外文翻譯---香農(nóng)和“香農(nóng)公式”

1、<p><b>　　畢業(yè)設(shè)計(jì)/論文</b></p><p>　　外 文 文 獻(xiàn) 翻 譯</p><p>　　系　 別 通信工程與技術(shù)系 </p><p>　　專 業(yè) 班 級(jí) 0703 　　 </p><p>　　姓　 名

2、 </p><p>　　評(píng) 分 </p><p>　　指 導(dǎo) 教 師 </p><p>　　20 年 月 日</p><p><b>　　香農(nóng)和“香農(nóng)公式”</b&g

3、t;</p><p>　　Lars Lundheim{挪威科學(xué)技術(shù)大學(xué)，通信系，教授}</p><p>　　在科學(xué)技術(shù)史上，十九世紀(jì)中葉和二十世紀(jì)中葉是一個(gè)非常突出的時(shí)期。在這個(gè)時(shí)期，一些發(fā)明和創(chuàng)作的實(shí)現(xiàn)消除了許多對(duì)個(gè)人和社會(huì)的有影響的限制。特別是在通信領(lǐng)域，迅速的發(fā)展就好像高架鐵路，蒸汽輪船，航空和電信一樣。值得人注意的是一些有趣的事情。由于一些限制被拆除，一些基本的或主要的隨即建立了新

4、的限制。例如，卡羅表明，從熱機(jī)提煉出來的熱能量是有一個(gè)極限值的。后來，這一定律被推廣到了第二定律熱力學(xué)。就像愛因斯坦的狹義相對(duì)論，一個(gè)機(jī)制的速度被發(fā)現(xiàn)。其他例子包括Kelvin的絕對(duì)零度，Heissenberg的測(cè)不準(zhǔn)原理和哥德爾不完備性定理的數(shù)學(xué)。在1948年出版的香農(nóng)信道編碼定理似乎是最后一個(gè)被發(fā)現(xiàn)的限制，人們或者會(huì)奇怪，為什么所有的限制都是在這個(gè)有限的時(shí)間內(nèi)完成的。有一個(gè)原因可以解釋。當(dāng)一個(gè)學(xué)者還年輕的時(shí)候，他總是試著去發(fā)現(xiàn)一些東

5、西——一些他們不能確定的東西。由于通信是在科學(xué)發(fā)展中時(shí)間最短的一個(gè)，很自然的它基本的法則是建立在后期階段。在本論文中，我們會(huì)盡量除開一些事態(tài)的發(fā)展對(duì)香農(nóng)公式的影響。</p><p><b>　　香農(nóng)公式</b></p><p>　　有時(shí)候，一個(gè)“天才之舉”科學(xué)成果的產(chǎn)生會(huì)讓人無法想象是來自于一個(gè)私人科學(xué)家。更常見的是有幾個(gè)獨(dú)立的小組在一個(gè)特定的成熟的時(shí)機(jī)逐步研究出來的

6、。在本文中我們將著眼于一個(gè)特定的概念，一個(gè)頻帶有限的信息傳輸通道的通道容量加白噪聲，高斯噪音。這就是所謂的“香農(nóng)公式”：</p><p>　　C = W log2(1 + P/N) bits/s</p><p>　　我們逐步發(fā)現(xiàn)，一方面，二戰(zhàn)過后這么些年，當(dāng)時(shí)已經(jīng)是一個(gè)成熟的時(shí)機(jī)，另一方面，香農(nóng)公式是一個(gè)非常特殊的發(fā)現(xiàn)，很多有識(shí)之士都非常的認(rèn)同這個(gè)觀點(diǎn)。許多數(shù)學(xué)表達(dá)式都用有關(guān)香農(nóng)的名字來命

7、名。這個(gè)例子并不是最著名的，但在通信領(lǐng)域中或許是最知名的。這也是當(dāng)時(shí)最迅速最有價(jià)值的理解之一了。</p><p>　　對(duì)香農(nóng)工作的介紹，這個(gè)問題在N. Knudtzon的論文上看到了。</p><p>　　香農(nóng)公式：如果信息源的信息速率R小于或者等于信道容量C，那么，在理論上存在一種方法可使信息源的輸出能夠以任意小的差錯(cuò)概率通過信道傳輸。 </p><p>　　該定

