1、Random Processessimultaneous adj.同时的,同时发生的collection n.集合,集 dentical adj.同一的,同样的 individual adj.个别的,单独的 ensemble n.集,总体 ensemble average 集平均variable n.变量;变元 random variable 随机变量 stationary adj.平稳的ergodic adj.各态历经的,遍历的 ergodic process 各态历经过程，遍历过程Random Processesdeterministic adj.确定性的normalize vt.使标准化
2、,使规格化normalized adj.规格化的，归一化的expectation n.期望(值)product n.乘积truncate vt.截短；截断periodic adj.周期的covariance n.协方差uncorrelated adj.不相关的uniform adj.均匀的，一致的overlap v.(部分)重叠separation n.间隔，距离spacing n.间隔，间距random process 随机过程on the average 平均,按平均数计算;一般地说Random Processesmake measurement 量度sample function 样本函
3、数statistical average 统计平均(值)probability density function 概率密度函数statistical characteristic 统计特性autocorrelation function 自相关函数in connection with 关于，与有关 physical significance 物理意义Fourier transform 傅里叶变换power spectral density 功率谱密度be at liberty to(do)被允许;可随意gaussian process 高斯过程physical fact 外界存在的事实uppe
4、rfrequency limit 频率上限Random Processesfall off 下降,减少in the limit 在极限情况下delta function 函数，狄拉克函数random pulse 随机（杂乱）脉冲time invariant 时不变的Parsevals theorem 巴塞瓦尔定理statistically independent 统计独立successive pulse 连续脉冲be representative of 表示，代表，代表的特征Random Processes To determine the probabilities of the vario
5、us possible outcomes of an experiment,it is necessary to repeat the experiment many times.Suppose then that we are interested in establishing the statistics associated with the tossing of a die.We might proceed in either of two ways.On one hand,we might use a single die and toss it repeatedly.Altern
6、atively,we might toss simultaneously a very large number of dice.Intuitively,we would expect that both methods would give the same results.Thus,we would expect that a single die would yield a particular outcome,on the average,of 1 time out of 6.Similarly,with many dice we would expect that 1/6 of th
7、e dice tossed would yield a particular outcome.Analogously,let us consider a random process such as a noise waveform n(t).To determine the statistics of the noise,we might make repeated measurements of the noise voltage output of a single noise Random Processessource,or we might,at least conceptuall
8、y,make simultaneous measurements of the output of a very large collection of statistically identical noise sources.Such a collection of sources is called an ensemble,and the individual noise waveforms are called sample functions.A statistical average may be determined from measurements made at some
9、fixed timet=t1 on all the sample functions of the ensemble.Thus to determine,say,n2(t),we would,at t=t1,measure the voltages n(t1)of each noise source,square and add the voltages,and divide by the(large)number of sources in the ensemble.The average so determined is the ensemble average of n2(t1).Now
10、 n(t1)1is a random variable and will have associated with it a probability density function.The ensemble averages will be identical with the statistical averages and may be represented by the same symbols.Thus the statistical or ensemble average of n2(t1)maybe written Random ProcessesEn2(t1)=n2(t1).
11、The averages determined by measurements on a single sample function at successive times will yield a time average,which we represent asn2(t).Exercises.Please translate the following words and phrases into Chinese.1.sample function2.ensemble average 3.physical significance4.a Fourier transform pair5.
12、deterministic waveform6.in the limit 7.time invariant8.an upperfrequency limit9.Parsevals theorem 10.random pulses样本函数总体均值物理意义傅立叶变换对确定性波型在极限情况下时不变的频率上限巴塞瓦尔定理随机脉冲Exercises.Please translate the following words and phrases into English.1.随机过程2.统计平均3.随机变量4.自相关函数5.傅里叶变换6.功率谱密度7.概率密度函数8.高斯过程9.平稳过程10.统计独立r
13、andom process statistical average random variable autocorrelation function Fourier transform power spectral density probability density function gaussian process a stationary process statistically independent 11.时间平均(值)12.统计特性13.各态历经过程14.狄拉克函数time average statistical characteristic ergodic process d
14、elta function Exercises.Fill in the blanks with the missing word(s).1.The ensemble averages will be identical the statistical averages and may be represented by the same symbols.2.The averages determined by measurements a single sample Function successive times will yield a time average,which we rep
15、resent as n2(t1).3.Suppose,for example,that the statistical characteristics of the sample Functions the ensemble were changing with time.4.For it may happen that while each sample function is stationary the individual sample functions may differ statistically one another.5.As an extension of that re
16、sult we shall define the power spectral density of a random process the same way.withonatinfrominExercises6.It is interest to inquire whether G(f)defined Eq.(1.2)for a random process has a physical significance whichcorresponds to the physical significance of G(f)for deterministic waveforms.8.Hence,
17、if we should select some sample function,a knowledge of the value of n(t)at time t would be no assistance improving our ability to predict the value attained by that same sample function at time t+.9.Hence,whenever we make an observation or measurement of the pulse waveform which extends a duration
18、long enough so that the average observed pulse shape,such as their amplitudes,widths,and spacings are representative of the waveform generally,we shall find that Eq.(1.12)applies.ofinofinoverExercises10.Let us select a section of this waveform which extends -T/2 -T/2.11.Since we have assumed an ergo
19、dic process,we are at liberty to (perform,performing)the averaging any sample function of the ensemble,since every sample function will yield the same result.tofromperformoverExercises.Answer the following questions according to the text.1.Please describe the relationship between the ergodic process
20、 and the stationary process.2.What are random pulses?3.What is the relationship between the power spectral density G(f)and the autocorrelation function R().4.What is referred to as ergodic?An ergodic process is stationary,but a stationary process is not An ergodic process is stationary,but a station
21、ary process is not necessarily ergodic.necessarily ergodic.The random pulses are of the same form but have random amplitudes The random pulses are of the same form but have random amplitudes and statistically independent random times of occurrenceand statistically independent random times of occurre
22、nceThe power spectral density and the autocorrelation function The power spectral density and the autocorrelation function constitute a Fourier transform pair.constitute a Fourier transform pair.When the nature of a random process is such that ensemble and time aveWhen the nature of a random process
23、 is such that ensemble and time averages are identical,the process is referred to as ergodic.rages are identical,the process is referred to as ergodic.Exercises5.What is white noise?Why do we call it“white”?White noise has a power spectral density which is uniform over all White noise has a power sp
24、ectral density which is uniform over all frequencies.Such noise is referred to as frequencies.Such noise is referred to as“whitewhite”noise in analogy with noise in analogy with the consideration that white light is a combination of all colors,that the consideration that white light is a combination of all colors,that is,colors of all frequenciesis,colors of all frequencies
