第五版：自适应滤波器理论详解——语音与图像处理

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《自适应滤波器(第五版－－英文版)》是由Simon Haykin编著，出自McMaster University的Communications Research Laboratory。本书专注于语音和图像处理领域的自适应滤波技术，是该领域的重要参考文献。作者以深入浅出的方式探讨了滤波问题的本质及其在通信工程中的应用。首先，章节"Filtering Problem"介绍了滤波的基本概念，解释了为何在信号处理中需要过滤掉噪声或提取有用信息。章节"Linear Optimum Filters"着重于线性最优滤波器的设计，这些滤波器在理想情况下可以提供最小化的误差，但在实际应用中可能受限于计算复杂性和实时性。 "Adaptive Filters"部分是本书的核心，讲述了自适应滤波器的理论基础，即它们如何通过学习和调整参数来适应输入信号的变化，如自适应均值滤波、自适应加权均值滤波和自适应无限 impulse response (IIR)滤波器。这里涵盖了自适应算法，如LMS (Least Mean Square) 和 RLS (Recursive Least Squares) 等。 "Linear Filter Structures"详细讨论了不同类型的线性滤波结构，包括但不限于无限冲激响应(IIR)滤波器和有限冲激响应(FIR)滤波器，以及它们的优缺点和适用场景。 "Approaches to the Development of Linear Adaptive Filters"则深入探讨了设计自适应滤波器的各种方法和技术，包括步进算法、增量算法以及基于模型的方法，帮助读者理解如何根据具体应用选择合适的算法。 "Adaptive Beamforming"章节涉及自适应阵列处理，这是一种在无线通信和声纳系统中广泛使用的自适应滤波应用，用于提高信号的方向性和抗干扰能力。 "Four Classes of Applications"列举了自适应滤波技术在多个领域的实际应用，包括信号处理、通信系统、音频处理、图像处理和生物医学信号分析等，展示了其广泛的实用价值。 "Historical Notes"部分回顾了自适应滤波技术的发展历程，介绍了关键的研究里程碑和重要人物，为读者提供了研究背景和历史视角。整个书籍通过实例、理论分析和实用技巧，为读者提供了全面而深入的自适应滤波器理论知识和实践指导，对于从事相关专业研究和工程实践的学生和工程师来说，是一部不可或缺的参考书。

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20 Background and Preview

where t denotes continuous time, h(t) designates the impulse response, A is an ampli-

tude-scaling factor, d(t) is the Dirac delta function (or unit impulse function), and t

denotes the propagation delay incurred in the course of transmitting the signal over

the channel. Equation (1) is the time-domain description of an ideal transmission

medium. Equivalently, we may characterize it in the frequency domain by writing

H1j

2= A

exp

1-j

(2)

where j is the square root of –1, v denotes angular frequency, H(jv) is the fre-

quency response of the transmission medium, and exp (·) stands for the exponen-

tial function. In practice, it is impossible for any physical channel to satisfy the

stringent requirements embodied in the idealized time-domain description given

by Eq. (1) or the equivalent frequency-domain description set forth in Eq.(2):

The best that we can do is to approximate Eq. (2) over a band of frequencies

representing the essential spectral content of the transmitted signal, which makes

the physical channel dispersive. In a digital communication system, this channel

impairment gives rise to intersymbol interference—a smearing of the successive

pulses (representing the transmitted sequence of 1’s and 0’s) into one another with

the result that they are no longer distinguishable.

 • Noise. Some form of noise is present at the output of every communication chan-

nel. The noise can be internal to the system, as in the case of thermal noise gener-

ated by an amplifier at the front end of the receiver, or external to the system due

to interfering signals originating from other sources.

The net result of the two impairments is that the signal received at the channel output

is a noisy and distorted version of the signal that is transmitted. The function of the

receiver is to operate on the received signal and deliver a reliable estimate of the original

message signal to a user at the output of the system.

As another example involving the use of filter theory, consider the situation

depicted in Fig. 2, which shows a continuous-time dynamic system whose state at time t

is denoted by the multidimensional vector x(t). The equation describing evolution of the

state x(t) is usually subject to system errors. The filtering problem is complicated by the

fact that x(t) is hidden and the only way it can be observed is through indirect measure-

ments whose equation is a function of the state x(t) itself. Moreover, the measurement

equation is subject to unavoidable noise of its own. The dynamic system depicted in

Fig. 2 may be an aircraft in flight, in which case the position and velocity of the aircraft

constitute the elements of the state x(t), and the measurement system may be a tracking

radar. In any event, given the observable vector y(t) produced by the measuring system

Dynamic

system

Measuring

system

Estimator

State

x(t)

Observation

y(t)

Estimate

of the state

x(t)

System

errors

Measurement

errors

Prior

information

FIGURE 2 Block diagram depicting the components involved in state estimation, namely

1t2.

