随机信号与应用卡尔曼滤波导论（第4版）

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"Introduction to Random Signals and Applied Kalman Filtering 4th edition" 是一本由 Robert Grover Brown 和 Patrick Y. C. Hwang 合著的经典卡尔曼滤波书籍，专注于应用层面的讲解，并在第四版中增加了 MATLAB 练习题目。本书主要介绍了随机信号处理和卡尔曼滤波的基本理论与实践应用。前三个章节为读者提供了随机过程理论的基础，涵盖了线性系统对随机输入的响应。第四章详细讨论了维纳滤波，后续内容则全面深入地探讨了各种类型的卡尔曼滤波技术，特别强调其实际应用。每章末尾的星标问题设计为计算机练习，鼓励读者通过编程实现和分析具体例子，以增进对工程实践中关键概念的理解。卡尔曼滤波是一种有效的估计理论方法，尤其适用于处理存在噪声的动态系统中的数据。它利用系统的状态方程和测量方程，通过递推方式更新状态估计，以最小化估计误差的均方值。在随机信号处理领域，卡尔曼滤波器是解决非理想环境中数据滤波、预测和融合的关键工具。书中结合 MATLAB 进行练习，有助于读者更好地掌握理论知识，并将其转化为实际技能。MATLAB 是一种广泛使用的数值计算软件，对于模拟和分析复杂系统非常有用，特别是在处理信号处理和控制系统的设计上。通过对本书的学习，读者可以了解到如何应用随机过程理论来理解和分析随机信号，以及如何设计和实现卡尔曼滤波器以优化系统性能。这不仅对学术研究有重要价值，而且对于从事通信、导航、控制工程、航空航天、自动控制、图像处理等领域的工程师来说，也是必备的技能之一。总结起来，"Introduction to Random Signals and Applied Kalman Filtering 4th edition" 是一本深入浅出的卡尔曼滤波教程，它通过实例和 MATLAB 练习，帮助读者建立起对随机信号处理和高级滤波技术的深刻理解，是学习和提升相关技能的理想教材。

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Probability and Random

Variables: A Review

1.1

RANDOM SIGNALS

Nearly everyone has some notion of random or noiselike signals. One has only to

tune a radio away from a station, turn up the volume, and the result is static, or noise.

If one were to look at a recording of such a signal, it would appear to wander on

aimlessly with no apparent order in its amplitude pattern, as shown in Fig. 1.1.

Signals of this type cannot be described with explicit mathematical functions such

as sine waves, step functions, and the like. Their description must be put in

probabilistic terms. Early investigators recognized that random signals could be

described loosely in terms of their spectral content, but a rigorous mathematical

description of such signals was not formulated until the 1940s, most notably with

the work of Wiener and Rice (1, 2).

Probability plays a key role in the description of noiselike or random signals. It

is especially important in Kalman ﬁltering, which is also sometimes referred to as

statistical ﬁltering. This is a bit of a misnomer though, because Kalman ﬁltering is

based on probabilistic descriptors of the signals and noise, and these are assumed to

be known at the beginning of the ﬁltering problem. Recall that in probability we

assume that we have some a priori knowledge about the likelihood of certain

elemental random outcomes. Then we wish to predict the theoretical relative

frequency of occurrence of combinations of these outcomes. In statistics we do

just the reverse. We make observations of random outcomes. Then, based on these

observations, we seek mathematical models that faithfully represent what we have

observed. In Kalman ﬁltering it may well be that some statistical methods were used

in arriving at the necessary probabilistic descriptors. But, nevertheless, once the

descriptors are determined (or assumed) it is all probability the rest of the way.

Our treatment of probability must necessarily be brief and directed toward the

speciﬁc needs of subsequent chapters. We make no apology for this, because many

ﬁne books have been written on probability in a rigorous sense. Our main objective

here is the study of random signal analysis and Kalman ﬁltering, and we wish to

move on to this as quickly as possible. First, though, we must at least review the bare

essentials of probability with special emphasis on random variables.

C01 12/09/2011 9:40:52 Page 4

1.2

INTUITIVE NOTION OF PROBABILITY

Most engineering and science students have had some acquaintance with the

intuitive concepts of probability. Typically, with the intuitive approach we ﬁrst

consider all possible outcomes of a chance experiment as being equally likely,

and then the probability of a particular combination of outcomes, say, event A,is

deﬁned as

PðAÞ¼

Possible outcomes favoring event A

Total possible outcomes

(1.2.1)

where we read P(A) as “probability of event A.” This concept is then expanded to

include the relative-frequency-of-occurrence or statistical viewpoint of probability.

With the relative-frequency concept, we imagine a large number of trials of some

chance experiment and then deﬁne probability as the relative frequency of

occurrence of the event in question. Considerations such as what is meant by

“large” and the existence of limits are normally avoided in elementary treatments.

This is for good reason. The idea of limit in a probabilistic sense is subtle.

Although the older intuitive notions of probability have limitations, they still

play an important role in probability theory. The ratio-of-possible-events concept is

a useful problem-solving tool in many instances. The relative-frequency concept is

especially helpful in visualizing the statistical signiﬁcance of the results of

probability calculations. That is, it provides the necessary tie between the theory

and the physical situation. An example that illustrates the usefulness of these

intuitive notions of probability should now prove useful.

Figure 1.1 Typical noise waveform.

EXAMPLE 1.1

In straight poker, each player is dealt 5 cards face down from a deck of 52 playing

cards. We pose two questions:

(a) What is the probability of being dealt four of a kind, that is, four aces, four

kings, and so forth?

(b) What is the probability of being dealt a straight ﬂush, that is, a continuous

sequence of ﬁve cards in any suit?

