1、Channel CapacityNew words and phrases 1abstraction n.抽象；抽象观念2encompass vt.包括，包含3devise vt.设计,发明4whereby adv.凭那个；由此5arbitrarily adv.任意地6theorem n.定理7infrequent adj.很少发生的,罕见的8override vt.压倒,制服，凌驾9thereby adv.由此,因此,从而10gaussian adj.高斯的 Gauss高斯(17771855,德国数学家、天文学家)11unity n.一，单一12complementary adj.补充的；互
2、补的Channel Capacity13derivation n.推导,导出，公式推导14formidable adj.困难的，棘手的,可怕的15undertake vt.进行,从事16presumed adj.假定的,推测的 presumably adv.推测起来,大概,估计,可能17likelihood n.可能,可能性likely adj.可能，可能的18interval n.间隔，时间间隔19rounding n.舍入(成整数);四舍五入20abrupt adj.不连续的，突然的，急剧的，陡的21reliably adv.可靠地，安全地，确实地22heuristic adj.启发式的,
3、渐进的23intuitive adj.直觉的;直观的 intuitively adv.直觉地,直观地24contemplate vt.预期(料)Channel Capacity25vice versa adv.反之亦然（viceversa)26quantize vt.量化27reception n.接收28extreme adj.极端的，极度的,偏激的29attenuate vt.衰减，使衰减30distort vt.失真，使失真distortion n.失真,畸变31equalizer n.均衡器32recoverable adj.可恢复的1 enter into 参加，成为一部分，涉及2
4、due to 由于;起因于;归功于3 be something of 有一点4 in principle 原则上,大致上on principle 按照原则（或道德标准）Channel Capacity5 provided that 假如,设若6 probability of error(error probability)误码率，误差概率 7 be close to 接近,靠近8 bandlimited gaussian channel 限带高斯信道9 physical system 物理系统,实际系统10 turn out （常与to,that连用）结果，结果是11 lower bound
5、下界,下限12 upper limit 上限13 rootmeansquare 均方根，均方根值；均方根(的)rootmeansquare error 均方根(有效值)误差14 amount of information （information content;quantity of information;information quantity）信息量15 ideal lowpass filter 理想低通滤波器Channel Capacity16 rise time 上升时间17 as a matter of convenience 为方便起见18 white Gaussian noi
6、se(white gaussian noise)高斯白噪声 19 on the other hand 另一方面20 trade off 交替换位，折衷选择(tradeoffn.折衷，权衡)21 signaltonoise ratio 信号噪声比，信噪比（SNR,S/N）22 be free to 随意,任意,不受拘束23 make up for 补偿Channel Capacity The importance of the concept of information rate is that it enters into a theorem due to Shannon which is
7、fundamental to the theory of communications.This theorem is concerned with the rate of transmission of information over a communication channel.While we have used the term communication channel on many occasions，it is well to emphasize at this point,that the term,which is something of an abstraction
8、,is intended to encompass all the features and component parts of the transmission system which introduce noise or limit the bandwidth.Shannons Theorem,Channel Capacity Shannons theorem says that it is possible,in principle,to devise a means whereby a communications system will transmit information
9、with an arbitrarily small probability of error provided that the information rate R is less than or equal to a rate C called the channel capacity.To put the matter more formally,we have the following:Channel CapacityTheorem Given a source of M equally likely messages,with M1,which is generating info
10、rmation at a rate R.Given a channel with channel capacity C.Then,ifRCthere exists a coding technique such that the output of the source may be transmitted over the channel with a probability of error in the received message which may be made arbitrarily small.The important feature of the theorem is
11、that it indicates that for RC transmission may be accomplished without error in the presence of noise.This result is surprising.For in our consideration of noise,say,gaussian noise,we have seen that the probability density of the noise extends to infinity.We should then imagine that there will be so
12、me times,however infrequent,when the noise must override the signal thereby resulting in errors.However,Shannons theorem says that this need not cause a message to be in error.Channel CapacityThere is a negative statement associated with Shannons theorem.It states the following:Theorem Given a sourc
13、e of M equally likely messages,with M1,which is generating information at a rate R;then if RC the probability the probability of error is close to unity for every possible set of M transmitter signals.This negative theorem states that if the information rate R exceeds a specified value C,the error p
14、robability will increase toward unity as M increases,and that also,generally,in this case where RC,increasing the complexity of the coding results in an increase in the probability of error.Capacity of a Gaussian Channel A theorem which is complementary to Shannons theorem and applies to a channel i
15、n which the noise is gaussian is known as the ShannonHartley theorem.Channel CapacityTheorem The channel capacity of a white,bandlimited gaussian channel is C=Blog2(1+S/N)bits/s(2.1)where B is the channel bandwidth,S is the signal power,and N is the total noise within the channel bandwidth,that is,N
16、=B,with 1/2 the(two-ided)power spectral density.This theorem,although restricted to the gaussian channel,is of fundamental importance.First,we find that channels encountered in physical systems generally are,at least approximately,gaussian.Second,it turns out that the results obtained for a gaussian
17、 channel often provide a lower bound on the performance of asystem operating over a nongaussian channel.Thus,if a particular encoderdecoder is used with a gaussian channel and an error probability Pe results,then with a nongaussian channel another encoderdecoder can be designed so that the Pe will b
18、e smaller.We may note that channel capacity equations corresponding to Eq.(2.1)have been derived or a number of nongaussian channels.Channel CapacityThe derivation of Eq.(2.l)for the capacity of a gaussian channel is rather formidable and will not be undertaken.However,the result may be made to appe
