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首页Haykin与Widrow合著:2003年LMS自适应滤波权威指南
《Haykin 和 Widrow - 2003 - 最小均方自适应滤波器》是一本由自适应滤波技术的原创发明者S.Haykin和B.Widrow共同编辑的权威著作。该书是领域内顶尖专家的贡献结晶,专为深入探讨Least-Mean-Square (LMS) 自适应滤波理论和技术而编写。作为业界唯一的综合性教材,它将LMS自适应滤波这一核心主题进行了全面且详尽的阐述。 LMS自适应滤波是一种在线学习算法,其核心理念在于通过最小化预测误差来不断调整滤波器系数,使其能够适应输入信号的动态变化。这种算法在信号处理、通信系统、控制系统等多个领域有着广泛应用,如噪声抑制、信号增强、无线通信中的干扰消除等。书中不仅介绍了LMS的基本原理,包括梯度下降法、权值更新公式(如mu-law准则)和收敛性分析,还涵盖了高级话题,如增量式LMS、复数域LMS、多输出系统以及LMS在神经网络和生物启发算法中的应用。 书中详细讲解了如何设计有效的初始化策略,如何选择合适的步长因子(mu),以及如何处理稳态误差等问题。此外,它还讨论了LMS算法的局限性和改进方法,比如RLS(Recursive Least Squares)算法,以及LMS在实际工程环境中的性能评估和优化技巧。 《Haykin 和 Widrow - 2003 - 最小均方自适应滤波器》不仅适合从事信号处理领域的研究人员,也对工程师和学生提供了深入理解LMS自适应滤波器理论和实践操作的重要参考。书中所包含的大量实例和仿真结果,使读者能够更好地理解和掌握这一关键的适应性信号处理技术。对于希望在这个快速发展的领域保持竞争力的专业人士来说,这无疑是一本不可或缺的专业读物。
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In the next section, we will briefly analyze the performance of the exact least
squares solution when the weights are obtained with a finite data sample. Then in
Sections 1.3 and 1.4, we will analyze the idealized LMS/Newton algorithm and, in
Section 1.5, show, at least heuristically, that its performance is equivalent to that of
an exact least squares algorithm. Based on this heuristic argument, we will view the
LMS/Newton process as an “optimal” gradient search algorithm.
In Section 1.6 we will define a class of nonstationary problems: problems in
which an unknown plant PðzÞ varies in a certain random way. Once again, the
adaptive filter will perform a modeling task. For this class of frequently encountered
problems, we will analyze and compare the performance of LMS/Newton with that
of the conventional steepest descent LMS algorithm. We will show that both
perform equivalently (in the mean square sense) for this class of nonstationary
problems. In Section 1.7, we will examin e the MSE learning curves and the transient
behavior of adaptive algorithms. The excess error energy will be defined to be the
area under the excess MSE curve. The LMS and LMS/ Newton algorithms will be
shown to perform, on average, equivalently with respect to this important criterion if
they both start learning from random initial conditions that have the same variance.
In Sections 1.8 and 1.9, we will conclude this chapter by summarizing the vario us
comparisons made between the LMS algorithm and the ideal LMS/Newton
algorithm.
1.2 LEARNING WITH A FINITE NUMBER OF DATA SAMPLES
Suppose that the adaptive linear combiner in Figure 1.2 is fed N independent zero-
mean K 1 training vectors uð1Þ; uð2Þ; ...; uðNÞ and their respective scalar desired
responses dð1Þ; dð2Þ ; ...; dðNÞ, all drawn from a wide-sens e stationary process.
Keeping the weights fixed, a set of N error equations can be written as
eðnÞ¼dðnÞu
T
ðnÞw; n ¼ 1; 2; ...; N: ð1:10Þ
The objective is to find a weight vector that minimizes the sum of the squares of the
error values based on the finite sample of N items of data.
Equation (1.10) can be written in matrix form as
e ¼ d Uw; ð1:11Þ
where U is an N K rectangular matrix
U ¼
4
½uð1Þ; uð2Þ; ...; uðNÞ
T
; ð1:12Þ
e is an N element error vector
e ¼
4
½eð1Þ; eð2Þ; ...; eðNÞ
T
; ð1:13Þ
1.2 LEARNING WITH A FINITE NUMBER OF DATA SAMPLES 5
![](https://csdnimg.cn/release/download_crawler_static/11227898/bg11.jpg)
and d is an N element vector of desired responses
d ¼
4
½dð1Þ; dð2Þ ; ...; dðNÞ
T
: ð1:14Þ
A unique solution of Eq. (1.11), a weight vector w that brings e to zero, exists only if
U is square and nonsingular. However, the case of greatest interest is that of N K.
