Linear MMSE Detector

While the linear decorrelating detector is designed to eliminate the MAI completely at the expense of enhancing the ambient noise, the linear MMSE detector, , is designed to minimize the total effect of the MAI and the ambient noise at the detector output. Specifically, the linear MMSE detector for user 1 is given by the solution to the following optimization problem:

Equation 2.19

Equation 2.20

Proof: First note that any linear detector must lie in the column space of S [i.e., m₁ range(S)]. This is because any component outside this space does not affect the signal components of the detector output [i.e., the first and second terms of (2.10], and it merely increases the noise level [i.e., the third term of (2.10)]. Therefore, we can write m₁ = Sx₁ for some where

Equation 2.21

Hence (2.20) is obtained.

The output of the linear MMSE detector is given by

Equation 2.22

Equation 2.23

where, using (2.20), we have

Equation 2.24

Equation 2.25

Note that unlike the decorrelator output (2.15), the linear MMSE detector output (2.22) contains some residual MAI. However, we will in general have ||m₁|| < ||d₁||, so that the effects of ambient noise are reduced by the linear MMSE detector.