8、理還指出：如果R>C，則沒有任何辦法傳遞這樣的信息，或者說傳遞這樣的二進(jìn)制信息的差錯(cuò)率為1/2。作為一個(gè)通信工程師，不論誰都應(yīng)該對(duì)這個(gè)定理很熟悉了，即使理解那個(gè)結(jié)果。這在1948年已經(jīng)是不爭(zhēng)的事實(shí)了。而帶寬和信號(hào)的力量因此出名，并且出現(xiàn)在第一期的“香農(nóng)”報(bào)上。概念的概率分布和隨機(jī)過程、潛在的假定的噪聲模型,一直用于研究社區(qū)幾年的一部分,而不是一個(gè)普通的電氣工程師的培訓(xùn)。</p><p>　　香農(nóng)公式的基本因

9、素是：</p><p><b>　　帶寬w</b></p><p><b>　　信號(hào)功率s</b></p><p><b>　　噪聲功率p</b></p><p><b>　　他們構(gòu)成對(duì)數(shù)函數(shù)</b></p><p>　　信道帶寬的

10、限制，通過設(shè)置一個(gè)符號(hào)，以便通過快速通道進(jìn)行信道傳輸。信噪比可以決定每個(gè)符號(hào)能代表多少信息。在結(jié)束時(shí)，信噪比可以用來計(jì)算通道接受段的信息量。因此,功率電平都是一種透射光力量和衰減信號(hào)在傳輸媒介上(頻道)的能量 最優(yōu)秀的夏儂論文可能是1948年至1949年它們的共性相結(jié)合時(shí)的獨(dú)特效果和共性闡述。它們服從一定的概率分布。同樣的，信道基本上表示從一個(gè)符號(hào)映射到另一個(gè)符號(hào)并且服從相應(yīng)的概率分布。結(jié)合起來，這個(gè)結(jié)論就適合任何的通信系統(tǒng)了，人造的或

11、者自然的，電子的或者手工的。</p><p>　　克勞德·埃爾伍德夏儂(1916-2001),信息理論的創(chuàng)始人,也是一個(gè)實(shí)踐和頑皮的一邊。這張照片顯示他發(fā)明的一個(gè)機(jī)械“老鼠",可以找到它的方式通過一個(gè)迷宮。他也被他的信息工作和數(shù)字工作累的瘦長(zhǎng)。</p><p><b>　　獨(dú)立發(fā)現(xiàn)：</b></p><p>　　一個(gè)重要指標(biāo)

12、,時(shí)機(jī)已經(jīng)成熟,理論在第一資料的傳輸了戰(zhàn)后的眾多的論文發(fā)表在這樣的理論嘗試。特別是三個(gè)公式相當(dāng)?shù)南嗨啤＿@些最著名的是公布維納于1949年的《控制論》。羅伯特維拉是一個(gè)麻省理工學(xué)院的教授，總所周知的是，他是一個(gè)有哲學(xué)傾向的心不在焉的數(shù)學(xué)教授。盡管如此，他深為關(guān)切關(guān)于數(shù)學(xué)在社會(huì)各個(gè)領(lǐng)域的應(yīng)用。這種哲學(xué)興趣正好讓他建立了科學(xué)的控制論。這個(gè)領(lǐng)域，這也許是最好的了[2]字幕定義：“控制與通信在動(dòng)物和機(jī)器“包括，除其他事項(xiàng)外，理論信息內(nèi)容在一個(gè)信

13、號(hào)，這些信息通過信道傳輸。維拉是這樣想的，但是思想交流不是他所能控制的，關(guān)于香農(nóng)公式，相關(guān)的信息和繁瑣的符號(hào)表現(xiàn)的并不明顯。參考與維拉的工作，更加證明了香農(nóng)公式的正確性蒂尤爾是麻省理工學(xué)院的電子研究實(shí)驗(yàn)室雇員在20世紀(jì)40年代后半期。1948年他在麻省理工學(xué)院的辯訴“關(guān)于信息4的傳播”獲得了極大的好評(píng)。在他的論文蒂尤爾開始參照Nyquist的和哈特利的工程（見下文）倚在采樣和一個(gè)帶限信號(hào)量化使用，他們認(rèn)為由一個(gè)帶符號(hào)間干擾限制渠道引進(jìn)原

14、則上可以淘汰，他的國家非常正確，在無噪聲條件下無限量信息可以傳輸?shù)倪@樣一個(gè)渠道?？紤]到噪音，他提供了一個(gè)參數(shù)</p><p>　　H ≤ 2BT log(1 + C/N)</p><p>　　這句話，和香農(nóng)公式驚人的相似，大多數(shù)讀者會(huì)視為等同。有趣的是，注意，在推導(dǎo)（2）蒂尤爾承擔(dān)使用PCM編碼。一個(gè)不是由香農(nóng)引用的工作成為克拉唯愛紙。在一個(gè)類似的方式蒂尤爾，開始出與哈特利的工作，并假設(shè)使

