Wireless Communication Systems: Advanced Techniques for Signal Reception (paperback)
7.9.1 Linear MMSE Detector and RLS Blind Adaptation Rule
Consider the following received signal model: Equation 7.151
where A K , b K and s K denote, respectively, the received amplitude, data bit, and the spreading waveform of the K th user ; i denotes the NBI signal; and Equation 7.152
where R r is the autocorrelation matrix of the received discrete signal r : Equation 7.153
The output SINR is given by Equation 7.154
where Equation 7.155
The mean output energy associated with w , defined as the mean-square output value of w applied to r , is Equation 7.156
where the last equality follows from (7.155) and the matrix inversion lemma. The mean-square error (MSE) at the output of w is Equation 7.157
The exponentially windowed RLS algorithm selects the weight vector w [ i ] to minimize the sum of exponentially weighted output energies:
where 0 < l < 1 is a forgetting factor (1 - l << 1). The purpose of l is to ensure that the data in the distant past will be forgotten in order to provide tracking capability in nonstationary environments. The solution to this constrained optimization problem is given by Equation 7.158
where Equation 7.159
A recursive procedure for updating w [ i ] is as follows: Equation 7.160
Equation 7.161
Equation 7.162
Equation 7.163
In what follows we provide a convergence analysis for the algorithm above. In this analysis, we make use of three approximations/assumptions: (a) For large i , R r [ i ] is approximated by its expected value [111, 301]; (b) the input data r [ i ] and the previous weight vector w [ i “1] are assumed to be independent [175]; (c) some fourth-order statistic can be approximated in terms of a second-order statistic [175]. 7.9.2 Convergence of the Mean Weight Vector
We start by deriving an explicit recursive relationship between w [ i ] and w [ i “1]. Denote Equation 7.164
Premultiplying both sides of (7.161) by s T , we have Equation 7.165
From (7.165) we obtain Equation 7.166
where Equation 7.167
Substituting (7.161) and (7.166) into (7.162), we can write Equation 7.168
where Equation 7.169
is the a priori least-squares estimate at time i . It is shown below that Equation 7.170
Equation 7.171
Substituting (7.161) and (7.170) into (7.168), we have Equation 7.172
Premultiplying both sides of (7.172) by R r [ i ], we get Equation 7.173
where we have used (7.159) and (7.169). Let q [ i ] be the weight error vector between the weight vector w [ i ] at time n and the optimal weight vector w : Equation 7.174
Then from (7.173) we can deduce that Equation 7.175
Therefore, Equation 7.176
where Equation 7.177
in which we have used (7.171) and (7.169). It has been shown [111, 301] that for large i , the inverse autocorrelation estimate Equation 7.178
Using this approximation , we have Equation 7.179
Therefore, for large i , Equation 7.180
where we have used (7.170) and (7.179). For large i , R r [ i ] and R r [ i “1] can be assumed almost equal, and thus approximately [111, 301] Equation 7.181
Substituting (7.181) and (7.180) into (7.176), we then have Equation 7.182
Equation (7.182) is a recursive equation that the weight error vector q [ i ] satisfies for large i . In what follows we assume that the present input r [ i ] and the previous weight error q [ i “1] are independent. In this application of interference suppression, this assumption is satisfied when the interference signal consists of only MAI and white noise. If, in addition, there is NBI present, this assumption is not satisfied but is nevertheless assumed, as is the common practice in the analysis of adaptive algorithms [111, 175, 301]. Taking expectations on both sides of (7.182), we have
where we have used the facts that s T w = s T w [ i ] = 1, s T q [ i ] = s T w [ i ] “ s T w = 0 and Equation 7.183
Therefore, the expected weight error vector always converges to zero, and this convergence is independent of the eigenvalue distribution. Finally, we verify (7.170) and (7.171). Postmultiplying both sides of (7.163) by r [ i ], we have Equation 7.184
On the other hand, (7.160) can be rewritten as Equation 7.185
Equation (7.170) is obtained by comparing (7.184) and (7.185). Multiplying both sides of (7.166) by s T k [ i ], we can write Equation 7.186