Solution to Question (a) This problem is relatively complicated if you think in

terms of the sequence of chance events that can take place when the cards are dealt

one at a time. Yet the problem is relatively easy when viewed in terms of the ratio

of favorable to total number of outcomes. These are easily counted in this case.

There are only 48 possible hands containing 4 aces; another 48 containing 4 kings;

etc. Thus, there are 13  48 possible four-of-a-kind hands. The total number of

4 CHAPTER 1 PROBABILITY AND RANDOM VARIABLES: A REVIEW

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1.3

AXIOMATIC PROBABILITY

It should be apparent that the intuitive concepts of probability have their limitations.

The ratio-of-outcomes approach requires the equal-likelihood assumption for all

outcomes. This may ﬁt many situations, but often we wish to consider “unfair”

chance situations as well as “fair” ones. Also, there are situations where all possible

outcomes simply cannot be enumerated. The relative-frequency approach is intui-

tive by its very nature. Intuition should never be ignored; but, on the other hand, it

can lead one astray in complex situations. For these reasons, the axiomatic

formulation of probability theory is now almost universally favored among both

applied and theoretical scholars in this area. As we would expect, axiomatic prob-

ability is compatible with the older, more heuristic probability theory.

Axiomatic probability begins with the concept of a sample space. We ﬁrst

imagine a conceptual chance experiment. The sample space is the set of all possible

outcomes of this experiment. The individual outcomes are called elements or points

in the sample space, We denote the sample space as S and its set of elements as

; s

; ...g. The number of points in the sample space may be ﬁnite, countably

inﬁnite, or simply inﬁnite, depending on the experiment under consideration.

It should be noted that elements of a sample space must always be mutually

exclusive or disjoint. On a given trial, the occurrence of one excludes the occurrence

of another. There is no overlap of points in a sample space.

possible poker hands of any kind is obtained from the combination formula for “52

things taken 5 at a time” (3). This is given by the binomial coefﬁcient



52!

5!ð52  5Þ!

52  51  50  49  48

5  4  3  2  1

¼ 2; 598; 960 (1.2.2)

Therefore, the probability of being dealt four of a kind is

Pðfour of a kind Þ¼

13  48

2; 598; 960

624

2; 598; 960

 :00024 (1.2.3)

Solution to Question (b) Again, the direct itemization of favorable events is the

simplest approach. The possible sequences in each of four suits are: AKQJ10,

KQJ109, . . . , 5432A. (Note: We allow the ace to be counted either high or low.)

Thus, there are 10 possible straight ﬂushes in each suit (including the royal ﬂush of

the suit) giving a total of 40 possible straight ﬂushes. The probability of a straight

ﬂush is, then,

PðStraight flushÞ¼

2; 598; 960

 :000015 (1.2.4)

We note in passing that in poker a straight ﬂush wins over four of a kind; and,

rightly so, since it is the rarer of the two hands.

1.3 AXIOMATIC PROBABILITY 5

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In axiomatic probability, t he term event has special meaning and should not

be used interchangeably with outcome. An event is a special subset of the sample

space S. We usually wish to consider various events deﬁned on a sample space,

and they will be denoted with uppercase letters such as A, B, C,...,orperhaps

, A

, . . . , etc. Also, we will have occasion to consider the set of operations of

union, intersection, and complement of our de ﬁned events. Thus, we must be

careful in our deﬁnition of events to make the set sufﬁciently complete such that

these set operations also yield properly deﬁned events. In discrete problems, this

can always be done by deﬁning the set of events under consideration to be all

possible subsets of the sample space S. We will tacitly assume that the null set is a

subset of every set, and that every set is a subset of itself.

One other comment about events is in order before proceeding to the basic

axioms of probability. The event A is said to occur if any point in A occurs.

The three axioms of probability may now be stated. Let S be the sample space

and A be any event deﬁned on the sample space. The ﬁrst two axioms are

Axiom 1: PðAÞ0 (1.3.1)

Axiom 2: PðSÞ¼1 (1.3.2)

Now, let A

, A

, . . . be mutually exclusive (disjoint) events deﬁned on S. The

sequence may be ﬁnite or countably inﬁnite. The third axiom is then

Axiom 3: PðA

[ A

[ ...Þ

¼ PðA

ÞþPðA

Þþ

(1.3.3)

Axiom 1 simply says that the probability of an event cannot be negative. This

certainly conforms to the relative-frequency-of-occurrence concept of probability.

Axiom 2 says that the event S, which includes all possible outcomes, must have a

probability of unity. It is sometimes called the certain event. The ﬁrst two axioms

are obviously necessary if axiomatic probability is to be compatible with the older

relative-frequency probability theory. The third axiom is not quite so obvious,

perhaps, and it simply must be assumed. In words, it says that when we have

nonoverlapping (disjoint) events, the probability of the union of these events is the

sum of the probabilities of the individual events. If this were not so, one could easily

think of counterexamples that would not be compatible with the relative-frequency

concept. This would be most undesirable.

We now recapitulate. There are three essential ingredients in the formal

approach to probability. First, a sample space must be deﬁned that includes all

possible outcomes of our conceptual experiment. We have some discretion in what

we call outcomes, but caution is in order here. The outcomes must be disjoint and

all-inclusive such that P(S) ¼1. Second, we must carefully deﬁne a set of events on

the sample space, and the set must be closed such that me operations of union,

intersection, and complement also yield events in the set. Finally, we must assign

probabilities to all events in accordance with the basic axioms of probability. In

physical problems, this assignment is chosen to be compatible with what we feel to

be reasonable in terms of relative frequency of occurrence of the events. If the

sample space S contains a ﬁnite number of elements, the probability assignment is

usually made directly on the elements of S. They are, of course, elementary events

6 CHAPTER 1 PROBABILITY AND RANDOM VARIABLES: A REVIEW

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