电子通信专业英语之信号线性系统篇

1、Signals, Linear Systems, and Convolution 课文讲解n Instead, we must find some way of making a finite number of measurements that allow us to infer how the system will respond to other inputs that we have not yet measured 译文译文因此，必须找到一些方法，通过一定数量的测试就能可以推断出系统对那些没有测试过的输入会有怎样的响应。 nThe type of signal which dep

2、ends on one independent variable, namely, time is called the one-dimensional signal. It can be represented by x(t). 译文译文一维信号是指取决于某一独立变量，即时间t的一类信号，可用x(t)来表示 。 nRepresenting signals with impulses（用脉冲函数表示信号）. Any signal can be expressed as a sum of scaled and shifted unit impulses（任何信号都可以表述为许多经过缩放、位移的脉

3、冲函数的总和）. We begin with the pulse or “staircase”（阶梯） approximation to a continuous signal, as illustrated in Figure4.1. Conceptually, this is trivial（简单的）: for each discrete sample of the original signal, we make a pulse signal. Then we add up all these pulse signals to make up the approximate signal

4、. Each of these pulse signals can in turn （依次）（依次） be represented as a standard pulse scaled（缩放）（缩放） by the appropriate value and shifted（位移）（位移） to the appropriate place. In mathematical notation（符号）: 用脉冲函数表示信号：任何信号都可以表述为许多经过缩放、位移的脉冲函数的总和。连续信号x(t) 可以用脉冲或阶梯函数来近似，如图4.1。从概念上来讲这很简单：原始信号的每一个离散采样都可以用一个脉冲

5、函数来表示，然后将所有采样累加起来近似原始信号。每一个冲激信号依次依次可以表示为将一个标准脉冲进行适当的缩放和位移后得到。用数学公式表示如下：nIn other words, we can represent any signal as an infinite sum of shifted and scaled unit impulses. A digital compact disc, for example, stores whole complex pieces of music as lots of simple numbers representing（表示）（表示） very sh

6、ort impulses, and then the CD player adds all the impulses back together one after another to recreate（重（重构）构） the complex musical waveform.n 也就是说也就是说，任何信号均可以由无限多个经过缩放、位移的单位冲激函数的和来表示。例如，数字数字CD机机即是利用许多小的脉冲来储存完整而复杂的音乐，而这些冲激函数在CD机中是由大量简单的数字来表示。播放时，CD机依次将这些脉冲迭加起来重构重构复杂的音乐波形。 nSystems come in a wide vari

7、ety of types. One important class is known as linear systems. To see whether a system is linear, we need to test whether it obeys certain rules that all linear systems obey. The two basic tests（特性）（特性） of linearity（线性） are homogeneity（齐次性，同种，heterogeneity：异种，异质） and additivity（叠加）.n系统包括相当多的类型，线性系统就是

8、其中一种非常重要的类型。判断一个系统是否为线性系统，要看它是否遵循线性系统的法则。线性的两个基本特性是齐次性和叠加性。nHomogeneity. As（当时候） we increase the strength of the input to a linear system, say（比方说） we double it, then we predict that the output function will also be doubled. For example, if the current injected（注射）（注射） to a passive（被动的）（被动的） neural

9、membrane is doubled, the resulting（由此产生的） membrane potential fluctuations will double as well（也）（也）. This is called the scalar rule or sometimes the homogeneity of linear systems.n齐次性齐次性：当增大输入信号的强度时，比如说将输入加倍，那输出也必然要加倍。例如，当加在被动神经膜上的电流加倍时，产生的膜电压的波动也会加倍。这称作“比例法则”或者有时称作线性系统的齐次性。nAdditivity. Suppose we m

10、easure how the membrane potential fluctuates（v.波动） over time in response to a complicated time-series of injected current . Next, we present（提出） a second (different) complicated time-series . The second stimulus（刺激） also generates fluctuations(n.波动) in the membrane potential which we measure and wri

11、te down. Then, we present the sum of the two currents and see what happens. Since the system is linear, the measured membrane potential fluctuations will be just the sum of the fluctuations to each of the two currents presented separately.n叠加性叠加性：假设假设要测量膜电压对一个复杂的时序输入电流会有怎样的影响；接着，我们换另一个不同的复杂时序信号，它同样会

12、使膜电压产生波动，对它进行测量并记录下来。然后，将输入电流变成前两者之和看看会发生什么。因为系统是线性的，测量得到的膜电压将会是两个输入信号分别作用时的电压之和。nSuperposition（重合性）（重合性）. Systems that satisfy both homogeneity and additivity are considered to be linear systems. These two rules, taken together, are often referred to as（被称作）（被称作） the principle of superposition. Mat

13、hematically, the principle of superposition is expressed as:n重合性重合性：同时满足齐次性和叠加性的系统就称作线性系统。以上两个准则合起来称为重合性。它可以用数学公式表述如下：nShift-invariance. Suppose that we inject a pulse of current and measure the membrane potential fluctuations. Then we stimulate（刺激，激励） again with a similar pulse at a different point

