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英文翻译
英文翻译

英文

Signal and System

1. Mathematical Representation of Signals

Anything that bears information can be considered a signal. Signals may describe a wide variety of physical phenomena. For example, speech, music, interest rates, the speed of an automobile are signals. Although signals can be represented in many ways, in all cases the information in a signal is contained in a pattern of variations of some form. For example, consider the human vocal mechanism, which produces speech by creating fluctuations in acoustic pressure.

The signal represents acoustic pressure variations as a function of time for the spoken words “should we chase.” The top line of the figure corresponds to the word “should,” the second line to the word “we,” and the last two lines to the word “chase.”

Signals are presented mathematically as functions of one or more independent variables. For example, a speech signal can be represented mathematically by acoustic pressure as a function of time, and a picture can be presented by brightness as a function of two spatial variables. For convenience, we will generally refer to independent variable as time, although it may not in fact represent time in specific time in specific applications. For example in geophysics, signals representing variations with depth of physical quantities such as density, porosity, and electrical resistivity are used to study the structure of the earth.

Two basic types of signals will be considered: continuous-time signal and discrete-time signal. In the case of continuous-time signals the independent variable is continuous, and thus these signals are defined for a continuum of values of the independent variable. On the other hand, discrete-time signals are defined only at discrete times, foe these signals, the independent variable takes on only a discrete set of values. A speech signals as a function of time and atmospheric pressure as a function of altitude are example of continuous-time signals. The weekly Dow-Jones stock market index is an example of a discrete-time signal. Other examples of discrete-time signals can be found in demographic studies in which various attributes, such as average budget, crime rate, or pounds of fish caught, are tabulated against such discrete variable as family size, total population, or type of fishing vessel, respectively.

To distinguish between continuous-time signals and discrete-time signals, we will use the symbol t to denote the continuous-time independent variable and n to denote the discrete-time independent. In addition, for continuous-time signals we will enclose the independent variable in parentheses (.), whereas for discrete-time signals we will use brackets [.] to enclose the independent variable. We will also have frequent occasions when it will be useful to represent signals graphically. It is important to note that the discrete-time signal x[n] is defined only for integer values of the independent variable. For further emphasis we will on occasion refer to x[n] as discrete-time sequence.

A discrete-time signal x[n] may represent a phenomenon for which the independent variable is inherently discrete. Signal such as demographic data are example of this .On the

other hand, a very important class of discrete-time signals arises from the sampling of continuous-time signals. In this case, the discrete-time signal x[n] represent successive samples of an underlying phenomenon for which the independent variable is continuous. No matter what source of the data, however, the signal x[n] is defined only for integer values of n. It makes no matter sense to refer to the 7/2th sample of a digital speech signal than it does to refer to the average budget of family with 5/2 family members.

2. Mathematical Representation of Systems

Physical systems in the broadest sense are an interconnection of components, devices, or subsystems. In contexts ranging from signal processing and communications to electromechanical motors, automotive vehicles, and chemical-processing plants, a system can be viewed as a process in which input signals are transformed by the system or cause the system to respond in some way, resulting in other signals as outputs, For example, a high fidelity system takes a recorded audio signal and generates a reproduction of that signal. If the hi-fi system has tone controls, we can change the tonal quality of the reproduced signal. Similarly, the circuit can be viewed as a system with input voltage x (t) and output voltage v (t), while the automobile can be thought of as a system with input equal to the force f (t) and output equal to the velocity v(t) of the vehicle An image-enhancement system transforms an input image into an output image that has some desired properties, such as improved contrast.

A continuous-time system is a system in which continuous-time input signals are applied and result in continuous-time output signals. Such as system will be represented pictorially as in FIG . 3-4(a), where x(t) is the input and y(t) is the output. Alternatively, we will often represent the input-output relation of a continuous-time system by the notation

X (t)→y(t)

Similarly, a discrete-time system-that is, a system that transforms discrete-time inputs into discrete-time outputs-will be depicted as in FIG .3-4(b) and will sometimes be represented symbolically as

X (n)→y(n)

X (t)

y (t) X (t) y (t)

Fig.3-4 (a) Continuous-time system; (b) Discrete-time system; Many real systems are built as interconnections of several subsystems. Be viewing such a system as an interconnection of its components, we can use our understanding of the components systems and of how they are interconnected in order to analyze the operation and behavior of the overall system. In addition, by describing a system in terms of an interconnection of simpler subsystems, we may in fact be able to define useful ways in which Continuous-time System Discrete-time System

to synthesize complex systems out of simpler, basic building blocks.

