外文資料翻譯---信號與系統(tǒng)

1、<p><b>　　外文資料</b></p><p>　　Signals and System</p><p>　　Signals are scalar-valued functions of one or more independent variables. Often for convenience, when the signals are one-

2、dimensional, the independent variable is referred to as “time” The independent variable may be continues or discrete. Signals that are continuous in both amplitude and time (often referred to as continuous -time or analo

3、g signals) are the most commonly encountered in signal processing contexts. Discrete-time signals are typically associated with sampling of continuous-time si</p><p>　　Discrete-time signals, also referred to

4、 as sequences, are denoted by functions whose arguments are integers. For example , x(n) represents a sequence that is defined for integer values of n and undefined for non-integer value of n . The notation x(n) refers t

5、o the discrete time function x or to the value of function x at a specific value of n .The distinction between these two will be obvious from the contest .</p><p>　　Some sequences and classes of sequences p

6、lay a particularly important role in discrete-time signal processing .These are summarized below. </p><p>　　The unit sample sequence, denoted by δ(n)=1 ,n=0 ，δ(n)=0,otherwise (1)</p><p&g

7、t;　　The sequence δ(n) play a role similar to an impulse function in analog analysis .</p><p>　　The unit step sequence ,denoted by u(n), is defined as </p><p>　　U(n)=1 , n≧0 u(n)=0 ,otherwise

8、 (2)</p><p>　　Exponential sequences of the form </p><p>　　X(n)= (3)</p><p>　　Play a role in discrete time signal processing simi

9、lar to the role played by exponential functions in continuous time signal processing .Specifically, they are eigenfunctions of discrete time linear system and for that reason form the basis for transform analysis techniq

10、ues. When ︳α ︳=1, x(n) takes the form </p><p>　　x(n)= A (4) </p><p>　　Because the variable n is an integer ,complex expo

11、nential sequences separated by integer multiples of 2π in ω(frequency) are identical sequences ,I .e:</p><p>　　(5) </p><p>　　This fact forms the core of

12、 many of the important differences between the representation of discrete time signals and systems .</p><p>　　A general sinusoidal sequence can be expressed as </p><p>　　x(n)=Acos(n +Φ)

13、 (6)</p><p>　　where A is the amplitude , the frequency, and Φ the phase .</p><p>　　In contrast with continuous time sinusoids, a discrete time sinusoidal signal is not necessa

14、rily periodic and if it is the periodic and if it is ,the period is 2π/ω0 is an integer .</p><p>　　In both continuous time and discrete time ,the importance of sinusoidal signals lies in the facts that a bro

15、ad class of signals and that the response of linear time invariant systems to a sinusoidal signal is sinusoidal with the same frequency and with a change in only the amplitude and phase .</p><p>　　Systems:In

16、 general, a system maps an input signal x(n) to an output signal y(n) through a system transformation T{.}.The definition of a system is very broad . without some restrictions ,the characterization of a system requires a

17、 complete input-output relationship knowing the output of a system to a certain set of inputs dose not allow us to determine the output of a system to other sets of inputs . Two types of restrictions that greatly simplif

18、y the characterization and analysis of a system ar</p><p>　　T{ax1(n)+bx2(n)}=ay1(n)+by2(n) (7)</p><p>　　Where T{x1(n)}=y1(n) , T{x2(n)}=y2(n), and a and b are any scalar con

19、stants.</p><p>　　Time invariance of a system is defined as Time invariance</p><p>　　T{x(n-n0)}=y(n-n0) (8)</p><p>　　Where y(n)=T{x(n)} andis a integer line

20、arity and time inva riance are independent properties, i.e ,a system may have one but not the other property ,both or neither .</p><p>　　For a linear and time invariant (LTI) system ,the system response y(n)

21、 is given by </p><p>　　y(n)= (9)</p><p>　　where x(n) is the input and h(n) is the response of the system when the input is δ(n).Eq(9) is the convolution sum .</p><p&

22、gt;　　As with continuous time convolution ,the convolution operator in Eq(9) is commutative and associative and distributes over addition:</p><p>　　Commutative :</p><p>　　x(n)*y(n)= y(n)* x(n)

23、 (10)</p><p>　　Associative:</p><p>　　[x(n)*y(n)]*w(n)= x(n)*[ y(n)*w(n)] (11)</p><p>　　Distributive:</p><p>　　x(n)*[y(n)+w(n)]=x(n)*y

