6.9.1 Proofs for Section 6.3.3Proof of Proposition 6.1The 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 6.6.2To 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
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