Reference no: EM13860202

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