14、 in time, and again we measure the membrane potential fluctuations. If we havent damaged the membrane with the first impulse then we should expect that the response to the second pulse will be the same as the response to the first pulse. The only difference between them will be that the second pulse

15、 has occurred later in time, that is, it is shifted in time. When the responses to the identical stimulus（n.激励） presented shifted in time are the same, except for the corresponding shift in time, then we have a special kind of linear system called a shift-invariant linear system3. Just as not all sy

16、stems are linear, not all linear systems are shift-invariant. n时不变特性：假设我们输入一个电流脉冲并测量膜电压的波动。然后在另外的时刻再次输入相同的脉冲，并且记录下这时的膜电压波动。如果膜没有在我们第一次输入电流时被破坏，那么可以预料第二次的响应应该与第一次的完全相同。它们之间唯一的不同在于第二次的脉冲在时间上要比第一次发生的要晚，也就是说，它在时间上发生了位移。如果系统对经过时移的相同激励产生相同的响应，只是响应也发生相应的时移，那么我们就得到一种特殊的线性系统，称为“线性时不变系统”。正如并非所有系统都是线性的，也并非所有线性

17、系统都是时不变的。 nTo get a better feeling for linearity, think about（设想） a technician trying to determine if(是否) an electronic device is linear. The technician would attach（附上，系上） a sine wave generator to the input of the device, and an oscilloscope（示波器） to the output. With a sine wave input, the technicia

18、n would look to see if the output is also a sine wave. For example, the output cannot be clipped（省略一部分的，clip:修剪） on the top or bottom, the top half cannot look different from the bottom half, there must be no distortion where the signal crosses zero, etc. n为了得到对线性系统更进一步的理解，我们设想由一个技师来测定一个电子设备是否为线性的。技

19、师会在输入端接上一个正弦波发生器，在输出端接上一个示波器。输入正弦波后，技师会去看输出端是否也是正弦波。比如，波形在波峰或波谷的位置不能失真，波形的上半部分不能与下半部分不同，在零点处不能有失真，等等。 nNext, the technician would vary the amplitude of the input and observe the effect on the output signal. If the system is linear, the amplitude of the output must track the amplitude of the input. L

20、astly, the technician would vary the input signals frequency, and verify that the output signals frequency changes accordingly. As the frequency is changed, there will likely be amplitude and phase changes seen in the output, but these are perfectly permissible in a linear system. n然后，技师要变换输入信号的幅值来观

21、察输出信号的效果。如果系统是线性的，那么输出信号的幅值会跟随输入信号的幅值。最后，技师要变换输入信号的频率，并且检验输出信号的频率是否产生相应的变化。当频率发生变化的时候，输出信号的幅值和相位可能会发生变化，但这些都是线性系统允许发生的。nAt some frequencies, the output may even be zero, that is, a sinusoid with zero amplitude. If the technician sees all these things, he will conclude that the system is linear. Whil

22、e this conclusion is not a rigorous mathematical proof, the level of confidence is justifiably high4. n在某些频率上，输出信号可能为零，也就是幅值为零的正弦信号。如果技师全部检验过这些后，他就可以会认定系统为线性的。尽管该结论并非严格的数学证明，但其可信度很高。 nConvolution. Homogeneity, additivity, and shift invariance may, at first, sound a bit abstract but they are very use

23、ful. To characterize a shift-invariant linear system, we need to measure only one thing: the way the system responds to a unit impulse. This response is called the impulse response function of the system. n卷积卷积：最初，齐次性、叠加性和时不变特性听起来有一点抽象，但它们很有用。描述一个线性时不变特性，只需要测量一样东西，即系统对单位冲激函数的响应，称作系统冲激响应函数。nOnce weve

24、 measured this function, we can (in principle) predict how the system will respond to any other possible stimulus. In the following, we show that the response of a shift-invariant linear system can be written as a sum of scaled and shifted copies of the systems impulse response function.n一旦有了这个函数，就能

25、从理论上预测系统对其他激励的响一旦有了这个函数，就能从理论上预测系统对其他激励的响应。接下来，我们将说明线性时不变系统的响应如何表示成应。接下来，我们将说明线性时不变系统的响应如何表示成一组经过缩放、位移的系统冲激响应函数的和。一组经过缩放、位移的系统冲激响应函数的和。nNotice what this last equation means. For any shift-invariant linear system T, once we know its impulse response h(t) (that is, its response to a unit impulse), we

26、can forget about T entirely, and just add up scaled and shifted copies of h(t) to calculate the response of T to any input whatsoever5. Thus any shift-invariant linear system is completely characterized by its impulse response h(t) .n注意公式（4）。对任意一个线性时不变系统T，只要已知冲激响应h(t) （即它对单位冲激函数的响应），就可以完全忘记T，只要将经过缩放和位移变换的h(t)相加就可以得到系统T对任意输入的响应。因此，任何线性时不变系统都能用其冲激响应h(t)完全表述。nConvolution as a series of weighted sums. While superposition and convolution may sound a little abstract, there is an equivalent stat