Question

(1) Consider a noisy wireless channel which flips each bit ( 0 becomes 1 or 1 becomes 0 ) with probability p. To guarantee the information is received reliably, each bit is transmitted m times (called repetition coding). For example, if m = 3 and “0” needs to be sent, you transmit “000” instead of “0”.

(1) Consideranoisywirelesschannelwhichflipseachbit(0becomes1or1becomes0)withprobabilityp. To guarantee the information is received reliably, each bit is transmitted m times (called repetition coding). For example, if m = 3 and “0” needs to be sent, you transmit “000” instead of “0”.

Assume m = 3, how many possible sequences the receiver may get? Among these sequences, how many have more “0”s than “1”s?

The receiver decodes the information by the majority rule. In other words, the receiver claims the bit is “0” if the received sequence has more “0”s than “1”s. Assume p = 0.4, what is the probability the receiver decodes the bit correctly?

The receiver decodes the information by the majority rule. In other words, the receiver claims the bit is “0” if the received sequence has more “0”s than “1”s. Assume p = 0.4, what is the probability the receiver decodes the bit correctly?

Answer 1

No. of possible sequences when m=3

The possible sequences are:

{000, 001, 010, 100, 110, 101, 011, 111}

From above we see that there are 4 sequences which have more "0"s than "1"s.

Probability of decoding the bit correctly

To make the situation simple let us suppose that 0 was transmitted (the answer is not affected by this choice). Now, the signal is decoded correctly by the receiver if 000 or 001 or 010 or 100 is received by the receiver.

Thus,

which is our required answer.

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