24、(n)+x(n)*w(n) (12)</p><p>　　In continuous time systems, convolution is primarily an analytical tool. For discrete time system ,the convolution sum. In addition to being important in the analysi

25、s of LTI systems, namely those for which the impulse response if of finite length (FIR systems).</p><p>　　Two additional system properties that are referred to frequently are the properties of stability and

26、causality .A system is considered stable in the bounded input-bounder output(BIBO)sense if and only if a bounded input always leads to a bounded output. A necessary and sufficient condition for an LTI system to be stable

27、 is that unit sample response h(n) be absolutely summable</p><p>　　For an LTI system,</p><p>　　Stability (13)</p><p>　　Because of Eq.(13),an absolutely

28、summable sequence is often referred to as a stable sequence.</p><p>　　A system is referred to as causal if and only if ,for each value of n, say n, y(n) does not depend on values of the input for n<n0.A n

29、ecessary and sufficient condition for an LTI system to be causal is that its unit sample response h(n) be zero for n<0.For an LTI system. Causality：</p><p>　　h(n)=0 for n <0

30、 (14)</p><p>　　Because of Eq.14.a sequence that is zero for n<0 is often referred to as a causal sequence.</p><p>　　1.Frequency-domain representation of signals</p><p>　　In this

31、 section, we summarize the representation of sequences as linear combinations of complex exponentials, first for periodic sequence using the discrete-time Fourier series, next for stable sequences using the discrete-time

32、 Fourier transform, then through a generalization of discrete-time Fourier transform, namely, the z-transform, and finally for finite-extent sequence using the discrete Fourier transform. In section 1.3.3.we review the u

33、se of these representation in charactering LIT syste</p><p>　　Discrete-time Fourier series</p><p>　　Any periodic sequence x(n) with period N can be represented through the discrete time series(D

34、FS) pair in Eqs.(15)and (16)</p><p>　　Synthesis equation :</p><p>　　= (15) </p><p>　　Analysis equation:<

35、/p><p>　　= (16) </p><p>　　The synthesis equation expresses the periodic sequence as a linear combination of harmonically related complex exp

36、onentials. The choice of interpreting the DFS coefficients X(k) either as zero outside the range 0≦k≦(N-1) or as periodically accepted convention , however ,to interpret X(k) as periodic to maintain a duality between the

37、 analysis and synthesis equations.</p><p>　　2.Discrete Time Fourier Transform </p><p>　　Any stable sequence x(n) (i.e. one that is absolutely summable ) can be represented as a linear combinatio

38、n of complex exponentials. For a periodic stable sequences, the synthesis equation takes the form of Eq.(17),and the analysis equation takes the form of Eq.(18)</p><p>　　synthesis equation:</p><p

39、>　　x(n)= (17) </p><p>　　analysis equation:</p><p>　　X(ω)= (18) <

40、/p><p>　　To relate the discrete time Fourier Transform and the discrete time Fourier Transform series, consider a stable sequence x(n) and the periodic signal x1(n) formed by time aliasing x(n),i.e </p>

41、<p>　　(19) </p><p>　　Then the DFS coefficients of x1(n) are proportional to samples spaced by 2π/N of the Fourier Transform x(n). Specifically, </p><p>　　X1(k

42、0=1/N X(ω) (20)</p><p>　　Among other things ,this implies that the DFS coefficients of a periodic signal are proportional to the discrete Fourier Transform of one period .</p><

43、p>　　3.Z Transform </p><p>　　A generalization of the Fourier Transform, the z transform ,permits the representation of a broader class of signals as a linear combination of complex exponentials, for which

44、the magnitudes may or may not be unity.</p><p>　　The Z Transform analysis and synthesis equations are as follows:</p><p>　　synthesis equations :</p><p>　　x(n)=

45、 (21) </p><p>　　analysis equations :</p><p>　　X(z)= (22) </p><p>　　From Eqs.(18) and (2

46、2) ,X(ω) is relate to X(z) by X(ω)= X(z) z= ,I.e ,for a stable sequence, the Fourier Transform X(ω) is the Z Transform evaluated on the contour |z|=1,referred to as the unit circle .</p><p>　　Eq.(22) conver

