Need help with Binary detector & prob. error. Urgent!!!

Hello guyz, I need you guys help me with these two problems. I have tried to solve but coulndn't get through. I am really appreciated you guy for trying to help me out. I need it soon...Again, thank you very much...

Re: Need help with Binary detector & prob. error. Urgent

hm.. looks like you are trying to pass on your homework..

The first symbol will be recieved as a gaussian distributed value with mean at H1*SQRT(E) with variance SIG1^2 and second will have mean as -H2*SQRT(E) and variance SIG2^2. The optimum detector should evaluate
p(t|r) = p(r|t)*p(t)/p(r)

where t is the transmitted signal with prob p(s1) and p(s2), r is the recieved signal. and p(r|t) is a guassian distributed variable with mean and variance as described above. Since we are going to do a comparison of prob for fixed r, we can ignore p(r) in this comparison, as it is going to be same independent of us assuming t = s1 or t =s2.

Thus we evaluate the guassian pdf at r for mean = H1S1 and dev = SIG1^2, mutiply with p(s1) and compare it against the guassian pdf at r for mean = H2S2 and dev = SIG2^2, mutiply with p(s2). We declare t = s1 if first expression is bigger, and second if not.

The probability of error will be s1 is decoded as s2 and viceversa.

in the case SIG1= SIG2 and p(s1) =p(s2), it would be enough to see whether the recieved signal is negative or positive.

- b

Re: Need help with Binary detector & prob. error. Urgent

the above soln seems correct

Re: Need help with Binary detector & prob. error. Urgent

Hi,
It's a binary hypothesis testing problem (but a little complicated becuase of multiple receivers).

1 <?> prob(√E|r1,r2)/prob(-√E|r1,r2) = [prob(r1,r2| √E)prob(√E)]/[prob(r1,r2| -√E)prob(-√E)]

if two noises are independent : prob(r1,r2|√E) = prob(r1|√E)prob(r2|√E) and the same result is true for -√E

prob(-√E)/prob(√E) <?> exp(-4√E[H1.r1/sig1^2 + H2.r2/sig2^2])
-ln(prob(-√E)/prob(√E))/4√E <?> H1.r1/sig1^2 + H2.r2/sig2^2

if equation state with < s1 have been transmitted and vice versa.
--------------------------------------------
for the second part of quastion1:
r1 + r2 <?> 0
but if we have only a receiver
r1 <?> 0

I can't discuss more about what this equations are, and what I am talking about.
but hope these help you.
regards