Lesson 2 Aperiodic Signal

1、Lesson 2 Aperiodic SignalvBackgroundsvText tour vLanguage in useVocabulary Structure 2021-12-131测控技术与仪器英语教程BackgroundsvTerminology The real part 实部The imaginary part 虚部Amplitude 振幅Spectrum 频谱The unit step function 单位阶越函数The exponentially decaying 指数衰减阶跃函数Lasplace transform 拉普拉斯变换Pole 极点Zero 零点Argand

2、 diagram 阿根图 2021-12-132测控技术与仪器英语教程Text tourvOutline2.1 Introduction2.2 The Exponential Form of the Fourier Series2.3 The Fourier Transform2.4 The Laplace Transform2021-12-133测控技术与仪器英语教程2.1 IntroductionvSignals may not generally be assumed to exist for all time past, present, and future, and it is i

3、mportant to understand the effects which time-limitation has upon their frequency spectra.vAn important class of signal are simply not repetitive in nature.2021-12-134测控技术与仪器英语教程2.2 The Exponential Form of the Fourier SeriesvThe Exponential Form of the Fourier Series =vanalysis)exp()2exp(1112tjatjaf

4、)2exp()exp(1110tjatjaammtjma)exp(12021-12-135测控技术与仪器英语教程01111( )cossinnnnnf tAAntBntcos()/ 2,sin()/ 2ixixixixxeexi ee 2.3 The Fourier Transform2.3.1 Derivation2.3.2 Examples of Continuous Spectra2.3.3 Symmetry of the Fourier Integral Equations2.3.4 Limitations of the Fourier Transform2021-12-136测控技术

5、与仪器英语教程2.3.1 DerivationThe Fourier transform (the Fourier integral) does for the non-repetitive signal waveform what the Fourier series does for the repetitive one. mmtjmatf)exp()(1dtetfGtj)()(2021-12-137测控技术与仪器英语教程The first tell us how the energy of the waveform is continuously distributed in the f

6、requency range between , whereas the second shows how, in effect, the waveform may be synthesized an infinite set of exponential functions the form , each weighted by the relevant value of .)exp()()(1tjmTGtf)exp(2)(11tjmGdeGtftj)(21)( i te( )G2.3.2 Examples of Continuous Spectravthe spectrum of the

7、isolated pulse waveform Figure 2.1 (a) A Signal isolated Pulse and (b) its Frequency Spectrum )sin(211)(tjtjejdteG2021-12-139测控技术与仪器英语教程2.3.2 Examples of Continuous SpectravThe spectrum of this pulse waveform illustrates a number of important points about time-limited signals.vA waveform which is ve

8、ry time-limited will occupy a wide band of frequencies, and vice versa.2021-12-1310测控技术与仪器英语教程2.3.2 Examples of Continuous Spectrat=0tateFigure2.32021-12-1311测控技术与仪器英语教程2.3.2 Examples of Continuous Spectravevaluate the spectrum of the exponentially decaying sinusoidal waveform =dtetfgtj)()(00sindtet

9、etjat)exp()exp(21sin000tjtjjt2)(22020jG2021-12-1312测控技术与仪器英语教程2.3.3 Symmetry of the Fourier Integral EquationsvThe description of Symmetry of the Fourier integral equations Figure2.4 Symmetry between time and frequency domains. (a)just as a retangular pulse waveform has a (sinx/x) spectrum, so (b)a

10、time function of (sinx/x)from has a rectangular spectrum2021-12-1313测控技术与仪器英语教程2.3.4 Limitations of the Fourier TransformvAn important limitation may be inferred from the equation vThe Fourier transform will clearly only exist if the right-hand side of the equation is finite. Furthermore, since the

11、magnitude is always unity it is sufficient for the transform to exist ifdtetfGtj)()(dttf)(2021-12-1314测控技术与仪器英语教程i te2.3.4 Limitations of the Fourier Transformt=0lttet=0t(a)(b)Figure2.5 (a)The unit step function, and (b) an expotentially decaying stepfunction2021-12-1315测控技术与仪器英语教程2.4 The Laplace Tr

12、ansform2.4.1 Relationship with the Fourier Transform2.4.2 Use of the Laplace Transform2.4.3 The Pole-zero Description of a Signal2021-12-1316测控技术与仪器英语教程2.4.1 Relationship with the Fourier TransformSome terms of the Laplace transform give rise to not only the sine and cosine waves of the Fourier meth

13、od, but also growing and decaying sine and cosine waves and growing and decaying exponentials. 2021-12-1317测控技术与仪器英语教程2.4.2 Use of the Laplace TransformvExample) the decaying exponential waveform ) the sinusoidal wave )(11)()(0)()(0sesdtedtetfsGtssst)()exp(sin)(00000jsjsdtstsG2021-12-1318测控技术与仪器英语教程

14、2.4.2 Use of the Laplace Transformthe decaying exponential waveform and the sinusoidal wave t=0ttet=0tt0sin0/(a)(b)Figure 2.62021-12-1319测控技术与仪器英语教程2.4.2 Use of the Laplace Transformthe magnitude and phase of the decaying exponential wave (putting , ) Figure 2.7 Magnitude function and (b) phase func

15、tion of the spectrum of the waveform of figure 2.6(a) )(1)(jGjs 2021-12-1320测控技术与仪器英语教程2.4.2 Use of the Laplace Transformvthe inverse transform is defined asSolution: ) table of transform ) partial fractionsvthe Laplace transform may be used to solve a great many problems other than those of signal

16、analysis. jjstdsesGjtf)(21)(2021-12-1321测控技术与仪器英语教程2.4.3 The Pole-zero Description of a SignalvThe pole-zero description of a signal It is general property of such functions that there are values of the complex frequency variables which make G(s) tend to infinity and values of s which make G(s) zero

