6.9 Appendix

6.9.1 Proofs for Section 6.3.3

Proof of Proposition 6.1

The proof of this result is based on the following lemma.

Lemma 6.1: Let X be a K x K positive-definite matrix. Denote by X _k the ( K “ 1) x ( K “ 1) submatrix obtained from X by deleting the k th row and k th column. Also, denote by x _k the k th column of X with the k th entry x _kk removed. Then we have

Equation 6.213

Proof: Since X _k is a principal submatrix of X , and X is positive definite, X _k is also positive definite. Hence exists.

Denote the above-mentioned partitioning of the symmetric matrix X with respect to the k th column and row by

In the same way, we partition its inverse:

Now from the fact that XY = I _K , it follows that

Equation 6.214

Equation 6.215

Solving for y _kk from (6.214) and (6.215), we obtain (6.213).

Proof of (6.60)

Using (6.62) and (6.63), by definition we have

Equation 6.216

From (6.62) and (6.216) it is immediate that (6.60) is equivalent to

Equation 6.217

Equation 6.218

Partition the three matrices above with respect to the k th column and the k th row to get

By (6.213), (6.218) is then equivalent to

Equation 6.219

Since

Equation 6.220

Equation 6.221

we then have

Equation 6.222

Therefore, to show (6.219), it suffices to show that

Equation 6.223

which is in turn equivalent to [189]

Equation 6.224

where X Y means that the matrix X “ Y is positive definite. Since by assumption, 0 < l ₂ ( b _j [ i ]) < , we have . It is easy to check that

Equation 6.225

Equation 6.226

Hence (6.224) holds and so does (6.60).

6.9.2 Derivation of the LLR for the RAKE Receiver in Section 6.6.2

To obtain the code-bit LLR for the RAKE receiver, a Gaussian assumption is made on the distribution of y _k [ i ] in (6.182); that is, we assume that

Equation 6.227

where m _k [ i ] is the equivalent signal amplitude and is the equivalent noise variance.

As in typical RAKE receivers [396], we assume that the signals from different paths are orthogonal for a particular user . Conditioned on b _k [ i ], using (6.133) and (6.134), the mean m _k [ i ] in (6.227) is given by

Equation 6.228

where the expectation is taken with respect to channel noise and all the code bits other than b _k [ i ]. The variance in (6.227) can be computed as

Equation 6.229

Using the orthogonality assumption, it is easy to check that A = 0 in (6.229). Since term B and term C in (6.229) are two zero-mean independent random variables , the variance is then given by

Equation 6.230

Due to the orthogonality assumption, the second term in (6.230) is given by

Equation 6.231

For simplicity, we assume that the time delay t _l,k is an integer multiple of the chip duration. Then the first term in (6.230) can be written as

Equation 6.232

where

Equation 6.233

Assuming that the signature waveforms contain i.i.d. antipodal chips, r ( k,l ),( k',l' )[ n ] is an i.i.d. binary random variable taking values of ±(1/ N ) with equal probability. Since a _l,k [ i ], a _l',k' [ i' ], and r _{( k,l ),( k',l' )} [ n ] are independent, (6.232) can be written as

Equation 6.234

Substituting (6.231) and (6.234) into (6.230), we have

Equation 6.235

Hence the LLR of b _k [ i ] can be written as

Equation 6.236