47、ge only for some value of z and not others ,The range of values of z for which X(z) converges, i.e, the region of convergence(ROC) ,corresponds to the values of z for which x(n)z-n is absolutely summable.</p><

48、;p>　　We summarize the properties of the z-transform but also of the ROC. For example, the two sequences and </p><p>　　Have z-transforms that are identical algebraically and that differ only i

49、n the ROC .</p><p>　　The synthesis equation as expressed in Eq.(21) is a contour integral with the contour encircling the origin and contained within the region of convergence. While this equation provides a

50、 formal means for obtaining x(n) from X(z),its evaluation requires contour integration. Such an integer tedious and usually unnecessary . When X(z) is a rational function of z , a more typically approach is to expand X(z

51、) using a partial fraction of equation. The inverse z-transform of the individual simpler term</p><p>　　There are a number of important properties of the ROC that, together with properties of the time domain

52、 sequence, permit implicit specification of the ROC. This properties are summarized as follows:</p><p>　　Propotiey1. The ROC is a connected region .</p><p>　　Propotiey2. For a rational z-trans

53、form, the ROC does not contain any poles and is bounded by poles.</p><p>　　Propotiey3. If x(n) is a right sided sequence and if the circle │z│=r0 is in the ROC, then all finite values of z for which 0<│z

54、│<r0 will be in the ROC.</p><p>　　Propotiey4. If x(n)is a left sided sequence and if the circle │z│=r0 is in the ROC, then all values of z for which 0<│z│<r0 will be in the ROC.</p><p&g

55、t;　　Propotiey5. If x(n)is a stable and casual sequence with a rational z-transform , then all the poles of X(z) are inside the unit circle .</p><p>　　4.Discrete Fourier Transform </p><p>　　In

56、section 1.3.2.1 ,we discussed the representation of periodic sequences in terms of the discrete Fourier series. With the correct interpretation, the same representation can be applied to finite duration sequences. The r

57、esulting Fourier representation for finite duration sequences is referred to as the Discrete Fourier Transform (DFT)</p><p>　　The DFT analysis and synthesis equations are </p><p>　　Analysis eq

58、uations:</p><p>　　X(k)= , 0≦k≦N-1 (23) </p><p>　　Synthesis equations:</p><p>　　x(n)= ,0≦n≦N-1 (24)</p>&l

59、t;p>　　The fact that X(k)=0 for k outside the interval 0≦k≦N-1 and that x(n)=0 for outside the interval 0≦k≦N-1 is implied but not always stated explicitly .</p><p>　　The DFT is used in a variety of signal

60、 processing applications, so it is of considerable interest to efficiently compute the DFT and inverse DFT. A straight forward computation of N-Point DFT or inverse DFT requires on the order of arithmetic operations (mul

61、tiplications and additions). The number of arithmetic operations required is significantly reduced through the set of Fast Fourier transform (FFT) algorithms. most FFT algorithms are based on the simple principle that an

62、 N-point DFT can be a </p><p>　　4.Frequency-domain representation of LTI system</p><p>　　In section 132 we reviewed the representation of signal as a linear combination of complex exponentials o

63、f the form or more generally .For linear systems, the response is then the same linear combination of the response to the individual complex exponentials .</p><p>　　If in addition the system is time invarian

64、t,the complex exponentials are eigenfunctions ,Consequently, the system can be characterized by the spectrum of eigen-values, corresponding to the frequency response if the signal decomposition is in terms of complex exp

65、onentials with unity magnitude or ,more generally ,to the system function in the contexts of the more general complex exponential .</p><p>　　The eigenfunction property follows directly from the convolution s

66、um and state that with x(n)= ,the output y(n) has the form </p><p>　　y(n)=H(z) (25) </p><p><b>　　where </b><

67、/p><p>　　H(z)= (26) </p><p>　　The system function H(z) is eigenvalue associated with the eigenfunction also, from Eq.(26).H(z) is the z-tra

68、nsform of the system unit sample response .when z= ,it correspond to the Fourier transform of the unit sample response .</p><p>　　Since Eq.(17) or (21) corresponds to a decomposition of x(n) as a linear co

69、mbination of complex exponentials .we can obtain the response y(n),using linearity and the eigenfunction property ,by multiplying the amplitudes of the eigenfunctions </p><p>　　In Eq.(22) by the eigenva

