Linear Decorrelating Detector

Linear Decorrelating Detector

A linear decorrelating detector for user 1, , is such that when correlated with the received signal r[i], it results in zero MAI [i.e., the second term in (2.10) is zero]. In particular, the linear decorrelating detector d₁ for user 1 satisfies

Equation 2.11

Equation 2.12

Denote by e_k a K-vector with all entries zeros except for the kth entry, which is 1. Assume that the user signature sequences are linearly independent [i.e., the matrix has full column rank, rank(S) = K]. Let be the correlation matrix of the user signature sequences. Then R is invertible. The following result gives the expression for the linear decorrelating detector.

Proposition 2.1: The linear decorrelating detector for user 1 is given by

Equation 2.13

Proof: It is easily verified that

Equation 2.14

Therefore, (2.11) and (2.12) hold.

The output of the linear decorrelating detector is given by

Equation 2.15

with

Equation 2.16

where, by (2.13),

Equation 2.17

and where in (2.17), [A]_i,j denotes the (i, j)th element of the matrix A. Note that by the Cauchy–Schwartz inequality, we have

Equation 2.18

Since ||s₁|| = 1 and , it then follows that ||d₁|| 1. Hence, by (2.16), we have Var{v₁[i]} s² (i.e., the linear decorrelating detector enhances the output noise level).