While one can construct a variety of system interconnections, there are several basic ones that are frequently encountered. A series or cascade information of two systems is illustrated in Fug.3-5(a). Diagrams such as this are referred to as block diagrams. Here, the output of System 1 is the input to System 2 and the overall system transforms an input by processing it first by System 1 and then by System 2 . An example of a series interconnection is a radio receiver followed by an amplifier. Similarly, one can define a series interconnection of three or more systems.

Input Output

(a)

(b)

(c)

Fig.3-5 Interconnection of two systems (a) series (cascade) interconnection;

(b) Parallel interconnection; (c) series-parallel interconnection

A parallel interconnection of two systems is illustrated in Fig.3-5(b).Here, the same input signal is applied t o systems 1to 2.the symbol ”⊕” in the figure denotes addition, so that the output of the parallel interconnection is the sum of the outputs of Systems 1 and 2. An example of a parallel interconnection is a simple audio system with several microphones feeding into a signal amplifier and speaker system. In addition to the simple parallel interconnection in Fig.3-8(b), we can define parallel interconnections more than two systems, and we can combine both cascade and parallel interconnection to obtain more complicated interconnections. An example of such an interconnection is given in Fig.3-5(c).

Another important type of system interconnection is a feedback interconnection, an example of which is illustrated in Fig.3-6. Here, the output of system 1 is the input to system 2, while the output of system 2 is fed back and added to the external input to produce the actual input to system 1. Feedback systems arise in a wide variety of applications. For example, a cruise control system on an automobile senses sense the vehicle’s velocity and adjusts the fuel Input

Output

System 1 System 2 ⊕ System 1 System 2 ⊕ System 1 System4 System 2 System 4 Input Output

Iutput flow in order to keep the speed at the desired level.

Fig.3-6 Feedback interconnection

3. Fourier Transforms and Frequency-Domain Description

Signals encountered in practice are mostly continuous-time signals and can be denoted as x (t), where t is a continuum. Although some signals such as stock markets, savings account and inventory are inherently discrete time, most discrete-time signals are obtained from continuous-time signals by sampling and can be denoted as x[n]:=x(n T), where T is sampling period and n is the time index and can assume only integers. Both x (t) and x[n] are functions of time and are called the time-domain description. In signal analysis, we study frequency contents of signals. In order to do so, we must develop a different but equivalent description, called the frequency-domain description. From the description, we can more easily determine the distribution of power in frequencies.

In digital processing of a continuous-time signal x (t), the first step is to select a sampling period T and then to sample x (t) to yield x(n T).It is clear that the smaller T is, the closer x (n T) is to x(t). However, a smaller T also requires more computation. Thus an important task in DSP is to find the largest possible T so that all information (if not possible, all essential information) of x(t) is retained in x(x T). Without the frequency-domain description, it is not possible to find such a sampling period. Thus computing the frequency content of signals is a first step in digital signal processing.

The frequency-domain description is developed from the Fourier transform. If the Fourier transform of a signal is defined, the transform is called the frequency spectrum of the signal that is

Fourier transform ←→frequency spectrum

The continuous-time Fourier transform is defined by the following pair of equations: Forward Continuous- Time Fourier Transform

X(j w)= e -jwt dt (3-1) And Inverse Continuous-Time Fourier Transform

X(t)=1/2错误!未找到引用源。e jwt

dt (3-2) ⊕ System 1 System 2 Output

F Equation (3-1) and (3-2) are referred to as the Fourier transform pair, with the function X(jw) referred to as Fourier Transform or Fourier integral of x(t) and eq. (3-2)as the inverse Fourier Transform equation. X(jw) is commonly referred to as the frequency-domain representation or the spectrum of the signal, as it provides us with the information needed for describing x(t) as a liner combination (specifically, an integral) of sinusoidal signals at different frequencies. Likewise, x(t) is the time-domain representation of the signal. We indicate this relationship between the two domains as

Time-Domain Frequency-domain

X (t) X (jw)

The notation F signifies that it is possible to go back and forth uniquely between the time-domain and the frequency-domain.

If we are given x(t) as a mathematical function, we can determine the corresponding spectrum function X(jw) by evaluating the integral in (3-1). In other words, (3-1) defines a mathematical operation for transforming x(t) into a new equivalent representation X(jw). It is common to say that we take the Fourier transform of x(t),meaning that we determine X(jw) so that we can use the frequency-domain representation of the signal.