70、lues H(z),i.e.,</p><p>　　y(n)= (27) </p><p>　　Eq.(27) then becomes the synthesis equation for the output ,i.e. </p><p>　　Y(z)=H(z)X(z)

71、 (28) </p><p>　　Eq.(28) corresponds to the z-transform convolution n property.</p><p>　　System characterized by linear constant –coefficient difference equations</p><p>　　A

72、particularly important class of discrete time system are those characterized by linear constant-coefficient difference equations (LCCDE) of the form </p><p>　　(29)

73、 </p><p>　　Where the and the are constants .Eq.(29) is typically referred to as an Nth-order difference equation .</p><p>　　A system characterized by an Nth-order difference equation of th

74、e form in Eq.(29) represents a linear time-invariant system only under an appropriate choice of the homogeneous to the equation itself ,Even under these additional constrains ,the system is not restricted to be casual .&

75、lt;/p><p>　　5.Solution of linear constant coefficient difference equation</p><p>　　Assuming the system is linear ,time-invariant ,and casual, the response of a system characterized by Eq.(29) can b

76、e obtained recursively </p><p>　　Specifically, we can rewrite Eq.(29) as </p><p>　　y(n)= - (30) </p><p>　　Since we assu

77、ming that the system is linear and casual ,if x(n)=0 for n<n0 then y(n)=0 for n<n0. with this assumed zero state ,y(n) can be generated recursively from Eq.(30).,where represents unit delay .while the recursion in

78、 Eq.(30) will generate the correct output sequence and .in fact, represents a specific algorithm for computing the output .The result will not be in any analytical convenient form .A convenient procedure to obtain the so

79、lution analytically is through the z-transform .Speci</p><p>　　H(z)= (31) </p><p>　　Eq(31) specifies the algebraic expres

80、sion for the system function , which we note is a rational function of z. It does not, however, explicitly specify the ROC. If we assume that the system is casual , then the ROC associated with Eq.(31) will be region out

81、side a circle passing through the outmost pole of H(z).If we do not impose causality ,then in general there are many choices for the ROC and correspondingly for the system impulse response .</p><p><b>

82、　　中文譯文</b></p><p><b>　　信號與系統(tǒng)</b></p><p>　　信號是一個獨立或多個獨立的變量的向量函數(shù)，為了方便，當(dāng)信號是一維的時候，該變量是時間函數(shù)，這個獨立的變量可能是連續(xù)的或是離散的。在幅度和時間上連續(xù)的信號(通常被認為時間連續(xù)或模擬信號)，這些都是在數(shù)字信號處理中常遇到的。離散時間信號與連續(xù)時間信號的抽樣信號有著密切的聯(lián)

83、系。在一個數(shù)字處理系統(tǒng)的數(shù)字設(shè)備中，信號幅度的量化是必要的。盡管不是在每一個領(lǐng)域都是十分精確，離散時間信號處理通常被認為是數(shù)字信號處理過程。</p><p>　　離散時間信號，通常又被稱為序列，用一些幅度為整數(shù)的函數(shù)來表征。例如X(n)代表：對于給定一個特征值n,離散函數(shù)或函數(shù)值x,這兩者之間的區(qū)別將在此中得到說明。</p><p>　　一些序列或一個序列數(shù)，在離散數(shù)字信號處理中充當(dāng)特別的

84、角色。它們概括如下：</p><p>　　δ(n)=1 n=0 δ(n)=0 (1)</p><p>　　序列δ(n)扮演的角色如同沖擊函數(shù)在模擬系統(tǒng)分析中的一樣。</p><p>　　單位階躍序列，用u(n)來表征，被定義如下：</p><p>　　u(n)=1 n≧0

85、 u(n)=0 (2)</p><p><b>　　指數(shù)序列的形式為：</b></p><p>　　X(n)= (3)</p><p>　　它在離散時間信號處理中的作用如同指數(shù)函數(shù)在連續(xù)時間信號處理過程中的作用一樣。特別的，它們是離散時間線性

