Haykin与Widrow合著：2003年LMS自适应滤波权威指南

自适应滤波

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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

eðnÞ¼dðnÞu

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

½uð1Þ; uð2Þ; ...; uðNÞ

; ð1:12Þ

e is an N element error vector

e ¼

½eð1Þ; eð2Þ; ...; eðNÞ

; ð1:13Þ

1.2 LEARNING WITH A FINITE NUMBER OF DATA SAMPLES 5

where

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

excess

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

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

7ðnÞ: ð1:26Þ

The parameter

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:

77ðnÞ¼2eðnÞuðnÞ: ð1:27Þ

8 ON THE EFFICIENCY OF ADAPTIVE ALGORITHMS

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