telephone channel carry data if the SNR were increased massively, from 1,000 to 10,000? (This would not be realistic in practice.)

1.3 With an SNR of 1,000, how fast could a telephone channel carry data if the bandwidth were increased to 4 kHz? Show your work or no credit.

1.4 What did you learn from these three analyses?

Solution

1.1

Shannons channel capacity riteris for noisy channel

with B hz band idth and S/N signal to noise ratio

then

maximum channel capacity

C= B log (1+ S/N ) , log base 2

here B = 3.1 k Hz = 3.1 * 1000 = 3100 Hz

then

C = 3100* log(1+1000)

= 3100* 9.96 = 30876 bps

= 30.876 kpps

1.2

S/N = 10000

then

C = 3100* log(1+10000) = 3100*13.28 =41168 bps = 41.168 kbps

if S/R = 1000, C= 30.87

if S/R = 10000, C= 41.168

means incresed by 1/3 of prevoius

1.3

now S/R = 1000 and B= 4 KHz

C= B log (1+ S/N ) , log base 2

here B = 4 k Hz = 3.1 * 1000 = 4000Hz

C = 4 * log (1+ 1000) = 4* (9.96 ) =39.84 kbps

then maximum speed = 39.84 which is approximated equal to above (1.2 example) speed.

1.4

( increse band width by 1 KHz) is equal to taking 10 times S/N ratio )

change in speed of the communication channel is same if we either increse band width by 1 KHz or multiply S/N with 10( 10 times)