17、; such values are known as the pole and zeros of G(s) respectively.)(1)(ssG2021-12-1322测控技术与仪器英语教程2.4.3 The Pole-zero Description of a SignalvIt is common practice to represent the poles and zeros of G(s) graphically by drawing their positions on a so-called Argand diagram , the real part of a compl

18、ex variable is plotted along the abscissa, and the imaginary part along the ordinate. 2021-12-1323测控技术与仪器英语教程Language in usevVocabulary precious, appropriate, demonstrate, transform, decayvStructure2021-12-1324测控技术与仪器英语教程Vocabulary2021-12-1325测控技术与仪器英语教程precious in dictionaryv昂贵的：价钱或价值很高的；值钱的v珍贵的：受到

19、高度评价的；珍藏的v亲爱的；被热爱的v矫揉造作的：矫饰地优美或过分文雅的v十足的；纯粹的2021-12-1326测控技术与仪器英语教程precious in textvIn the previous chapter we have seen how a periodic signal may be expressed as the sum of a set of sinusoidal waves which are harmonically related.v在前一章中，我们已经看到周期信号是如何被表示成一组正弦谐波的叠加。 2021-12-1327测控技术与仪器英语教程precious in

20、 usevprecious mannerisms.v矫揉造作的习气va precious mess.v一团糟2021-12-1328测控技术与仪器英语教程appropriate in dictionaryv拨出：拨出作特殊用途v挪用，窃用：为自己独自占有或使用，常未经许可2021-12-1329测控技术与仪器英语教程appropriate in textveither as a set of sine and cosine waves of appropriate amplitude and frequency v或者表达为适当振幅和频率的一组正弦和余弦波 2021-12-1330测控技术与仪

21、器英语教程appropriate in usevappropriating funds for education.v拨款作教育基金vLee appropriated my unread newspaper and never returned it.v李擅自拿走我尚未阅读的报纸并从不归还2021-12-1331测控技术与仪器英语教程visualize in dictionaryv形成思维图像；设想v使可见2021-12-1332测控技术与仪器英语教程visualize in textvIt is indeed hard to visualize a wave such as an isola

22、ted pulse being composed of an infinite set of waves of infinitely small amplitudes .v的确很难想象像这些单个脉冲是由无限个无穷小振幅的波组成 .2021-12-1333测控技术与仪器英语教程visualize in usevtried to visualize the scene as it was described.v尽力设想所描绘的场景2021-12-1334测控技术与仪器英语教程demonstrate in dictionaryv表明：清楚细致地显示；表明v证实，论证：通过推理或引证说明真实性；证明v

23、演示，说明：通过实验、例子或实际应用表明；解释并说明v展示，示范：向可能的购买者示范（某物品的）使用2021-12-1335测控技术与仪器英语教程demonstrate in textvIt may seem strange that the pulse contains no energy at such frequencies, but it is not hard to demonstrate. v似乎很奇怪，脉冲在这样的频率下没有能量，然而这是不难说明。 2021-12-1336测控技术与仪器英语教程demonstrate in usevdemonstrated her skill a

24、s a gymnast; demonstrate affection by hugging.v显示她的体操技巧；拥抱以示亲热vdemonstrate a proposition.v证明一个命题vdemonstrated the laws of physics with laboratory equipment.v用实验装置演示物理定律vThe salesperson plugged in and demonstrated the vacuum cleaner.v售货员插上电源，示范真空吸尘器的操作2021-12-1337测控技术与仪器英语教程transform in dictionaryv使变

25、形：明显地改变外观或形式v改变的性质、职能或状况v使变换v使（一个结构）发生转换v变压：使经过变压器的转换v转变：使（一个细菌细胞）发生转化2021-12-1338测控技术与仪器英语教程transform in textvSo far it has been implied that the Fourier transform may be successfully used for any no-repetitive signal, but there are in fact limitations and difficulties in its applications. v到现在为止，似

26、乎傅立叶变换适用于任何的非重复的信号，但是事实上在它的应用中有一些限制和困难。 2021-12-1339测控技术与仪器英语教程transform in usevA thick, fibrous fog had transformed the trees into ghosts and the streetlights into soft, haloed moonsv “浓密厚重的雾使树木看起来象鬼魂，街灯看起来象是带着柔和光晕的月亮”2021-12-1340测控技术与仪器英语教程decay in dictionaryv腐烂：分解成各组成部分；腐烂v衰变：由于放射性衰变而解体或缩小v衰减：指电压

27、或电流的逐渐减弱v能量衰减：指人造卫星在轨道上运行时能量消减v衰亡，毁坏：v使衰退：从正常、优秀或繁荣的状态衰败下去，恶化2021-12-1341测控技术与仪器英语教程decay in textvSuch terms give rise to not only the sine and cosine waves of the Fourier method, but also growing and decaying sine and cosine waves and growing and decaying exponentials.v在大多数例子中，一个函数由许多简单的线性因子组成，部分分式

28、法就是要我们把一个函数表示许多比较简单的函数之和，而每个简单的函数都出现在变换表中。2021-12-1342测控技术与仪器英语教程decay in useva civilization that had begun to decay.v已经开始衰落的文明2021-12-1343测控技术与仪器英语教程Structure2021-12-1344测控技术与仪器英语教程vIt is possible to derive frequency spectra for such signals using as a starting point of the work we have already done on the Fourier series.v我们可以通过傅立叶级数作为研究这类信号频谱的出发点。2021-12-1345测控技术与仪器英语教程vInstead of using the simple trigonometric form，we