86、系統(tǒng)的表征函數(shù)，也由于此，構(gòu)成變換分析技術(shù)的基礎(chǔ)。當(dāng)X(n)呈現(xiàn)復(fù)雜的指數(shù)序列的形式的時候，被表達為：</p><p>　　x(n)= A (4)</p><p>　　由于變量n是一個整數(shù)，復(fù)雜的指數(shù)序列在ω(頻率)上以2π的整數(shù)倍被分解為唯一的指數(shù)序列，例如：</p><p><b>　　(5)

87、</b></p><p>　　這個事實構(gòu)成了表征離散時間信號與系統(tǒng)和連續(xù)時間信號與系統(tǒng)不同特征的核心。一個規(guī)則的正弦曲線序列能被表達為：</p><p>　　x(n)=Acos(n +Φ) (6)</p><p>　　A為幅度，為頻率，Φ為相位，與連續(xù)的正弦時間信號相比，離散時間正弦信號不必是周期性的

88、，如果是，也只有在2π/是一個整數(shù)時，周期才為2π/，對于連續(xù)和離散時間信號，正弦信號的重要性在于這樣一個事實：大量的信號族能被這樣的正弦信號線性組合，同時正弦信號通過線性時不變系統(tǒng)的響應(yīng)是一個正弦的，它們有相同的頻率，僅僅只是在幅度和相位上有改變。</p><p>　　系統(tǒng)：通常，系統(tǒng)通過一個系統(tǒng)傳遞函數(shù)T{.}映射輸入函數(shù)X(n)，輸出信號Y(n)，這種對系統(tǒng)的定義太過于寬泛，如果沒有約束條件，系統(tǒng)的特征需要

89、一個完全的輸入——輸出系統(tǒng)。</p><p>　　如果對于一個特定的系統(tǒng)輸入有特定的輸出，無法推出與之不同輸入有著相同的輸出的結(jié)果。兩種類型的限制大大簡化了對系統(tǒng)的描述和分析，它們是線性和時不變的，時不變性又稱移不變性。幸運的是，許多系統(tǒng)在實際中能夠用一個線性的，時不變的系統(tǒng)逼近。</p><p><b>　　線性：</b></p><p>　

90、　T{ax1(n)+bx2(n)}=ay1(n)+by2(n) (7)</p><p>　　這里：T{x1(n)}=y1(n) , T{x2(n)}=y2(n) a,b 為常量</p><p>　　時不變系統(tǒng)可以定義如下：</p><p>　　T{x(n-n0)}=y(n-n0)

91、 (8)</p><p>　　這里：y(n)=T{x(n)} n0為任意常數(shù)。線性時不變有獨立的特性的特征，例如一個系統(tǒng)可能有這個特征但不具備另一個特征，或兼而有之或都不具備。</p><p>　　對于一個線性時不變系統(tǒng)(LTI)來說，系統(tǒng)響應(yīng)y(n)被定義為：</p><p>　　y(n)= (9)&l

92、t;/p><p>　　這里x(n)是輸入，h(n)是輸入為δ(n)時的系統(tǒng)響應(yīng)，等式(9)是卷積和。</p><p>　　由于是連續(xù)時間卷積，這個卷積運算等式(9)具有交換律，結(jié)合律，分配律的性質(zhì)。</p><p><b>　　交換律： </b></p><p>　　x(n)*y(n)= y(n)* x(n)

93、 (10)</p><p><b>　　結(jié)合律： </b></p><p>　　[x(n)*y(n)]*w(n)= x(n)*[ y(n)*w(n)] (11)</p><p><b>　　分配律： </b></p><p>　　x

94、(n)*[y(n)+w(n)]=x(n)*y(n)+x(n)*w(n) (12)</p><p>　　在連續(xù)時間系統(tǒng)中，卷積運算是一種分析工具。對于離散時間系統(tǒng)，卷積和除了在分析線性和時不變系統(tǒng)時很重要，在實現(xiàn)一種特殊的線性和時不變系統(tǒng)即有限沖擊響應(yīng)系統(tǒng)時，也因為其清晰的概念顯得很重要。另外兩個與頻率有關(guān)的系統(tǒng)特征是穩(wěn)定性與因果性。一個系統(tǒng)當(dāng)且僅當(dāng)一個有限的輸入導(dǎo)致一個有限的輸出時，

