The document discusses bit error rate and the average probability of error in receiving digital signals over noise channels. It explains the two types of errors that can occur, shows the model of the receiver, and provides calculations and plots to show how the average probability of error decreases exponentially as the signal-to-noise ratio increases. Examples are included to illustrate applying the concepts.

1. Error rate due to noise
1

2. Bit error rate
2
x(t) y(t) Y

3. Bit error rate
3

4. Receiver model
The structure of the receiver used to
perform this decision making process is
shown below
4

5. Possible errors
When the receiver detects a bit there are
two possible kinds of error
1. Symbol 1 is chosen when a 0 was actually
transmitted; we refer to this error as an error
of the first kind
2. Symbol 0 is chosen when a 1 was actually
transmitted; we refer to this error as an error
of the second kind
5

6. Average probability of error
calculations
6

7. Average probability of error
calculations
7

8. Average probability of error
calculations
8

9. Average probability of error
calculations
9

10. 10

11. 11

12. 12

13. 13

14. 14

15. 15

16. 16

17. Expression for the average
probability of error
17

18. 18

19. 19

20. 20

21. 21

22. A plot of the average probability of error as
a function of the signal to noise energy is
shown below
22

23. As it can be seen from the previous figure,
the average probability of error decreases
exponential as the signal to noise energy
is increased
For large values of the signal to noise
energy the complementary error function
can be approximated as
23

24. Example1
24
Solution

25. Example1 solution
25

26. Example 2
26

27. Example 2 solution
27

28. Example 2 solution
28

29. Example 2 solution
29

30. Example 2 solution
30

31. Example 3
31

32. Example 3 solution
32

33. Example 3 solution
33