1. Suppose that a receiver receives the following signal:
Y = X + N,
where X ∈{-1, 1} is the transmitted signal and N is the Gaussian background noise of mean zero and unit variance. It is known that
Pr(X = -1) = Pr(X=1) 1/2
(a) For the signal detection, if the receiver uses the following decision rule, find the average error probability:
Decide X =1, if Y≥ 0;
Decide X = -1, if Y < 0
(b) Find the average error probability for the decision rule in (a) if Pr(X = -1)=1/4 and Pr(X=1) = 3/4.
(c) Suppose that the same signal is transmitted 3 times in order to improve the performance. The receiver can make 3 decisions from the 3 received signals as in (a). If the majority rule is used to combine 3 decisions in order to make a final decision, what is the error probability?
Q-function Table
| x |
Q(x)=1/√2Π x∫0exp (-(z2/2))dz |
|
0
|
0.5000
|
|
0.1000
|
0.4602
|
|
0.2000
|
0.4207
|
|
0.3000
|
0.3821
|
|
0.4000
|
0.3446
|
|
0.5000
|
0.3085
|
|
0.6000
|
0.2743
|
|
0.7000
|
0.2420
|
|
0.8000
|
0.2119
|
|
0.9000
|
0.1841
|
|
1.0000
|
0.1587
|
|
1.1000
|
0.1357
|
|
1.2000
|
0.1151
|
|
1.3000
|
0.0968
|
|
1.4000
|
0.0808
|
|
1.5000
|
0.0668
|
|
1.6000
|
0.0548
|
|
1.7000
|
0.0446
|
|
1.8000
|
0.0359
|
|
1.9000
|
0.0287
|
|
2.0000
|
0.0228
|
|
2.1000
|
0.0179
|
|
2.2000
|
0.0139
|
|
2.3000
|
0.0107
|
|
2.4000
|
0.0082
|
|
2.5000
|
0.0062
|
|
2.6000
|
0.0047
|
|
2.7000
|
0.0035
|
|
2.8000
|
0.0026
|
|
2.9000
|
0.0019
|
|
3.0000
|
0.0013
|
|
3.1000
|
0.0010
|
|
3.2000
|
0.0007
|
|
3.3000
|
0.0005
|
|
3.4000
|
0.0003
|
|
3.5000
|
0.0002
|