Least-Squares Regression and Linear Decorrelator

Least-Squares Regression and Linear Decorrelator

Consider the synchronous signal model (4.2). Denote q_K A_k b_k. Then (4.2) can be rewritten as

Equation 4.5

Equation 4.6

where . Consider the linear regression problem of estimating the K unknown parameters q₁, q₂, . . . , q_K from the N observations r₁, r₂, . . . , r_N in (4.5). Classically, this problem can be solved by minimizing the sum of squared errors (or squared residuals) [i.e., through the least-squares (LS) method]:

Equation 4.7

Equation 4.8

It is easily seen from (4.8) that the maximum-likelihood estimate of q under the i.i.d. Gaussian noise assumption is given by the LS solution in (4.7). Upon differentiating (4.7), is then the solution to the following linear system equations

Equation 4.9

Equation 4.10

Define the cross-correlation matrix of the signature waveforms of all users as R S^TS. Assuming that the user signature waveforms are linearly independent (i.e., S has a full column rank K), R is invertible, and the LS solution to (4.9) or (4.10) is given by

Equation 4.11

We observe from (4.11) that the LS estimate is exactly the output of the linear decorrelating multiuser detector for the K users (cf. Proposition 2.1). This is not surprising, since the linear decorrelating detector gives the maximum likelihood estimate of the product of the amplitude and the data bit q_k = A_kb_k in Gaussian noise [296]. Given the estimate , the estimated amplitude and the data bit are then determined by

Equation 4.12

Equation 4.13