Information Theory - Digital Communication: Principles and System Modelling - page 176

Repeater1 Repeater2 Rx

C C C

C W W

W C W

W W C

Therefore, the probability of erroneous detection is 1 / 2 .
The number 1 / 2 is really interesting. The number can be expressed as

1
2

1 1

2

( 2 − 1 )

1
2 =

1
2 ×

1
2 =

2 C 1 ×

2 × × (8.27)

For a 3 repeater cascade, the table would be as follows:

Repeater1 Repeater2 Repeater3 Rx

W W W W

W W C C

W C W C

W C C W

C W W C

C W C W

C C WW

C C C C

Here, the probability of erroneous detection is 4/8 1/2. This is a priory probability.

But, out of these 4 wrong (W) reception, in 3 cases there are 1 W repeater, therefore,
probability is

=

1
2

1 1

2

( 3 − 1 )

3
4 ×

1
2 =

1
2 ×

1
4 =

3 C 1 ×

6 × × (8.28)

Hence, Eqs. ( 8.27 ) and ( 8.28 ) suggests the distribution of error probability model
will be

n C x p x n − x

P e = ( 1 − p ) (8.29)

This is clearly a binomial distribution .