非单位联合块对角化在卷积盲源分离中的应用与优化
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"这篇研究论文探讨了快速非单位ary联合块对角化技术在卷积盲源分离（CBSS）中的应用，并提出了通过简并解消除来解决算法收敛到异常解的问题。"
正文：
卷积盲源分离（CBSS）是信号处理领域中的一个关键问题，它涉及到从混合信号中恢复出原始独立源信号，而无需预先知道混合过程的具体信息。在CBSS中，信号通常会经历非线性的卷积过程，这使得分离任务变得复杂。传统的解决方法包括基于正交块对角化的算法，但这些方法在处理非线性卷积问题时可能会遇到效率和性能上的限制。
这篇由Wei-Tao Zhang, Shun-Tian Lou和Hong-Min Lu共同撰写的论文，"快速非单位ary联合块对角化与简并解消除用于卷积盲源分离"，提出了一个新的非单位ary联合块对角化（JBD）算法，该算法在处理CBSS问题时具有优势。非单位ary JBD允许更灵活的矩阵变换，能够更好地适应非线性卷积环境，然而，它也面临着一个问题：容易收敛到奇异或病态的解，即简并解。
为了解决这个问题，作者引入了一种简并解消除策略。在JBD过程中，当算法接近可能的简并解时，他们会采用特定的方法来识别和排除这些不理想的解决方案。这有助于确保算法能够稳定地收敛到满足分离要求的正确对角化形式，从而提高源信号的分离质量和效率。
文章中详细介绍了算法的设计原理，包括如何定义和寻找循环最小化器，以及如何利用最小二乘准则来优化块对角化过程。此外，作者还提供了算法的实现细节和性能评估，证明了新方法在实际应用中的有效性和鲁棒性。
关键词包括：联合块对角化（JBD）、卷积盲源分离（CBSS）、循环最小化器和最小二乘准则。这些关键词揭示了研究的核心内容和方法论，表明该论文不仅贡献了一个新的算法，还对CBSS领域的理论基础进行了深入探索。
这篇论文为卷积盲源分离提供了一个改进的非单位ary JBD框架，通过简并解消除策略克服了算法可能存在的不稳定因素，对于提升CBSS性能具有重要意义，特别是在处理复杂卷积环境下的信号分离问题时。
Author's personal copy
810 W.-T. Zhang et al. / Digital Signal Processing 22 (2012) 808–819
where I
n
r
is
n
r
× n
r
identity matrix, the semi-unitary constraint
U
H
U = I
n
r
imposedin(5)stemsfromthederivationoftheJBD
NU
algorithm in which the partitions of the block diagonalizer are ob-
tained from an eigenvalue decomposition (EVD), so they are semi-
unitary. Now let us consider the matrix B given by
B =[U
1
, U
2
,...,U
r−1
, U
r−1
], (6)
where U
i
∈ U
W
, ∀i ∈ I
r−1
. It is readily to show that
C(B) =
K
k=1
OffB
B
H
R
k
B
2
F
=
K
k=1
r
i=1
r
j=1, j=i
B
H
i
R
k
B
j
2
F
=
K
k=1
r
−1
i=1
r
−1
j=1, j=i
U
H
i
R
k
U
j
2
F
+
K
k=1
r
−1
i=1
U
H
r
−1
R
H
k
U
i
2
F
+
K
k=1
r
−1
j=1
U
H
j
R
k
U
r−1
2
F
. (7)
Since ∀i ∈ I
r−1
, ∀m ∈ I
n
r
,wehaveu
mH
i
W = 0 according to (5),
where u
m
i
denotes the
m
-th column of U
i
.Sothefirstterminthe
last equation at the right-hand side of (7) reads
K
k=1
r
−1
i=1
r
−1
j=1, j=i
U
H
i
R
k
U
j
2
F
=
r−1
i=1
r
−1
j=1, j=i
n
r
m=1
n
r
n=1
K
k=1
u
mH
i
WD
k
W
H
u
n
j
2
= 0. (8)
Similarly, it can be shown that C(B) = 0, which implies that the
matrix given by (6) is a global optimal solution of JBD
NU
algorithm.
