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1、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 in both am
2、plitude 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, quantization o
3、f 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 , x(n) rep
4、resents 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 sequences an
5、d 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 sequence ,denote
6、d 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 time linear
7、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 the core of
8、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 sinusoidal signa
9、l 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 a sinusoida
10、l 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 characterizati
11、on 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 system are lin
12、earity 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) , Tx2(n)=
13、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 a linear a
14、nd 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 associative and d
15、istributes 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 addition t
16、o 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 input-bounder o
17、utput(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 is often re
18、ferred 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 LTI system.
19、 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 periodic sequen
20、ce 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 section 1.3.3.we
21、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 equation expres
22、ses 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 duality between
23、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 the analys
24、is 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) Then the
25、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 Transform A general
26、ization 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 : x(n)= (2
27、1) 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 others ,The rang
28、e 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 identical algebrai
29、cally 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 requires contour
30、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 .There are a n
31、umber 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 contain any pol
32、es 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 i
33、n the ROC.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 .4.Discrete Fourier Transform In section 1.3.2.1 ,we discussed the representation of periodic sequences in terms of the discrete Fourier series. With the c
34、orrect interpretation, the same representation can be applied to finite duration sequences. The resulting Fourier representation for finite duration sequences is referred to as the Discrete Fourier Transform (DFT)The DFT analysis and synthesis equations are Analysis equations: X(k)= , 0kN-1 (23) Syn
35、thesis equations:x(n)= ,0nN-1 (24)The fact that X(k)=0 for k outside the interval 0kN-1 and that x(n)=0 for outside the interval 0kN-1 is implied but not always stated explicitly .The DFT is used in a variety of signal processing applications, so it is of considerable interest to efficiently compute
36、 the DFT and inverse DFT. A straight forward computation of N-Point DFT or inverse DFT requires on the order of arithmetic operations (multiplications and additions). The number of arithmetic operations required is significantly reduced through the set of Fast Fourier transform (FFT) algorithms. mos
37、t FFT algorithms are based on the simple principle that an N-point DFT can be a computed by two(N/2)-point DFTs, or three (N/3)-point DFTs, ect. Computation of N-point DFT or inverse DFT using FFT algorithms requires on the order of NN arithmetic operations.4.Frequency-domain representation of LTI s
38、ystemIn section 132 we reviewed the representation of signal as a linear combination of complex exponentials of the form or more generally .For linear systems, the response is then the same linear combination of the response to the individual complex exponentials .If in addition the system is time i
39、nvariant,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 exponentials with unity magnitude or ,more generally ,to the system function
40、 in the contexts of the more general complex exponential .The eigenfunction property follows directly from the convolution sum and state that with x(n)= ,the output y(n) has the form y(n)=H(z) (25) where H(z)= (26) The system function H(z) is eigenvalue associated with the eigenfunction also, from E
41、q.(26).H(z) is the z-transform of the system unit sample response .when z= ,it correspond to the Fourier transform of the unit sample response .Since Eq.(17) or (21) corresponds to a decomposition of x(n) as a linear combination of complex exponentials .we can obtain the response y(n),using linearit
42、y and the eigenfunction property ,by multiplying the amplitudes of the eigenfunctions In Eq.(22) by the eigenvalues H(z),i.e., y(n)= (27) Eq.(27) then becomes the synthesis equation for the output ,i.e. Y(z)=H(z)X(z) (28) Eq.(28) corresponds to the z-transform convolution n property.System character
43、ized by linear constant coefficient difference equationsA particularly important class of discrete time system are those characterized by linear constant-coefficient difference equations (LCCDE) of the form (29) Where the and the are constants .Eq.(29) is typically referred to as an Nth-order differ
44、ence equation .A system characterized by an Nth-order difference equation of the 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 .5.Solution of
