随机最优与自适应信号处理导论

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"An Introduction to Random Optimum and Adaptive Signal Processing - Doug Gray" 本书"随机最优与自适应信号处理简介"由Doug Gray撰写，是CSSIP领域的一份重要资料。书中主要探讨了以下几个关键主题： 1. 优化：本书专注于二次成本函数的优化问题。在信号处理领域，优化通常涉及寻找最佳滤波器参数或算法，以最小化某个性能指标（如均方误差）。 2. 自适应：讨论了一系列迭代收敛至最优解并跟踪最优解的算法。这些算法适用于不断变化的环境，例如在噪声或干扰下的信号处理。 3. 统计信号：对随机过程理论进行了深入讲解，这是理解和分析随机信号的基础。了解随机过程有助于预测和估计信号的行为。 4. 离散时间线性滤波器：作为主要考虑的对象，这些滤波器处理实信号，但也可应用于处理具有正交成分的复信号，如在多天线系统中的应用。 5. 谱分析：在约束优化和子空间方法的背景下进行，这在频谱估计和信号分类中有重要应用。 6. 最优线性预测：由于其在估计理论中的核心地位以及作为格子滤波器介绍的重要性，本书对此进行了详细讨论。最优线性预测涉及预测未来信号值，最小化预测误差。书中各章节详细展开如下： - 第1章：预备知识：介绍了优化和自适应的基本原理，讨论了自适应系统的特点和结构，包括具体应用的详细结构以及系统问题。 - 第2章：随机过程理论：涵盖了概率密度函数、平稳随机过程、集合平均、遍历性、协方差、高斯随机过程、无相关随机过程、带延迟线的协方差矩阵估计、平均输出功率、平稳随机过程的Z变换以及传统谱估计的统计特性。 - 第3章：最优离散时间线性滤波：讨论了均方误差（MSE）曲面，以及如何最小化该误差以设计最优滤波器。这部分内容对于理解滤波器性能和设计至关重要。通过深入学习这本书，读者将能够掌握随机过程和自适应信号处理的关键概念，从而在实际的通信、雷达、音频处理等应用中设计和分析高效的信号处理系统。

CHAPTER 2

Theory of Random Processes

2.1. Probability Density Functions

A random variable (r.v.) is measurable in the sense that a probability can be associ-

ated with it, i.e. p(x). Note formally p

(x) means the probability that the random

variable X takes the value x, but in these notes the abridged notation p(x) will be

used. The Gaussian, p(x) = (2πσ

)

−1/2

exp(−x

/2σ

), is a well-known example

and will be written using the terminology: x ∼ N(µ, σ

It is often better to deal with the distribution function, P (x), which denotes Pr[X <

x] and can be shown to be

P (x) =

−∞

p(u) du.

p(x) can be loosely thought of as the probability that random variable, X, lies

within a small region centred around x.

N Multiple random variables, or a real random vector, are deﬁned by a joint prob-

ability density function

p(x

, x

, . . . , x

)

where p(·) denotes the probability that each component of a random vector takes

the values x

, x

, . . . , x

respectively.

Note that a complex random variable can be treated as a two dimensional random

vector, e.g. a complex random variable z = u + jv has the p(z) deﬁned by p(u, v)

and denotes the probability of the real part of the random variable z lying close to

u and the imaginary part lying close to v.

EXERCISE. The components of the complex random variable z are uniformly dis-

tributed over the region |z| < 1. Find the probability that z lies in the region

|z| <

√

Statistical Independence: Two random variables x and y are statistically inde-

pendent if p(x, y) = p(x)p(y).

Example 2.1. N = 2, x and y are Gaussian statistically independent if

p(x, y) = 1/(2πσ

) exp

−1/2

(x/σ

)

+ (y/σ

)

ª¢

Example 2.2. 64 points are located on a square grid. Let e

and f

be the events

that a point is placed at random on the j

row and k

column.

Then p(e

) = p(f

) = 1/8 and p(e

, f

) = 1/64 = p(e

)p(f

Some good references for this section are

Davenport and Root, Chapters 2, 3, 4

Carlson, Chapters 4, 5

C. W. Therrien, Discrete Random Signals and Statistical Signal Processing

2.3. ENSEMBLE AVERAGING 18

Stationarity implies things not changing with time. For most of the random pro-

cesses that we will consider the sample values won’t be ﬁxed in time for a particu-

lar realisation so we need to come up with an appropriate deﬁnition of stationarity.

Again we use the joint pdf of our sampled random process and we deﬁne a random

process to be stationary if all joint pdf’s are independent of time. Formally this is

deﬁned as

p(x[k

] = u

, x[k

] = u

, . . .) = p(x[k

+ L]) = u

, x[k

+ L] = u

, . . .).

Note this has to hold for all N and for all L.

2.3. Ensemble Averaging

Given a random variable and functions of that random variable we want to deﬁne

quantities that measure properties of that random variable that are indendent of the

outcome of a particular experiment but reﬂect the outcome of all possible experi-

ments. This collection of all possible outcomes as deﬁned above is the ensemble

and when we average a quantity over all possible outcomes we form what we call

the ensemble average or expectation of a random variable. For our purposes if

f(x) is some function of our random variable then we formally deﬁne the expecta-

tion to be

{f(x)} =

∞

−∞

f(x)p(x) dx.

In practice, when there is no ambiguity as to the values of the random variable

over which we are integrating, we usually drop the subscript X. (N.B. The strong

intuitive concept of linear averaging here.)

Some examples are

(a) the mean m

= µ = E

{x} =

∞

−∞

xp(x) dx

(b) the variance, i.e., m

− µ

= E

{(x − µ)

} =

∞

−∞

(x − µ)

p(x) dx

} =

∞

−∞

p(x) dx

and

(d) the entropy = E

{log(p(x))} = −

∞

−∞

p(x) log(p(x)) dx.

The ensemble average of some function f(x[k], x[k + 1], . . . , x[k + N]) of the

samples of a random process is formally deﬁned by

E{f(x[k], x[k + 1], . . . , x[k + N])}

···

f(u

, u

, . . . u

)

× p(x[k] = u

, x[k + 1] = u

, . . . , x[k + N] = u

) du

···du

Since the joint pdf of the samples of a stationary random process are independent of

time it follows that ensemble averages of functions of a stationary random process

will be independent of time. For example, the mean of a stationary random process

is independent of time, i.e.,

E{x[k

]} =

p(x[k

] = u

) du

p(x[k

] = u

) du

= E{x[k

]}

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