1. Suppose that a receiver receives the following signal: Y = X + N , where X E {-1, 1} is the...

Question

1. Suppose that a receiver receives the following signal: Y = X + N , where X E {-1, 1} is the transmitted signal and N is the Gaussian background noise of \ 1 mean zero and unit variance. It is known that PO= —1) = PO = 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
[3 marks]
(b) Find the average error probability for the decision rule in (a) if \ 1 ¦ 3 PO= —1) = — and PO = 1) = — . 4 4
[3 marks]
(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?
[4 marks]

005_5k-egl40sx0.pdf

Robert · Accepted Answer

Solution Q1(a): 
Given that the values of the noise N follows the Gaussian probability distribution function 
 ( )  
 
√    
 
 (   )
Where,
and  
   
  
 

Probability of error in polar ASK signal is given by:
With this threshold, the probability of error given x=1 is transmitted is (the area in blue region): 
 (
 
   
)
√    
∫  
 (   ) 
   

 
 
√  
∫
   

 
 
√  
∫
   

  ( ) 
Similarly, 
With this threshold, the probability of error given x=-1 is transmitted is (the area in grean region): 
 (
 
    
)
√    
∫  
 (   ) 
   

 
 
√  
∫
   

 
 
√  
∫
   

  ( )
Total bit error probability is given by:
    (    )   ((   ) (    ))   (   )   ((   ) (   )) 
        ( )       ( )   ( )
Solution Q1(b):

1. Suppose that a receiver receives the following signal: Y = X + N , where X E {-1, 1} is the transmitted signal and N is the Gaussian background noise of \ 1 mean zero and unit variance. It is known...

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