For the CDMA communications system of Problem 11 3 8, a detection strategy known as decorrelation applies a transformation to Y to generate (SS) 1 SY P1 2X where (SS) 1SN is still a Gaussian noise vector with expected value E 0 Decorrelation separates the signals in that the ith component of is i p...

A short answer is that the decorrelator cannot be the same as the optimal maximum likelihood ML detector If they were the same that means we have redu ...

For the CDMA communications system of Problem 11.3.8, a detection strategy known as decorrelation applies a transformation

Question:

For the CDMA communications system of Problem 11.3.8, a detection strategy known as decorrelation applies a transformation to Y to generate
Ỹ = (SʹS)-1 SʹY = P1/2X +
where = (SʹS)-1SʹN is still a Gaussian noise vector with expected value E[] = 0. Decorrelation separates the signals in that the ith component of Ỹ is
Ỹi = √pi Xi + i,
which is the same as a single-user receiver output of the binary communication system of Example 11.6. For equally likely inputs Xi = 1 and Xi = - 1, Example 11.6 showed that the optimal (minimum probability of bit error) decision rule based on the receiver output Yi is
i = sgn (Ỹi).
Although this technique requires the code vectors S1,...,Sk to be linearly independent, the number of hypotheses that must be tested is greatly reduced in comparison to the optimal ML detector introduced in Problem 11.3.8. In the case of linearly independent code vectors, is the decorrelator optimal? That is, does it achieve the same bit error rate (BER) as the optimal ML detector?