Similarly, given X(jw) as a mathematical function, we can determine the corresponding time function x(t) using (3-2) by evaluating an integral, Thus, (3-2) defines the inverse Fourier transform operation that goes from the frequency-domain to the time-domain.

Armed with the powerful tool of Fourier transform, we will be able to (1) define a precise notion of bandwidth for a signal, (2) explain the inner workings of modern communication systems which are able to transmit many signals simultaneously by sharing the available bandwidth, and (3) define filtering operations that are needed to separate signals in such frequency-shared systems. There are many other applications of the Fourier transform, so it is safe to say that Fourier analysis provides the rigorous language needed to define and design modern engineering systems.

4. The Sampling Theorem

Under certain conditions, a continuous-time signal can be completely represented by and recoverable from a sequence of its values, or samples, at points equally space in time. This somewhat surprising property follows from a basic result that is referred to as the sampling theorem. This theorem is extremely important and useful. It is exploited, for example, in moving pictures, which consist of a sequence of individual frames, each of which represent an instantaneous view (i.e., a sample in time) of a continuously changing scene. When these samples are viewed in sequence at a sufficiently fast rate, we perceive an accurate representation of the original continuously moving scene.

Much of the important of the sampling theorem also lies in its role as a bridge between continuous-time signals and discrete-time signals. The fact that under certain conditions a continuous-time signal can be completely recovered from a sequence of its samples provides a mechanism for repressing a continuous-time signal by a discrete-time signal. In many contexts, processing discrete-time signals is more flexible and is often preferable to processing continuous-time signals. This is due in large part to the dramatic development of digital

technology over the past few decades, resulting in the availability of inexpensive, lightweight, and programmable, and easily reproducible discrete-time system. We exploit sampling to convert a continuous-time signal to a discrete-time signal, process the discrete-time signal using a discrete-time system, and then convert back to continuous-time signal.

Sampling theorem can be stated as follows:

Let x(t) be a band-limited signal with X(jw)=0 for |w|>wm. Then x(t) is uniquely determined by its samples x(nT),n=0,+1,-1 (i)

Ws>2w m

Where

W s=2错误!未找到引用源。/T

Given these samples, we can reconstruct x(t) by generating a periodic impulse train in which successive impulses have amplitudes that are successive sample values. This impulse train in which successive through an ideal low pass filter with gain T and cutoff frequency greater than w m and less than w s -2 w m. The resulting output signal will exactly equal x (t).

The frequency 2w m, which, under the sampling theorem, must be exceeded by the sampling frequency, is commonly referred to as the Nyquist rate.

In previous discussion, it was assumed that the sampling frequency was sufficiently high that the conditions of sampling theorem were met. With w s>2w m the spectrum of the sampled signal consists of scaled replications of the spectrum of x(t), and this forms the basis for the sampling theorem. When w s<2w m X(jw),the spectrum of x(t),is no longer replicated in X p(jw) and thus is no longer recoverable by lowpass filtering. The reconstructed signal will no longer be equal to x(t). This effect is referred to as aliasing.

Sampling has a number of important applications. One particularly significant set of applications relates to using sampling to process continuous-time signal with discrete-time systems, by means of minicomputers, or any of a variety of devices specifically oriented toward discrete-time signal processing.

中文

信号和系统

1信号的数学表示

承载信息的任何事物可以看作是一个信号。信号可以描述变化万千的物理现象。

例如,语言、音乐、利率、汽车的行驶速度等都是信号。虽然信号可以用许多方式来表示,但是在所有的情况下,信号所载有的信息总是包含在以某种形式变化的波形中。

例如,人的声道系统所产生的语音信号就是一种声压的起伏变化。

信号是描述的口语单词should we chase 声压随时间变化的函数,第一个波形对

应单词should,第二个是we,最后二个是chase。

信号在数学上表示为一个或多个自变量的函数。例如,一个语言信号在数学上可以用声压随时间变化的函数来表示,而一张照片可以表示为亮度随二维空间变量变化的函数。为了方便起见,通常用时间来表示自变量,尽管在某些具体应用中自变量不一定是时间。例如,在地球物理学研究中用于研究地球结构的一些物理量如密度,气隙度和电阻率就是随地球深度变化的信号。

信号分为二种基本类型:连续时间信号和离散时间信号。连续时间信号的自变量连续的,因此,这些信号定义为自变量的值是连续的一类信号,另一方面,离散时间信号只定义自变量是离散时刻的信号,这些信号的自变量只取一系列离散的值。作为时间函数的语言信号和随海拔高度变化的气压信号都是连续时间信号的例子。每周的道琼指数是离散时间信号的例子。在人口统计学的研究中可以找到其他离散信号的例子,例如像平均预算,犯罪率或捕鱼的重量等各种属性都可以分别对家庭大小、总人口数或捕鱼船的类型等离散变量列成表格。