As such, Eq. (1.11) would typically be overconstrained and one would generally
seek a best least squares solution. The sum of the squares of the errors is
e
T
e ¼ d
T
d þ w
T
U
T
Uw 2d
T
Uw: ð1:15Þ
This sum multiplied by 1/N is an estimate
^
jj
of the MSE
j
. Thus,
^
jj
¼
1
N
e
T
e ð1:16Þ
and
lim
N!1
^
jj
¼
j
: ð1:17Þ
Note that
^
jj
is a quadratic function of the weights. The parameters of the quadratic
form are related to properties of the N data samples. ðU
T
UÞ is square and is assumed
to be positive definite.
^
jj
is a small-sample-size MSE function.
j
is the large-sample-
size “true” MSE function, and it is also a quadratic function of the weights. Figure
1.3 shows a comparative sketch of these functions. Many small-sample-size data-
dependent curves are possible, but there is only one large-sample-size curve. The
unique large-sample-size curve is the average of the many small-sample-size curves.
Figure 1.3 Small- and large-sample MSE curves.
6
ON THE EFFICIENCY OF ADAPTIVE ALGORITHMS
![](https://csdnimg.cn/release/download_crawler_static/11227898/bg12.jpg)
The minimum of a small-sample-size function can be found by differentiating
Eq. (1.15) and setting the derivat ive to zero. The result is
w
LS
¼ðU
T
UÞ
1
U
T
d: ð1:18Þ
This is the exact least squares solution for the given data sample. The Wiener
solution w
o
is the expected value of w
LS
.
Each small-sample-size curve is an ensembl e member. Let the ensemble be
constructed in the following manner. Assume that the vectors uð1Þ; uð2 Þ; ...; uðNÞ
are the same for all ensemble members but that the associated desired responses
dð1Þ; dð2Þ; ...; dðNÞ differ from one ensemble member to another because of the
stochastic character of plant noise (refer to Fig. 1.2). Over this ensemble, therefore,
the U matrix is constant, while the desired response vector d is stochastic. In order to
evaluate the excess MSE due to adaptation with the finite amount of data available,
we have to find
j
excess
¼
1
N
E½
e
T
U
T
U
e
; ð1:19Þ
where the error in the weight vector is
e
¼
4
w
LS
w
o
: ð1:20Þ
Expectation is taken over the above-described ensemble. Equation (1.19) can be
written as
j
excess
¼
1
N
TrðE½
e
T
U
T
U
e
Þ ¼
1
N
E½Trð
e
T
U
T
U
e
Þ
¼
1
N
E½Trð
e e
T
U
T
UÞ ¼
1
N
TrðE½
e e
T
U
T
UÞ:
ð1:21Þ
The covariance matrix of the weight error vector
e
, E½
e e
T
, can be computed as
follows. First, note that
e
¼ðU
T
UÞ
1
U
T
d w
o
¼ðU
T
UÞ
1
U
T
ðd Uw
o
Þ: ð1:22Þ
Then, for small
e
,
2
E½
ee
T
¼ðU
T
UÞ
1
U
T
E½ðd Uw
o
Þðd Uw
o
Þ
T
UðU
T
UÞ
1
ðU
T
UÞ
1
j
min
;
ð1:23Þ
2
For the sake of simplicity, we assumed here that the plant noise vðnÞ in Figure 1.2 is white and that the
adaptive filter has enough weights to match the unknown plant.
1.2 LEARNING WITH A FINITE NUMBER OF DATA SAMPLES
7
![](https://csdnimg.cn/release/download_crawler_static/11227898/bg13.jpg)
where
j
min
is the minimum MSE, the minimum of the true MSE function (see
Fig. 1.3).
Substitution of Eq. (1.23) into Eq. (1.21) yields
j
excess
¼
K
N
j
min
: ð1:24Þ
It is important to note that this formula does not depend on U. The above-described
ensemble can be generalized to an ensemble of ensembles, each having its own U,
without changing Eq. (1.24). Hence, this formula is valid for a very wide class of
inputs.