19、ar reasonable by the following considerations.Suppose that,for the purpose of transmission over the channel,the messages are represented by fixed voltage levels.Then,as the source generates one message after another in sequence,the transmitted signal s(t)takes on a waveform similar to that shown in
20、Fig.2.1.Please translate the following words and phrases into Chinese.1.physical syste2.rise time 3.amount of information 4.in principle 5.gaussian channel 6.probability density 7.rootmeansquare 8.trade off 9.lower bound 1.equalizer 11.vice versa 12.upper limit Exercises物理系统，实际系统物理系统，实际系统上升时间上升时间信息量
21、信息量理论上，原则上理论上，原则上高斯信道高斯信道概率密度概率密度均方根值均方根值,均方根；均方根均方根；均方根(的的)交替换位，折衷选择交替换位，折衷选择下界下界,下限下限均衡器均衡器反之亦然反之亦然上限上限.Please translate the following words and phrases into English.1.通信理论2.香农定理3.信道带宽4.信号波形5.理想低通滤波器 6.自相关函数7.无噪声高斯信道 8.通信信道9.信息速率10.信噪比Exercisescommunication theory(theory of communications)Shannons
22、 theoremchannel bandwidthsignal waveformideal lowpass filterautocorrelation functionnoiseless gaussian channelcommunication channelinformation rateSignal to noise ratio（SNR,S/N）Exercises11.信道容量12.双边功率谱密度13.误码率14.奈奎斯特采样速率15.限带高斯信道16.高斯白噪声channel capacityTwo sided power spectral densityerror probabili
23、ty(probability of error)Nyquist sampling rateband limited gaussian channelwhite Gaussian noiseExercises.Fill in the blanks with the missing word(s).1.There is a negative statement associated Shannons theorem.2.the purpose of transmission over the channel,the messages are represented by fixed voltage
24、 levels.3.Since the transmission of any of the M messages is equally likely,H=log2M,thus our channel is transferring information a rate R=rH.4.For a fixed signal power and the presence of white gaussian noise the channel capacity approaches an upper limit with increasing bandwidth.5.It is great inte
25、rest to recognize that the tradeoff between bandwidth and signal to noise ratio is not limited by a lower limit bandwidth.withonatinofinExercises6.The signal is transmitted a channel which can be represented as a lowpass RC circuit with cutoff at 1 Hz.7.If there is no noise,then we are entirely free
26、 to make for the attenuation by the use of an amplifier and to correct the frequency distortion by the use of an equalizer.8.That is,we need to estimate the interval T which should be assigned to each message to allow the transmitted levels to be recognized Individually the receiver,even though the
27、bandwidth B of the channel is limited.9.Therefore,the 25 percent reduction bandwidth requires a 60 percent increase signal power.inasinofoverExercises10.While we have used the term communication channel many occasions,it is well to emphasize at this point,that the term,which is Something an abstract
28、ion,is intended to encompass all the features and componentparts of the transmission system which introduce noise or limit the bandwidth.11.The probability of error is close unity for every possible set of M transmitter signals.12.It turns out the results obtained for a gaussian channel often provid
29、e a lower bound the performance of a system Operating a nongaussian channel.inthatintoinas.Answer the following questions according to the text.1.What is ShannonHartley theorem?2.Please describe Shannons theorem in your own words in English.ExercisesShannonHartley theorem:The channel capacity of a w
30、hite,bandlimited gaussian channel is C=Blog2(1+S/N)bits/s,where B is the channel bandwidth,S is the signal power,and N is the total noise within thechannel bandwidth,that is,N=B,with/2 the(two sided)power spectral density.Given a source of M equally likely messages,with M1,which is generating inform
31、ation at a rate R.Given a channel with channel capacity C.Then,if RC,There exists a coding technique such that the output of the source may be transmitted over the channel with a probability of error in the received message which may be made arbitrarily small.3.Try to describe Shannons theorem by a
32、negative statement in English.4.Why is the ShannonHartley theorem important?ExercisesThe ShannonHartley theorem,although restricted to the gaussian channel,is of fundamental importance.First,we find that channels encountered in physical systems generally are,at least approximately,gaussian.Second,it
33、 turns out that the results obtained for a gaussian channel often provide a lower bound on the performance of a system operating over a nongaussian channel.Thus,if a particular encoderdecoder is used with a gaussian channel and an error probability Pe results,then with a nongaussian channel another
34、encoder decoder can be designed so that the Pe will be smaller.Given a source of M equally likely messages,with M1,which is generating information at a rate R;then if RC,the probability of error is close to unity for every possible set of M transmitter signals.Exercises5.How can we trade off bandwid
35、th for signaltonoise ratio according to the ShannonHartley principle?For example,if S/N=7 and B=4 kHz,we find C=12103bits/s.If the SNR is increased to S/N=15 and B decreased to 3 kHz,the channel capacity remains the same.With a 3 kHz bandwidth the noise power will be 3/4 as large as with 4 kHz.Thus the signal power will have to be increased by the factor B=3/415/7=1.6.Therefore,the 25 percent reduction in bandwidth requires a 60 percent increase in signal power.
