外文资料翻译信号与系统.docx

1、外文资料翻译信号与系统外文资料Signals and SystemSignals are scalar-valued functions of one or more independent variables. Often for convenience, when the signals are one-dimensional, the independent variable is referred to as “time” The independent variable may be continues or discrete. Signals that are continuous

2、 in both amplitude and time (often referred to as continuous -time or analog signals) are the most commonly encountered in signal processing contexts. Discrete-time signals are typically associated with sampling of continuous-time signals. In a digital implementation of signal processing system, qua

3、ntization of signal amplitude is also required . Although not precisely Correct in every context, discrete-time signal processing is often referred to as digital signal processing.Discrete-time signals, also referred to as sequences, are denoted by functions whose arguments are integers. For example

4、 , 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 to 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 .Some s

5、equences and classes of sequences play a particularly important role in discrete-time signal processing .These are summarized below. The unit sample sequence, denoted by (n)=1 ,n=0 ，(n)=0,otherwise (1)The sequence (n) play a role similar to an impulse function in analog analysis .The unit step seque

6、nce ,denoted by u(n), is defined as U(n)=1 , n0 u(n)=0 ,otherwise (2)Exponential sequences of the form X(n)= (3)Play a role in discrete time signal processing similar to the role played by exponential functions in continuous time signal processing .Specifically, they are eigenfunctions of discrete t

7、ime linear system and for that reason form the basis for transform analysis techniques. When =1, x(n) takes the form x(n)= A (4) Because the variable n is an integer ,complex exponential sequences separated by integer multiples of 2 in (frequency) are identical sequences ,I .e: (5) This fact forms t

8、he core of many of the important differences between the representation of discrete time signals and systems .A general sinusoidal sequence can be expressed as x(n)=Acos(n +) (6)where A is the amplitude , the frequency, and the phase .In contrast with continuous time sinusoids, a discrete time sinus

9、oidal signal is not necessarily periodic and if it is the periodic and if it is ,the period is 2/0 is an integer .In both continuous time and discrete time ,the importance of sinusoidal signals lies in the facts that a broad class of signals and that the response of linear time invariant systems to

10、a sinusoidal signal is sinusoidal with the same frequency and with a change in only the amplitude and phase .Systems:In 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 cha

11、racterization of a system requires a 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 simplify the characterization and analysis of a sys

12、tem are linearity and time invariance, alternatively referred as shift invariance . Fortunately, many system can often be approximated by a linear and time invariant system . The linearity of a system is defined through the principle of superposition:Tax1(n)+bx2(n)=ay1(n)+by2(n) (7)Where Tx1(n)=y1(n

13、) , Tx2(n)=y2(n), and a and b are any scalar constants.Time invariance of a system is defined as Time invariance Tx(n-n0)=y(n-n0) (8)Where y(n)=Tx(n) andis a integer linearity and time inva riance are independent properties, i.e ,a system may have one but not the other property ,both or neither .For

14、 a linear and time invariant (LTI) system ,the system response y(n) is given by y(n)= (9)where x(n) is the input and h(n) is the response of the system when the input is (n).Eq(9) is the convolution sum .As with continuous time convolution ,the convolution operator in Eq(9) is commutative and associ

15、ative and distributes over addition:Commutative : x(n)*y(n)= y(n)* x(n) (10)Associative: x(n)*y(n)*w(n)= x(n)* y(n)*w(n) (11)Distributive: x(n)*y(n)+w(n)=x(n)*y(n)+x(n)*w(n) (12)In continuous time systems, convolution is primarily an analytical tool. For discrete time system ,the convolution sum. In

16、 addition to being important in the analysis of LTI systems, namely those for which the impulse response if of finite length (FIR systems).Two additional system properties that are referred to frequently are the properties of stability and causality .A system is considered stable in the bounded inpu

17、t-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 is that unit sample response h(n) be absolutely summableFor an LTI system,Stability (13)Because of Eq.(13),an absolutely summable sequence

18、is often referred to as a stable sequence.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 nn0.A necessary and sufficient condition for an LTI system to be causal is that its unit sample response h(n) be zero for n0.For an

19、LTI system. Causality： h(n)=0 for n 0 (14)Because of Eq.14.a sequence that is zero for n0 is often referred to as a causal sequence.1.Frequency-domain representation of signalsIn this section, we summarize the representation of sequences as linear combinations of complex exponentials, first for peri

20、odic sequence using the discrete-time Fourier series, next for stable sequences using the discrete-time 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 sectio

21、n 1.3.3.we review the use of these representation in charactering LIT systems.Discrete-time Fourier seriesAny periodic sequence x(n) with period N can be represented through the discrete time series(DFS) pair in Eqs.(15)and (16)Synthesis equation : = (15) Analysis equation: = (16) The synthesis equa

22、tion expresses the periodic sequence as a linear combination of harmonically related complex exponentials. The choice of interpreting the DFS coefficients X(k) either as zero outside the range 0k(N-1) or as periodically accepted convention , however ,to interpret X(k) as periodic to maintain a duali

23、ty between the analysis and synthesis equations.2.Discrete Time Fourier Transform Any stable sequence x(n) (i.e. one that is absolutely summable ) can be represented as a linear combination of complex exponentials. For a periodic stable sequences, the synthesis equation takes the form of Eq.(17),and

24、 the analysis equation takes the form of Eq.(18)synthesis equation: x(n)= (17) analysis equation: X()= (18) 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 (19

25、) Then the DFS coefficients of x1(n) are proportional to samples spaced by 2/N of the Fourier Transform x(n). Specifically, X1(k0=1/N X() (20)Among other things ,this implies that the DFS coefficients of a periodic signal are proportional to the discrete Fourier Transform of one period .3.Z Transfor

26、m 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 the magnitudes may or may not be unity.The Z Transform analysis and synthesis equations are as follows:synthesis equations

27、 : x(n)= (21) analysis equations : X(z)= (22) From Eqs.(18) and (22) ,X() is relate to X(z) by X()= X(z) z= ,I.e ,for a stable sequence, the Fourier Transform X() is the Z Transform evaluated on the contour |z|=1,referred to as the unit circle .Eq.(22) converge only for some value of z and not other

28、s ,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.We summarize the properties of the z-transform but also of the ROC. For example, the two sequences and Have z-transforms that are identic

29、al algebraically and that differ only in the ROC .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 formal means for obtaining x(n) from X(z),its evaluation requir

30、es 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) using a partial fraction of equation. The inverse z-transform of the individual simpler terms can usually then be recognized by inspection .Th

31、ere are a number of important properties of the ROC that, together with properties of the time domain sequence, permit implicit specification of the ROC. This properties are summarized as follows:Propotiey1. The ROC is a connected region .Propotiey2. For a rational z-transform, the ROC does not cont

32、ain any poles and is bounded by poles.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 0zr0 will be in the ROC.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 0zr0 will be in the ROC.Propotiey5. If x(n)is a stable and casual sequence with a rational z-tra