Problem#1

A disadvantage of a broadcast medium is the wasted capacity due to multiple hosts attempting to access the channel at the same time. As a simplistic example, suppose that time is divided into descrete time slots and each host attempts to use the channel with the probability p during each slot. There are n host connected to the channel.
(a) What is the probability that only channel (say channel i) will try to transmit during a slot?
=> p(1-p)^n-1

(b) What is th eprobability that the exactly one channel (Any one) will try to transmit during a slot?
=> np(1-p)^n-1

(c) What is the probability that no channel will try to transmit during a slot?
=> (1-p)ⁿ

(d) What is the probability that there will be a collision?
There will be a collision when exactly one host is not transmitting and when there is some transmission taking place.
=> 1 - np(1-p)^n-1-(1-p)ⁿ

Problem#2

Suppose you are given a 4kHz channel with a signal to noise ratio of 20 db. A signal has the highest frequency component at 4kHz. How many bits per sample can be transmitted to utilize the channel effectively?

Shannon limit = 4log₂(101) = 26.63 kbps
Minimum sampling frequency required = 8kHz
To achieve the maximum data rate,
26.83 = 8 log₂V this gives V=3.32
Hence for effective channel utilization number of bits per sample=3.

Problem#3

You have been hired by a Deepali & Company to design communication equipment. You are to design multiplexing equipment to multiplex four signals in a single stage to be fed to a single channel. The signals are band limited to 1kHz, 500Hz, 5kHz, 10 kHz.

(a) What is the Nyquist sampling frequency?
=> 2 x 10 = 20kHz

(b) What is the time gap between two samples?
=> 1/(4x20) = 12.5 µsec

(c) If each sample is encoded into 8 bits what data rate channel must support?
=> Data rate = 8(4)(20)=640kbps