95、才被認為在有限輸入——有限輸出意義下是穩(wěn)定的。線性系統(tǒng)穩(wěn)定的充分必要條件是：h(n)絕對可和。</p><p>　　對于一個線性系統(tǒng)來說,</p><p><b>　　穩(wěn)定性：</b></p><p><b>　　(13)</b></p><p>　　由于等式(13)，一個絕對可和的連續(xù)性通常是一個

96、穩(wěn)定的連續(xù)性。一個系統(tǒng)有且僅有：對其任意一個n值，通常被稱作，若響應(yīng)y(n)并不決定于大于時候的值時，就稱該系統(tǒng)是因果的。一個線性系統(tǒng)是因果的充分必要條件是它的單位沖擊響應(yīng)h(n)在n<0時，輸出為0，對于一個線性時不變系統(tǒng)，</p><p><b>　　因果性：</b></p><p>　　h(n)=0 n<0

97、 (14)</p><p>　　因為等式(14)，當(dāng)n<0時，輸出值為0的連續(xù)性通常稱為因果連續(xù)性。</p><p>　　對信號頻率范圍的描述</p><p>　　在這部分，我們把序列的描述概括為一系列復(fù)雜的指數(shù)函數(shù)的疊加。首先對于周期的序列，用離散序列得到級數(shù)。對于恒等的序列用離散的復(fù)氏變換，即z變換。最后對于有限擴展序列用離散的復(fù)氏變換。在1

98、.3.3節(jié)中，我們回顧這些對LTI系統(tǒng)描述方法的應(yīng)用。</p><p>　　1.離散時間復(fù)氏系統(tǒng)</p><p>　　任意周期序列x(n)。周期為N,它可以用成對的等式(15)和(16)的復(fù)氏級數(shù)來描述。</p><p><b>　　合成方程：</b></p><p>　　=

99、(15)</p><p><b>　　分解方程：</b></p><p>　　= (16)</p><p>　　這個合成方程將周期序列表達為一系列有序相關(guān)復(fù)雜的指數(shù)函數(shù)的線性組合。DFS系數(shù)K的選擇范圍從0到N-1。不影響等式(15)周期性的重復(fù)序列。這通?？梢越邮?，但是把X(k)描述為周期的，這樣來保

100、持合成和分解等式的對偶性。</p><p><b>　　離散時間復(fù)氏變換</b></p><p>　　任何穩(wěn)定序列X(N)如：一個絕對可和的序列?？擅枋鰹橐幌盗袕?fù)雜指數(shù)函數(shù)的線性組合。對于一個周期穩(wěn)定序列。合成方程用式子(17)的形式表達，分解方程用式子(18)的形式來表達。</p><p><b>　　合成方程：</b>

101、</p><p>　　x(n)= (17)</p><p><b>　　分解方程：</b></p><p>　　X(ω)= (18)</p><p>　　為了聯(lián)立離散復(fù)氏變換和離散復(fù)氏級數(shù)，考慮一個穩(wěn)定序列x(n)和周期信號，如果

102、：</p><p><b>　　(19)</b></p><p>　　那么的DFS系數(shù)是抽樣間隔2π/N成常系數(shù)比的復(fù)氏變換x(n)。即</p><p>　　X1(k0=1/N X(ω) (20) </p><p

103、>　　對于其他，這意味著一個周期信號的DFS系數(shù)是與一個周期內(nèi)離散復(fù)氏變換成正比。</p><p><b>　　2.Z變換</b></p><p>　　復(fù)氏變換的普遍化，即Z變換，認為更加廣泛的信號響應(yīng)是一系列復(fù)雜的指數(shù)函數(shù)的線性組合，對它們來說，量值可能不唯一。</p><p>　　Z變換分解和綜合方程如下：</p>&l

104、t;p><b>　　綜合方程：</b></p><p>　　x(n)= (21)</p><p><b>　　分解方程：</b></p><p>　　X(z)= (22)</p><p>　　

105、從等式(18)和(21)，X(ω)通過X(ω)= (z) z=聯(lián)系X(z)</p><p>　　如：對于一個穩(wěn)定序列的復(fù)氏變換是在|z|=1時的Z變換值，Z映射為單位圓。 </p><p>　　等式(22)僅僅對于一部分Z值收斂。X(Z)收斂時Z的范圍，即收斂域，對應(yīng)x(n)z-n絕對可和的Z值。我們將在后面總結(jié)有關(guān)ROC的更多特征。</p><p>　　Z變換的