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 sT, 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 Rr[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 
behaves
like a quasi-deterministic quantity when N(1 -
l) << 1.
Therefore, for large i, we can replace
by its expected value, which is given by [7, 111, 301]
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, Rr[i] and Rr[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 sTw = sTw[i] = 1, sTq[i] = sTw[i]–sTw = 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 sTk[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 
and in
(7.191) we have used the following fact,
which is derived below:
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 
is the mean output energy defined in
(7.156).
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 kth 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 z1, z2, z3, and z4, 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]} 
0.
Derivation of (7.199)
Using the Gaussian moment factoring formula, we obtain
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