It is useful to consider a dimensionless ratio between the excess MSE and the
minimum MSE. This ratio is commonly called (see, e.g., [1, 2, 4]) the mis-
adjustment, M. For the exact least squares solution based on learning with a finite
data sample, we find the misadjustment from Eq. (1.24) as
M ¼
K
N
¼
ðnumber of weightsÞ
ðnumber of independent training samplesÞ
: ð1:25Þ
The misadjustment is a dimensionless measure of how far on average the optimal
finite-data solution deviates from the optimal infinite-data Wiener solution. There
are many ways to find the optimal finite-data solution, such as adaptive algorithms of
various types and analytical methods based on Eq. (1.18) and Eqs. (1.12) and (1.14).
The particularities of the algorithm for finding the optimal finite-data solution have
no effect on this solution or on the resulting misadjustment.
The misadjustment formula in Eq. (1.25) was first presented without detailed
proof by Widrow and Hoff [1] in 1960 and has been used for many years in pattern
recognition studies. For small valu es of M (less than 25 percent), it has proven to be
an excellent approximation. A formula similar to Eq. (1.25), based on somewhat
different assumptions, was derived by Davisson [43] in 1970. A more exact formula
was derived by Widrow and Walach [31, App. A].
1.3 STEADY FLOW LEARNING WITH THE LMS ALGORITHM
Gradient methods are commonly used to adjust adaptive parameters in order to
search the quadratic MSE performance function for its minimum. Most widely used
is the method of steepest descent. With this method, a sequence of changes is mad e
to the weight vector along the direction of the negative gradient. Thus, the next
weight vector, wðn þ 1Þ, is made equal to the present weight vector, wðnÞ, plus a
change proportional to the negative gradient at the nth iteration:
wðn þ 1Þ¼wðnÞ
m
7ðnÞ: ð1:26Þ
The parameter
m
controls stability and rate of convergence. An “instantaneous
gradient”
^
77ðnÞ, an estimate of the true gradient 7ðnÞ, can be found by differentiation
of Eq. (1.5) with respect to w:
^
77ðnÞ¼2eðnÞuðnÞ: ð1:27Þ
8 ON THE EFFICIENCY OF ADAPTIVE ALGORITHMS
![](https://csdnimg.cn/release/download_crawler_static/11227898/bg14.jpg)
This is a noisy but unbiased estimate of the gradient [5, p. 101]. Using this
instantaneous gradient in place of the true gradient in Eq. (1.26) yields the LMS
algorithm of Widrow and Hoff:
wðn þ 1Þ¼wðnÞþ2
m
eðnÞuðnÞ: ð1:28Þ
The behavior of this algorithm has been analyzed extensively in the literature (see,
e.g., [2–4, 44–51]). It was proved in [2] and [4] that if the adaptation constant
m
were chosen such that
0 ,
m
,
1
TrðRÞ
; ð1:29Þ
then the adaptive weights would relax from their initial condition to hover randomly
about the Wiener solution w
o
. The weight error vector
e
ðnÞ¼wðnÞw
o
ð1:30Þ
will then converge to zero in the mean, and its variance will be stable ([2– 4, 47, 52,
53]). The relaxation process will be gover ned by the relation
E½
e
ðn þ 1Þ ¼ ðI 2
m
RÞE ½
e
ðnÞ: ð1:31Þ
Therefore, there will be K different modes of convergence corresponding to K
eigenvalues of the autocorrelation matrix R. Using normal decomposition of the
matrix R:
R ¼ QLQ
T
ð1:32Þ
QQ
T
¼ I; L ¼
l
1
0
.
.
.
0
l
K
2
6
6
4
3
7
7
5
; ð1:33Þ
we can find the corresponding K time constants
t
i
of the weight relaxation process as
ml
i
! 1;
t
i
¼
1
2
ml
i
; 1 i K: ð1:34aÞ
As the weights relax toward the Wiener solution, the MSE, a quadratic function of
the weights, undergoes a geometric progression toward
j
min
. The “learning curve” is
a plot of MSE versus number of adaptation cycles. The natural modes of the learning
curve have time constants half as large as the corresponding time constants of the
1.3 STEADY FLOW LEARNING WITH THE LMS ALGORITHM 9
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