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Section 1 The Filtering Problem 21

over the interval [0, T], and given prior information, the requirement is to estimate the

state x(t) of the dynamic system.

Estimation theory, illustrated by the two examples just described, is statistical in

nature because of the unavoidable presence of noise or system errors contaminating the

operation of the system being studied.

Three Basic Kinds of Estimation

The three basic kinds of information-processing operations are filtering, smoothing, and

prediction, each of which may be performed by an estimator. The differences between

these operations are illustrated in Fig. 3:

 • Filtering is an operation that involves the extraction of information about a quan-

tity of interest at time t by using data measured up to and including t.

 • Smoothing is an a posteriori (i.e., after the fact) form of estimation, in that data

measured after the time of interest are used in the estimation. Specifically, the

smoothed estimate at time t′ is obtained by using data measured over the interval

[0, t′], where t′ < t. There is therefore a delay of t − t′ involved in computing the

smoothed estimate. The benefit gained by waiting for more data to accumulate is

that smoothing can yield a more accurate estimate than filtering.

 • Prediction is the forecasting side of estimation. Its aim is to derive information

about what the quantity of interest will be like at some time t + t in the future (for

some t > 0) by using data measured up to and including time t.

From the figure, it is apparent that both filtering and prediction are real-time operations,

whereas smoothing is not. By a real-time operation, we mean an operation in which the

estimate of interest is computed on the basis of data available now.

Span of available data used

in ltering at time t

Origin

Time

(a)

Span of available data used

in smoothing at time t

Span of available data used

in prediction at time t + t

tt

Origin

Time

(b)

tt + t

Origin

Time

(c)

FIGURE 3 Illustrating the three basic forms of estimation: (a) filtering; (b) smoothing;

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2. LINEAR OPTIMUM FILTERS

We may classify filters as linear or nonlinear. A filter is said to be linear if the filtered,

smoothed, or predicted quantity at the output of the filter is a linear function of the

observations applied to the filter input. Otherwise, the filter is nonlinear.

In the statistical approach to the solution of the linear filtering problem, we assume

the availability of certain statistical parameters (i.e., mean and correlation functions) of

the useful signal and unwanted additive noise, and the requirement is to design a linear

filter with the noisy data as input so as to minimize the effects of noise at the filter output

according to some statistical criterion. A useful approach to this filter-optimization prob-

lem is to minimize the mean-square value of the error signal defined as the difference

between some desired response and the actual filter output. For stationary inputs, the

resulting solution is commonly known as the Wiener filter, which is said to be optimum

in the mean-square-error sense. A plot of the mean-square value of the error signal ver-

sus the adjustable parameters of a linear filter is referred to as the error-performance

surface. The minimum point of this surface represents the Wiener solution.

The Wiener filter is inadequate for dealing with situations in which nonstationarity

of the signal and/or noise is intrinsic to the problem. In such situations, the optimum fil-

ter has to assume a time-varying form. A highly successful solution to this more difficult

problem is found in the Kalman filter, which is a powerful system with a wide variety of

engineering applications.

Linear filter theory, encompassing both Wiener and Kalman filters, is well devel-

oped in the literature for continuous-time as well as discrete-time signals. However, for

technical reasons influenced by the wide availability of computers and the ever increas-

ing use of digital signal-processing devices, we find in practice that the discrete-time

representation is often the preferred method. Accordingly, in subsequent chapters, we

only consider the discrete-time version of Wiener and Kalman filters. In this method of

representation, the input and output signals, as well as the characteristics of the filters

themselves, are all defined at discrete instants of time. In any case, a continuous-time

signal may always be represented by a sequence of samples that are derived by observing

the signal at uniformly spaced instants of time. No loss of information is incurred during

this conversion process provided, of course, we satisfy the well-known sampling theorem,

according to which the sampling rate has to be greater than twice the highest frequency

component of the continuous-time signal. We may thus represent a continuous-time

signal u(t) by the sequence u(n), n = 0, ;1, ;2, . . . , where for convenience we have nor-

malized the sampling period to unity, a practice that we follow throughout the book.

3. ADAPTIVE FILTERS

The design of a Wiener filter requires a priori information about the statistics of the

data to be processed. The filter is optimum only when the statistical characteristics of

the input data match the a priori information on which the design of the filter is based.

When this information is not known completely, however, it may not be possible to

design the Wiener filter or else the design may no longer be optimum. A straightforward

approach that we may use in such situations is the “estimate and plug” procedure. This

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