MATLAB基础：数字信号处理入门指南

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"《数字信号处理基础：MATLAB®应用教程》是由罗伯特·J·舒林和桑德拉·L·哈里斯合著的一本教材，专为那些希望深入了解数字信号处理(Digital Signal Processing, DSP)原理并掌握MATLAB这一强大工具的学生和工程师设计。本书旨在通过实例讲解和互动式学习，引导读者探索数字信号处理的基础理论和实践操作。在本书中，作者深入浅出地阐述了数字信号处理的核心概念，如离散时间信号、傅立叶变换、滤波器设计、采样定理等，并通过MATLAB的函数和工具箱来实现这些理论。读者将学习如何使用MATLAB进行信号分析、频域表示、时域表示转换，以及如何利用其强大的可视化功能来理解复杂信号处理过程。章节内容涵盖了数字信号的采集、预处理、特征提取、滤波、频谱分析等多个关键环节，同时提供了一系列实际项目，帮助读者将理论知识应用于实际工程场景。此外，书中还包含了对MATLAB编程的最佳实践和技巧的指导，确保读者能够在处理实时信号和解决实际问题时得心应手。值得注意的是，这本书的目标受众是教育机构和学习者，它强调的是学习和研究的目的，而非商业用途。若读者在使用过程中发现版权问题，应及时与出版商联系删除或获取授权。版权信息明确指出，未经许可，书中的任何部分都不能以任何形式复制、传输或机械式存储使用，包括但不限于复印和电子版传播。《数字信号处理基础：MATLAB®应用教程》是一本极具价值的学习资源，适合那些想要在数字信号处理领域深化理解并提升MATLAB技能的专业人士，无论是学生还是从事相关工作的工程师，都能从中受益匪浅。"

Schilling-1120949 book November 12, 2010 9:33

12 Chapter 1 Signal Processing

Here, T > 0 is the time between samples or sampling interval in seconds. The sample spacingSampling interval

also can be speciﬁed using the reciprocal of the sampling interval which is called the samplingSampling frequency

frequency, f



Hz (1.2.2)

Here, the unit of Hz is understood to mean samples/second. Notice that the integer k in (1.2.1)

denotes discrete time or, more speciﬁcally, the sample number. The sampling interval T is

Discrete-time

left implicit on the left-hand side of (1.2.1) because this simpliﬁes subsequent notation. In

those instances where the value of T is important, it will be stated explicitly. An example of

a discrete-time signal generated by sampling the continuous-time signal in Figure 1.8 using

T = .25 seconds is shown in Figure 1.9.

Just as the independent variable can be continuous or discrete, so can the dependent variable

or amplitude of the signal be continuous or discrete. If the number of bits of precision used to

represent the value of x (k) is ﬁnite, then we say that x (k) is a quantized or discrete-amplitude

Quantized signal

signal. For example, if N bits are used to represent the value of x(k), then there are 2

distinct

values that x(k) can assume. Suppose the value of x(k) ranges over the interval [x

, x

]. Then

the quantization level, or spacing between adjacent discrete values of x(k),is

Quantization level

q =

− x

(1.2.3)

The quantization process can be thought of as passing a signal through a piecewise-constant

staircase type function. For example, if the quantization is based on rounding to the nearest N

bits, then the process can be represented with the following quantization operator.

Quantization

operator

(x)



= q · round





(1.2.4)

A graph of Q

(x) for x ranging over the interval [−1, 1] using N = 5 bits is shown in

Figure 1.10. A quantized discrete-time signal is called a digital signal. That is, a digital signal,

Digital signal

FIGURE 1.9: A

Discrete-time Signal

x(k) with T = .25

0 .5 1 1.5 2 2.5 3 3.5 4

0.5

1.5

2.5

3.5

x(k) = 10kT exp(−kT)

kT (sec)

x(k)

T = .25

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Schilling-1120949 book November 12, 2010 9:33

14 Chapter 1 Signal Processing

produce x

(k). Careful inspection of Figure 1.11 reveals that at some of the samples there are

noticeable differences between x

(k) and x

(kT). If rounding is used, then the magnitude of

the error is, at most, q/2.

Most of the analysis in this book will be based on discrete-time signals rather than digital

signals. That is, inﬁnite precision is used to represent the value of the dependent variable.

Finite precision, or ﬁnite word length effects, are examined in Chapters 6 and 7 in the context

of digital ﬁlter design. When digital ﬁlter are implemented in MATLAB using the default

double-precision arithmetic, this corresponds to 64 bits of precision (16 decimal digits). In

most instances this is sufﬁciently high precision to yield insigniﬁcant ﬁnite word length effects.

A digital signal x

(k) can be modeled as a discrete-time signal x(k) plus random quanti-

Quantization

noise

zation noise, v(k), as follows.

(k) = x(k) + v(k) (1.2.6)

An effective way to measure the size or strength of the quantization noise is to use average

power deﬁned as the mean, or expected value,ofv

(k). Typically, v(k) is modeled as a

Expected value

random variable uniformly distributed over the interval [−q/2, q/2] with probability density

p(x) = 1/q. In this case, the expected value of v

(k) is

E[v

] =



q/2

−q/2

p(x)x



q/2

−q/2

dx (1.2.7)

Thus, the average power of the quantization noise is proportional to the square of the quanti-

zation level with

E[v

] =

(1.2.8)

Example 1.1 Quantization Noise

Suppose the value of a discrete-time signal x(k) is constrained to lie in the interval [−10, 10].

Let x

(k) denote a digital version of x(k) using quantization level q, and consider the following

problem. Suppose the average power of the quantization noise, v(k), is to be less than .001.

What is the minimum number of bits that are needed to represent the value of x

(k)? The

constraint on the average power of the quantization noise is

E[v

] <.001

Thus, from (1.2.3) and (1.2.8), we have

− x

)

12(2

)

<.001

Recall that the signal range is x

=−10 and x

= 10. Multiplying both sides by 12, taking

the square root of both sides, and then solving for 2

yields

√

.012

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