But obviously, this solution is a singular solution because the last
two partitions of the block diagonalizer given by (6) are identical.
Actually, it should be pointed out that even if the matrix W
is nonsingular, the JBD
NU
algorithm may converge to an ill-
conditioned solution. This is because that the minimization of cost
function (3) cannot guarantee the independence between the par-
titions of block diagonalizer.
In addition, it should be stressed that the inherent ambiguities
exist in JD algorithms also appear in JBD algorithms. If
and
designate, respectively, a block permutation matrix and a nonsin-
gular block diagonal matrix, where a block permutation matrix is
also a permutation matrix whose structure is stated as follow. In
each block row and block column, there is only one nonzero square
block and the nonzero block is filled by an arbitrary permutation
matrix, e.g.
⎡
⎢
⎢
⎢
⎢
⎢
⎣
000100
001000
100000
010000
000010
000001
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Thus the relation
R
k
= AD
k
A
H
= (A)
H
−1
D
k
−H
H
H
A
H
(9)
shows that A (or the corresponding B) and the block diagonal ma-
trix set
D are not uniquely defined. Therefore, for the JBD problem,
what we can do is to estimate a joint block diagonalizer, say
ˆ
B,
such that
ˆ
B = B. (10)
In the sequel, we will derive our new algorithm to eliminate the
ill-conditioned solutions.
2.3. Minimization algorithm
As stated previously, ill-conditioned solutions obtained by min-
imizing criteria (3) stem from the lack of any constraint on the
joint block diagonalizer B. Therefore, the resulting algorithm can-
not guarantee that the partitions of B are uncorrelated with each
other. This motivates the idea that some penalty term should be
used to eliminate the ill-conditioned solutions. To this end, we pro-
pose to consider the following cost function:
J (B) = J
1
(B) − J
2
(B), (11)
where
J
1
(B) =
K
k=1
OffB
B
H
R
k
B
2
F
(12)
represents the off-diagonal-block-error term and
J
2
(B) = log
det(B)
(13)
represents the penalty term, which is expected to rectify B to some
well-conditioned solution.
Note that the penalty term (13) will tend to negative infin-
ity when B is close to singular, this correspondingly cause the
cost function (11) tend to positive infinity. So minimizing (11) at
least guarantees that the joint block diagonalizer B is updated in
a direction apart from singular, which is expected to result in a
well-conditioned solution.
In order to minimize (11) with respect to B,weadoptacyclic
minimizer technique [29] to complete the optimization task. The
basic idea behind this strategy is that in the optimization process
we divide the parameter space into multiple sets. At each itera-
tion of the algorithm we minimize the criterion with respect to
one set conditioned on the previously estimated sets of parame-
ters. The newly estimated set is then used to update the remaining
sets. This process continues until convergence is achieved. The ad-
vantage of using cyclic minimizer technique is that the resulting
algorithms usually have fast convergence and there are no param-
eters to adjust.
Let us first divide B into r partitions as B
=[B
1
, B
2
,...,B
r
],
where
∀i ∈ I
r
, B
i
=[b
1
i
, b
2
i
,...,b
n
i
i
] is a block matrix of dimension
M
× n
i
. Now we rewrite (12) as
J
1
(B) =
K
k=1
r
i=1
r
j=1, j=i
B
H
i
R
k
B
j
2
F
=
K
k=1
r
i=1
r
j=1, j=i
tr
B
H
i
R
k
B
j
B
H
j
R
H
k
B
i
=
r
i=1
tr
B
H
i
K
k=1
R
k
r
j=1, j=i
B
j
B
H
j
R
H
k
B
i
.
(14)
Let B
(i)
=[B
1
,...,B
i−1
, B
i+1
,...,B
r
] be the M × (M − n
i
) matrix
obtained by deleting the i-th block from B.Wehave
r
j=1, j=i
B
j
B
H
j
= BB
H
− B
i
B
H
i
= B
(i)
B
(i)H
. (15)
Substituting (15) into (14), then (14) can be written as
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