外文翻譯---香農和“香農公式”

1、<p><b>　　畢業(yè)設計/論文</b></p><p>　　外 文 文 獻 翻 譯</p><p>　　系　 別 通信工程與技術系 </p><p>　　專 業(yè) 班 級 0703 　　 </p><p>　　姓　 名

2、 </p><p>　　評 分 </p><p>　　指 導 教 師 </p><p>　　20 年 月 日</p><p><b>　　香農和“香農公式”</b&g

3、t;</p><p>　　Lars Lundheim{挪威科學技術大學，通信系，教授}</p><p>　　在科學技術史上，十九世紀中葉和二十世紀中葉是一個非常突出的時期。在這個時期，一些發(fā)明和創(chuàng)作的實現(xiàn)消除了許多對個人和社會的有影響的限制。特別是在通信領域，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，從熱機提煉出來的熱能量是有一個極限值的。后來，這一定律被推廣到了第二定律熱力學。就像愛因斯坦的狹義相對論，一個機制的速度被發(fā)現(xiàn)。其他例子包括Kelvin的絕對零度，Heissenberg的測不準原理和哥德爾不完備性定理的數(shù)學。在1948年出版的香農信道編碼定理似乎是最后一個被發(fā)現(xiàn)的限制，人們或者會奇怪，為什么所有的限制都是在這個有限的時間內完成的。有一個原因可以解釋。當一個學者還年輕的時候，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。由于通信是在科學發(fā)展中時間最短的一個，很自然的它基本的法則是建立在后期階段。在本論文中，我們會盡量除開一些事態(tài)的發(fā)展對香農公式的影響。</p><p><b>　　香農公式</b></p><p>　　有時候，一個“天才之舉”科學成果的產生會讓人無法想象是來自于一個私人科學家。更常見的是有幾個獨立的小組在一個特定的成熟的時機逐步研究出來的

6、。在本文中我們將著眼于一個特定的概念，一個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，當時已經(jīng)是一個成熟的時機，另一方面，香農公式是一個非常特殊的發(fā)現(xiàn)，很多有識之士都非常的認同這個觀點。許多數(shù)學表達式都用有關香農的名字來命

7、名。這個例子并不是最著名的，但在通信領域中或許是最知名的。這也是當時最迅速最有價值的理解之一了。</p><p>　　對香農工作的介紹，這個問題在N. Knudtzon的論文上看到了。</p><p>　　香農公式：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制信息的差錯率為1/2。作為一個通信工程師，不論誰都應該對這個定理很熟悉了，即使理解那個結果。這在1948年已經(jīng)是不爭的事實了。而帶寬和信號的力量因此出名，并且出現(xiàn)在第一期的“香農”報上。概念的概率分布和隨機過程、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,而不是一個普通的電氣工程師的培訓。</p><p>　　香農公式的基本因

9、素是：</p><p><b>　　帶寬w</b></p><p><b>　　信號功率s</b></p><p><b>　　噪聲功率p</b></p><p><b>　　他們構成對數(shù)函數(shù)</b></p><p>　　信道帶寬的

10、限制，通過設置一個符號，以便通過快速通道進行信道傳輸。信噪比可以決定每個符號能代表多少信息。在結束時，信噪比可以用來計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年它們的共性相結合時的獨特效果和共性闡述。它們服從一定的概率分布。同樣的，信道基本上表示從一個符號映射到另一個符號并且服從相應的概率分布。結合起來，這個結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,也是一個實踐和頑皮的一邊。這張照片顯示他發(fā)明的一個機械“老鼠",可以找到它的方式通過一個迷宮。他也被他的信息工作和數(shù)字工作累的瘦長。</p><p><b>　　獨立發(fā)現(xiàn)：</b></p><p>　　一個重要指標

12、,時機已經(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。特別是三個公式相當?shù)南嗨啤＿@些最著名的是公布維納于1949年的《控制論》。羅伯特維拉是一個麻省理工學院的教授，總所周知的是，他是一個有哲學傾向的心不在焉的數(shù)學教授。盡管如此，他深為關切關于數(shù)學在社會各個領域的應用。這種哲學興趣正好讓他建立了科學的控制論。這個領域，這也許是最好的了[2]字幕定義：“控制與通信在動物和機器“包括，除其他事項外，理論信息內容在一個信

13、號，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，關于香農公式，相關的信息和繁瑣的符號表現(xiàn)的并不明顯。參考與維拉的工作，更加證明了香農公式的正確性蒂尤爾是麻省理工學院的電子研究實驗室雇員在20世紀40年代后半期。1948年他在麻省理工學院的辯訴“關于信息4的傳播”獲得了極大的好評。在他的論文蒂尤爾開始參照Nyquist的和哈特利的工程（見下文）倚在采樣和一個帶限信號量化使用，他們認為由一個帶符號間干擾限制渠道引進原

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@樣一個渠道?？紤]到噪音，他提供了一個參數(shù)</p><p>　　H ≤ 2BT log(1 + C/N)</p><p>　　這句話，和香農公式驚人的相似，大多數(shù)讀者會視為等同。有趣的是，注意，在推導（2）蒂尤爾承擔使用PCM編碼。一個不是由香農引用的工作成為克拉唯愛紙。在一個類似的方式蒂尤爾，開始出與哈特利的工作，并假設使

15、用的PCM編碼，鍵盤基本上找到一個公式等價于（1）及（2）。第四個獨立的發(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>　　以上內容出自：http://citeseerx.ist.psu.edu/viewdoc/summa