and (7.167) can be rewritten as Equation 7.187
Equation (7.171) is obtained comparing (7.186) and (7.187). 7.9.3 Weight Error Correlation Matrix
We proceed to derive a recursive relationship for the time evolution of the correlation matrix of the weight error vector q [ i ], which is the key to analysis of the convergence of the MSE. Let K [ i ] be the weight error correlation matrix at time n . Taking the expectation of the outer product of the weight error vector q [ i ], we get Equation 7.188
We next compute the four expectations appearing on the right-hand side of (7.188). First term Equation 7.189
Equation 7.190
Equation 7.191
Equation 7.192
Equation 7.193
where in (7.189) we have used (7.183); in (7.193) we have used (7.152); in (7.190) and (7.192) we have used the fact that Equation 7.194
Second term Equation 7.195
where we have used (7.183) and the following fact, which is shown below: Equation 7.196
Therefore, the second term is a transient term. Third term The third term is the transpose of the second term, and therefore it is also a transient term. Fourth term Equation 7.197
Equation 7.198
where in (7.198) we have used (7.152), and in (7.197) we have used the following fact, which is derived below: Equation 7.199
where Now combining these four terms in (7.188), we obtain (for large i ) Equation 7.200
Finally, we derive (7.194), (7.196), and (7.199). Derivation of (7.194) We use the notation [ ·] mn to denote the ( m, n )th entry of a matrix and [ ·] k to denote the k th entry of a vector. Then Equation 7.201
Next we use the Gaussian moment factoring theorem to approximate the fourth-order moment introduced in (7.201). The Gaussian moment factoring theorem states that if z 1 , z 2 , z 3 , and z 4 , are four samples of a zero-mean, real Gaussian process, then [175] Equation 7.202
Using this approximation, we proceed with (7.201): Equation 7.203
Therefore,
where in the last equality we used (7.183) and the following fact: Equation 7.204
Derivation of (7.196) Similarly, we use the approximation by the Gaussian moment factoring formula and obtain
since E { q [ i ]} Derivation of (7.199) Using the Gaussian moment factoring formula, we obtain
7.9.4 Convergence of MSE
Next we consider the convergence of the output MSE. Let Equation 7.205
Equation 7.206
Since Equation 7.207
Since E { q [ i } Equation 7.208
Since l 2 + (1- l 2 ) < [ l + (1 - l )] 2 = 1, the term tr{ R r K [ i ]} converges. The steady-state excess mean-square error is then given by Equation 7.209
Again we see that the convergence of the MSE and the steady-state misadjustment are independent of the eigenvalue distribution of the data autocorrelation matrix, in contrast to the situation for the LMS version of the blind adaptive algorithm [183]. 7.9.5 Steady-State SINR
We now consider the steady-state output SINR of the RLS blind adaptive algorithm. At time i the mean output value is Equation 7.210
The variance of the output at time i is Equation 7.211
Let Equation 7.212
Therefore the steady-state SINR is given by Equation 7.213
where SINR * is the optimum SINR value given in (7.154). 7.9.6 Comparison with Training-Based RLS Algorithm
We now compare the preceding results with the analogous results for the conventional RLS algorithms in which the data symbols b [ i ] are assumed to be known to the receiver. This condition can be achieved by using either a training sequence or decision feedback. In this case, the exponentially windowed RLS algorithm chooses w [ i ] to minimize the cost function Equation 7.214
The RLS adaptation rule in this case is given by [175] Equation 7.215
Equation 7.216
where e p [ i ] is the prediction error at time i and k [ i ] is the Kalman gain vector defined in (7.160). Using the results from [111], we conclude that the mean weight vector w [ i ] converges to w (i.e., E { w [ i ]} Equation 7.217
The MSE Equation 7.218
The steady-state excess mean-square error is given by [111] Equation 7.219
where we have used the approximation that Equation 7.220
where the last equality follows from (7.156). The output MSE at time i is Equation 7.221
Therefore, Equation 7.222
Using (7.220) and (7.222), after some manipulation, we have Equation 7.223
Therefore, the output SINR in the steady state is given by Equation 7.224
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