15、用的PCM編碼，鍵盤基本上找到一個(gè)公式等價(jià)于（1）及（2）。第四個(gè)獨(dú)立的發(fā)現(xiàn)是在1948年由Laplume一出版。</p><p>　　On Shannon and “Shannon’s formula”</p><p>　　Lars Lundheim</p><p>　　Department of Telecommunication, Norwegian Univ

16、ersity of Science and</p><p>　　Technology (NTNU)</p><p>　　The period between the middle of the nineteenth and the middle of the twentieth century represents a</p><p>　　remarkable pe

17、riod in the history of science and technology. During this epoch, several discoveries and</p><p>　　inventions removed many practical limitations of what individuals and societies could achieve.</p>&l

18、t;p>　　Especially in the field of communications, revolutionary developments took place such as high speed</p><p>　　railroads, steam ships, aviation and telecommunications.</p><p>　　It is inte

19、resting to note that as practical limitations were removed, several fundamental or principal</p><p>　　limitations were established. For instance, Carnot showed that there was a fundamental limit to how</p

20、><p>　　much energy could be extracted from a heat engine. Later this result was generalized to the second law</p><p>　　of thermodynamics. As a result of Einstein’s special relativity theory, the ex

21、istence of an upper velocity</p><p>　　limit was found. Other examples include Kelvin’s absolute zero, Heissenberg’s uncertainty principle</p><p>　　and Gödel’s incompleteness theorem in math

22、ematics. Shannon’s Channel coding theorem, which was</p><p>　　published in 1948, seems to be the last one of such fundamental limits, and one may wonder why all of</p><p>　　them were discovered

23、during this limited time-span. One reason may have to do with maturity. When a</p><p>　　field is young, researchers are eager to find out what can be done – not to identify borders they cannot</p><

24、;p>　　pass. Since telecommunications is one of the youngest of the applied sciences, it is natural that the</p><p>　　more fundamental laws were established at a late stage.</p><p>　　In the pre

25、sent paper we will try to shed some light on developments that led up to Shannon’s</p><p>　　information theory. </p><p>　　“Shannon’s formula”</p><p>　　Sometimes a scientific result

26、comes quite unexpected as a “stroke of genius” from an individual</p><p>　　scientist. More often a result is gradually revealed, by several independent research groups, and at a</p><p>　　time wh

27、ich is just ripe for the particular discovery. In this paper we will look at one particular concept,</p><p>　　the channel capacity of a band-limited information transmission channel with additive white, Gaus

28、sian</p><p>　　noise. This capacity is given by an expression often known as “Shannon’s formula1”:</p><p>　　C = W log2(1 + P/N) bits/second. (1)</p><p>　　We intend to show that, on t

29、he one hand, this is an example of a result for which time was ripe exactly</p><p>　　a few years after the end of World War II. On the other hand, the formula represents a special case of</p><p>

30、;　　Shannon’s information theory2 presented in [1], which was clearly ahead of time with respect to the</p><p>　　insight generally established. Many mathematical expressions are connected with Shannon’s name.

31、 The one quoted here is not the</p><p>　　most important one, but perhaps the most well-known among communications engineers. It is also the</p><p>　　one with the most immediately understandable

32、significance at the time it was published.</p><p>　　2 For an introduction to Shannon’s work, see the paper by N. Knudtzon in this issue.</p><p>　　“Shannon’s formula” (1) gives an expression for

33、how many bits of information can be transmitted</p><p>　　without error per second over a channel with a bandwidth of W Hz, when the average signal power is</p><p>　　limited to P watt, and the si

34、gnal is exposed to an additive, white (uncorrelated) noise of power N with</p><p>　　Gaussian probability distribution. For a communications engineer of today, all the involved concepts</p><p>　　

35、are familiar – if not the result itself. This was not the case in 1948. Whereas bandwidth and signal</p><p>　　power were well-established, the word bit was seen in print for the first time in Shannon’s paper

36、. The</p><p>　　notion of probability distributions and stochastic processes, underlying the assumed noise model, had</p><p>　　been used for some years in research communities, but was not part o

37、f an ordinary electrical engineer’s</p><p><b>　　training.</b></p><p>　　The essential elements of “Shannon’s formula” are:</p><p>　　1. Proportionality to bandwidth W</