为了区别连续时间信号和离散信号,我们用符号t 来表示连续时间自变量,用n 来表示离散时间自变量。另外,给连续时间信号的自变量加上圆括弧(.),给离散时间信号的自变量加上方括弧[.]。我们常常用图解的方法来表示信号。值得强调的是,离散时间信号x[n]仅仅在自变量的整数值上有定义。为了更进一步强调,有时我们把X[n] 看作是离散时间序列。

一个离散时间信号x[n] 可以描述一个自变量固有离散的现象,比如人口数据就是这类信号的例子。另一方面,一类非常重要的离散时间信号来源于连续时间信号的取样。既然这样,离散时间信号x[n] 描述了自变量是连续的基本现象的连续采样。

不管数据是什么来源,信号x[n] 只在n 为整数时有定义。所谓的一个数字语音信号的第7/2个样本和所谓的具有5/2个家庭平均预算一样都是毫无意义的。

2.系统的数学表示

广义的物理系统是各组成部分、设备、或子系统的互联。从信号处理、通讯到电机马达、机动车和化学处理设备,一个系统可看作输入信号的变换器或对输入信号做出某种响应而产生出另外的输出信号。例如,一个高保真度的系统对输入音频信号进行录制,并重现原输入信号。如果高保真系统有音调控制,我们可以改变重现信号的音质。同样地,电路可看作是输入电压为x(t),输出电压为v(t)的系统,而汽车可看作是输入为动力f(t),输出为汽车速度v(t)的系统。图像增强系统可改变一幅输入图像,使输出图像具有某些所需要的性质,比如增强图像对比度。

一个连续时间系统是施加连续时间输入信号,而产生连续时间输出信号的系统。这个系统的框图表示如图3-4(a)所示,其中x(t)为输入,y(t)为输出。另一方面,我们常用符号来表示连续时间系统的输入输出关系:x(t)→y(t)。

同样地,一个离散时间系统,将离散时间输入转为离散时间输出的描述如图3-4(b)所示,有时用符号描述为:x[n]→y[n]。

许多实际的系统由多个子系统互联而成。把这样一个系统看作是它的各组成部分的互联,可以用对各部分系统以及它们之间互联方式的理解来分析整个系统的作用和性能。另外根据简单系统的互联关系来描述一个系统,事实上可以通过定义有用的方法来简单基本的模块中合成复杂的系统。

当我们构造多种系统互联的时候,经常碰到几种基本的互联方式。二个系统的串联或级联如图3-5(a)所示,这种图称为框图。这里系统1的输出是系统2 的输入,整个系统首先按系统1 ,然后系统2 来变换输入。一个级联系统的例子是收音机接一个扩音器。类似的,级联可以是三个或更多需要连接。

二个系统的并联如图3-5(b)所示,在这里相同的输入信号施加于系统1和系统2.图中符号“⊕”表示相加,因此并联系统的输出系统1和系统2的输出之和。简单的音频系统是一个系统并联的例子,几个麦克风输入到一个单一放大器和扬声器系统。除了图3-5(b)所示的简单并联结构外,系统的比例可以是二个以上系统的并联,我们还可以将级联与并联结合在一起得到一种更为复杂的系统。图3-5(c)所示就是这种结构的例子。

另一种重要的系统互联是反馈系统,如图3-6所示。这里系统1的输出是系统2 的输入,而系统2的输出反馈回来与外加信号,一起组成系统1的实际输入。反馈系统有很广泛的应用。例如,汽车感应系统的巡航控制系统感应汽车的速度并调整燃油流量来保持所希望的速度。

3.傅里叶变换和频域描述

实际遇到的信号大多是连续时间信号,这类信号可以用x(t)表示,其中t 是连续变量。虽然有些信号,例如股票市场、储蓄账户和库存本来就是离散信号,但大多数离散时间信号,是从连续时间信号采样而来,可以表示为:x[n]=x(nT),其中T为采样周期,n为只能取整数的时间量。x(t)和x[n]都是时间函数,称为时域描述。在信号分析中,我们研究信号的频谱。为了能做到这一点,必须开发不同等效的描述方法,称为频域描述法。从这种描述中很容易确定频率的能量分布。