38、p><p>　　2. Signal power S</p><p>　　3. Noise power P</p><p>　　4. A logarithmic function</p><p>　　The channel bandwidth sets a limit to how fast symbols can be transmitted o

39、ver the channel. The signal</p><p>　　to noise ratio (P/N) determines how much information each symbol can represent. The signal and noise</p><p>　　power levels are, of course, expected to be mea

40、sured at the receiver end of the channel. Thus, the</p><p>　　power level is a function both of transmitted power and the attenuation of the signal over the</p><p>　　transmission medium (channel)

41、.</p><p>　　The most outstanding property of Shannon’s papers from 1948 and 1949 is perhaps the unique</p><p>　　combination of generality of results and clarity of exposition. The concept of an i

42、nformation source is</p><p>　　generalized as a symbol-generating mechanism obeying a certain probability distribution. Similarly, the</p><p>　　channel is expressed essentially as a mapping from

43、one set of symbols to another, again with an</p><p>　　associated probability distribution. Together, these two abstractions make the theory applicable to all</p><p>　　kinds of communication syst

44、ems, man-made or natural, electrical or mechanical.</p><p>　　Claude Elwood Shannon (1916-2001), the founder of information theory, had also a practical and</p><p>　　a playful side. The photo sho

45、ws him with one of his inventions: a mechanical “mouse” that could</p><p>　　find its way through a maze. He is also known for his electronic computer working with roman</p><p>　　numerals and a g

46、asoline-powered pogo stick.</p><p>　　Independent discoveries</p><p>　　One indicator that the time was ripe for a fundamental theory of information transfer in the first postwar</p><p&

47、gt;　　years is given in the numerous papers attempting at such theories published at that time. In</p><p>　　particular, three sources give formulas quite similar to (1). The best known of these is the book en

48、titled</p><p>　　Cybernetics [2] published by Wiener in 1949. Norbert Wiener was a philosophically inclined and</p><p>　　proverbially absent-minded professor of mathematics at MIT. Nonetheless, h

49、e was deeply concerned</p><p>　　about the application of mathematics in all fields of society. This interest led him to founding the</p><p>　　science of cybernetics. This field, which is perhaps

50、 best defined by the subtitle of [2]: “Control and</p><p>　　Communication in the Animal and the Machine” included, among other things, a theory for</p><p>　　information content in a signal and t

51、he transmission of this information through a channel. Wiener was,</p><p>　　however, not a master of communicating his ideas to the technical community, and even though the</p><p>　　relation to

52、Shannon’s formula is pointed out in [2], the notation is cumbersome, and the relevance to</p><p>　　practical communication systems is far from obvious.</p><p>　　Reference to Wiener’s work was do

53、ne explicitly by Shannon in [1]. He also acknowledged the work by</p><p>　　Tuller3. William G. Tuller was an employee at MIT’s Research Laboratory for Electronics in the</p><p>　　second half of

54、the 1940s. In 1948 he defended a thesis at MIT on “Theoretical Limitations on the Rate</p><p>　　of Transmission of Information4”. In his thesis Tuller starts by referring to Nyquist’s and Hartley’s</p>

55、<p>　　works (see below). Leaning on the use of sampling and quantization of a band-limited signal, and</p><p>　　arguing that intersymbol interference introduced by a band-limited channel can in princi

56、ple be</p><p>　　eliminated, he states quite correctly that under noise-free conditions an unlimited amount of</p><p>　　information can be transmitted over such a channel. Taking noise into accou

57、nt, he delivers an argument</p><p>　　partly based on intuitive reasoning, partly on formal mathematics, arriving at his main result that the</p><p>　　information H transmitted over a transmissio

58、n link of bandwidth B during a time interval T with</p><p>　　carrier-to-noise-ratio C/N is limited by</p><p>　　H ≤ 2BT log(1 + C/N). (2)</p><p>　　This expression has a striking rese

59、mblance to Shannon’s formula, and would by most readers be</p><p>　　considered equivalent. It is interesting to note that for the derivation of (2) Tuller assumes the use of</p><p>　　PCM encodin

60、g.</p><p>　　A work not referenced by Shannon is the paper by Clavier [16]5. In a similar fashion to Tuller, starting</p><p>　　out with Hartley’s work, and assuming the use of PCM coding, Clavier

61、 finds a formula essentially</p><p>　　equivalent to (1) and (2). A fourth independent discovery is the one by Laplume published in 1948</p><p>　　以上內(nèi)容出自：http://citeseerx.ist.psu.edu/viewdoc/summa