连续时间信号的数字处理,第一步是选择一个采样周期T,然后采样x(t)产生x(nT)。很明显,周期T越小,x(nT)越接近x(t)。然而,T越小计算量越大。因此,数字信号处理的一项重要任务,是要找出最大可能T,使x(t)的所有的信任然保留在x(nT)中。没有频域描述,就不可能找到采样周期。因此,数字信号处理的第一步是计算信号的频谱。

频域描述来自于傅里叶变换。信号的傅里叶变换称为信号的频谱

傅里叶变换←→频谱

连续时间傅里叶变换定义为下面的等式:连续时间傅里叶正变换:

X(j w)= e-jwt dt

连续时间傅里叶反变换:

X(t)=1/2e jwt dt

式(3-1)和式(3-2)称为傅里叶变换对,函数X(jw)称为x(t)的傅里叶变换和傅里叶积分,而式(3-2)称为傅里叶反变换。X(jw)通常称为信号的频域表示或信号的频谱,因为X(jw)告诉我们这样的一个信息,就是x(t)可以描述成不同频率的正弦信号的线性组合。同样,X(t)是信号的时域描述,我们表述这二种域之间的关系如下:

时域频域

F

X (t) X (jw)

其中,符号F表示时域和频域变换是一一对应的。

已知数学函数x(t),可以通过式(3-1)的积分运算得到相应的频谱函数X(jw)换句话,式(3-1)定义了一个数学运算,将x(t)变换为新的等效表示X(jw)。通常说得到了x(t)的傅里叶变换,就意味着确定了X(jw),这样我们就可以利用信号的频域表示方法。

同样,已知函数X(jw),就是利用式(3-2)通过计算积分确定相应的时间函数x(t)。因此,式(3-2)得到了由频域到时域的傅里叶变换。

运用傅里叶变换这个强大工具,我们可以:1:确定一个信号带宽;2:解释通过共享可用带宽来同时发送多个信号的现代通信系统的内部工作机理;3:确定在这样的频率共享系统中分离信号的滤波方式。傅里叶变换有很多其他应用,也就是说,傅里叶分析为确定和设计现代工程系统提供了精确的方法。

4抽样理论

在一定的条件下,一个连续时间信号完全可以由该信号在时间等间隔点上的瞬时点上的瞬时值或样本值来表示,并且能用这些样本值恢复出信号来。这个性质来自于基本的结论,即抽样定理。这个定理非常重要并且得到了广泛的运用。例如,抽样定理在电影里得到了运用。电影由一组按时序排列的单个画面所组成,其中每个画面都代表着连续变化景象中的一个瞬时画面(即时间样本)。当这个画面以真够快的速率按顺序观看时,我们看到的是对原始连续影片的精确重现。

抽样理论的重要性同样在与它在连续时间信号和离散信号之间起了桥梁的作用。在一定条件下,可以用信号的时序样本值完全恢复出原连续时间信号,这就提供了用一个离散信号来表示一个连续时间信号的机理。在许多方面,处理离散时间信号要更加灵活些,因此往往比处理连续时间信号更为可取。这主要归功于在过去十几年高速发展的数字技术,使我们可以得到廉价,重量轻,可编程并且可容易复制的离散时间系统,我们利用抽样把连续信号转换成离散信号,通过离散时间系统来处理离散时间信号,然后再把离散时间信号转换成连续时间信号。

抽样定理如下:

假设x(t)是一个带宽受限信号,并且X(jw)=0,∣w∣>2w

m,其中w

s =2/T,

那么x(t)可以通过

它的样本值x(nT),n=0,……来唯一确定。

已知这些样本值,我们可以用以下方法重新构造x(t):产生一个周期的冲激串,其冲击激强度就是一次而来的样本值,然后将冲激串通过一个增益为T,截止频率大于w

m,而小于w s -2 w m的理想低通滤波器,该滤波器的输出就等于x(t)。

在抽样定理中,采样频率必须大于2w m,频率2w m通常称作奈奎斯特抽样率。

在前面的讨论中,假设抽样频率足够高,因而满足抽样定理的条件。由于w s>2w m,取样信号的频谱是x(t)频谱的周期性延拓。这构成了抽样定理的基础。当w s<2w m,X(jw)即x(t)的频谱不再在X p(jw) 中复制,因此通过低通滤波器也不能恢复。重构的信号不再等于x(t),这种现象称作频谱混叠。

抽样有许多重要的应用,一组特别重要的应用是利用微计算机,微处理器,或任何一个专门用于离散时间信号处理的器件,通过这些离散时间系统抽样来处理连续时间信号。

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