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1、2020/7/17,1,3.7 z-Transform Z变换,The DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systems Because of the convergence condition, in many cases, the DTFT of a sequence may not exist As a result, it is not possible to make use of such frequency-domain ch
2、aracterization in these cases,2020/7/17,2,z-TransformZ变换,A generalization of the DTFT defined by leads to the z-transform z-transform may exist for many sequences for which the DTFT does not exist Moreover, use of z-transform techniques permits simple algebraic manipulations,2020/7/17,3,z-TransformZ
3、变换,Consequently, z-transform has become an important tool in the analysis and design of digital filters For a given sequence gn, its z-transform G(z) is defined as where is a complex variable,2020/7/17,4,z-Transform Z变换,If we let , then the z-transform reduces to The above can be interpreted as the
4、DTFT of the modified sequence For r = 1 (i.e., |z| = 1), z-transform reduces to its DTFT, provided the latter exists,2020/7/17,5,z-TransformZ变换,The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle单位圆 Like the DTFT, there are conditions on the convergence of th
5、e infinite series For a given sequence, the set R of values of z for which its z-transform converges is called the region of convergence (ROC)收敛区域,2020/7/17,6,z-Transform Z变换,From our earlier discussion on the uniform convergence of the DTFT, it follows that the series converges if is absolutely sum
6、mable, i.e., if,2020/7/17,7,z-Transform Z变换,In general, the ROC收敛区域of a z-transform of a sequence gn is an annular region of the z-plane: where,2020/7/17,8,The z-transform is a form of the Cauchy-Laurent series and is an analytic function at every point in the ROC Let f (z) denote an analytic (or ho
7、lomorphic) function over an annular region centered at,Cauchy-Laurent Series柯西 - 洛朗序列,2020/7/17,9,Then f (z) can be expressed as the bilateral series being a closed and counterclockwise integration contour contained in,Cauchy-Laurent Series,where,2020/7/17,10,z-Transform,Example - Determine the z-tr
8、ansform X(z) of the causal sequence and its ROC Now The above power series converges to ROC is the annular region |z| |a|,2020/7/17,11,z-Transform,Example - The z-transform m(z) of the unit step sequence mn can be obtained from by setting a = 1: ROC is the annular region,m,2020/7/17,12,z-Transform,N
9、ote: The unit step sequence mn is not absolutely summable, and hence its DTFT does not converge uniformly Example - Consider the anti-causal sequence,2020/7/17,13,z-Transform,Its z-transform is given by ROC is the annular region,2020/7/17,14,z-Transform,Note: The z-transforms of the two sequences an
10、d are identical even though the two parent sequences are different Only way a unique sequence can be associated with a z-transform is by specifying its ROC,2020/7/17,15,z-Transform,The DTFT of a sequence gn converges uniformly if and only if the ROC of the z-transform G(z) of gn includes the unit ci
11、rcle The existence of the DTFT does not always imply the existence of the z-transform,2020/7/17,16,z-Transform,Example - The finite energy sequence has a DTFT given by which converges in the mean-square sense,2020/7/17,17,z-Transform,However, does not have a z-transform as it is not absolutely summa
12、ble for any value of r, i.e. Some commonly used z-transform pairs are listed on the next slide,2020/7/17,18,Table 3.8: Commonly Used z-Transform Pairs,2020/7/17,19,Rational z-Transforms,In the case of LTI discrete-time systems we are concerned with in this course, all pertinent z-transforms are rati
13、onal functions of That is, they are ratios of two polynomials in :,2020/7/17,20,Rational z-Transforms,The degree of the numerator polynomial P(z) is M and the degree of the denominator polynomial D(z) is N An alternate representation of a rational z-transform is as a ratio of two polynomials in z:,2
14、020/7/17,21,Rational z-Transforms,A rational z-transform can be alternately written in factored form as,2020/7/17,22,Rational z-Transforms,At a root of the numerator polynomial , and as a result, these values of z are known as the zeros零点 of G(z) At a root of the denominator polynomial , and as a re
15、sult, these values of z are known as the poles极点 of G(z),2020/7/17,23,Rational z-Transforms,Consider Note G(z) has M finite zeros and N finite poles If N M there are additional zeros at z = 0 (the origin in the z-plane) If N M there are additional poles at z = 0,2020/7/17,24,Rational z-Transforms,Ex
16、ample - The z-transform has a zero at z = 0 and a pole at z = 1,m,2020/7/17,25,Exercise 5.1.3. Producing Pole/Zero Plots.,% Exercise 5.1.3. Producing Pole/Zero Plots. clc; clear; close all; % Step (a). b = 1 3 3 1; % Numerator Coefficients. a = 1 .5 .3 .1; % Denominator Coefficients. % Produce and d
17、isplay the Poles/Zeros plot. figure(Name,Exercise 5.1.3. Producing Pole/Zero Plots); zplane(b,a); grid on; % Produce and display the frequency response. figure(Name,Exercise 5.1.3. Producing Pole/Zero Plots); freqz(b,a); % or better: % fvtool(b,a); % Step (b). % Find the numerical values of poles an
18、d zeros z p k = tf2zpk(b,a),2020/7/17,26,result,2020/7/17,27,Rational z-Transforms,A physical interpretation of the concepts of poles and zeros can be given by plotting the log-magnitude as shown on next slide for,2020/7/17,28,Rational z-Transforms,poles,zeros,2020/7/17,29,Rational z-Transforms,Obse
19、rve that the magnitude plot exhibits very large peaks around the points which are the poles of G(z) It also exhibits very narrow and deep wells around the location of the zeros at,2020/7/17,30,ROC of a Rational z-Transform,ROC of a z-transform is an important concept Without the knowledge of the ROC
20、, there is no unique relationship between a sequence and its z-transform Hence, the z-transform must always be specified with its ROC,2020/7/17,31,ROC of a Rational z-Transform,Moreover, if the ROC of a z-transform includes the unit circle, the DTFT of the sequence is obtained by simply evaluating t
21、he z-transform on the unit circle There is a relationship between the ROC of the z-transform of the impulse response of a causal LTI discrete-time system and its BIBO stability,2020/7/17,32,ROC of a Rational z-Transform,The ROC of a rational z-transform is bounded by the locations of its poles To un
22、derstand the relationship between the poles and the ROC, it is instructive to examine the pole-zero plot of a z-transform Consider again the pole-zero plot of the z-transform m(z),2020/7/17,33,ROC of a Rational z-Transform,In this plot, the ROC, shown as the shaded area, is the region of the z-plane
23、 just outside the circle centered at the origin and going through the pole at z = 1,2020/7/17,34,ROC of a Rational z-Transform,Example - The z-transform H(z) of the sequence is given by Here the ROC is just outside the circle going through the point,2020/7/17,35,ROC of a Rational z-Transform,A seque
24、nce can be one of the following types: finite-length, right-sided, left-sided and two-sided In general, the ROC depends on the type of the sequence of interest,2020/7/17,36,ROC of a Rational z-Transform,Example - Consider a finite-length sequence gn defined for , where M and N are non-negative integ
25、ers and Its z-transform is given by,2020/7/17,37,ROC of a Rational z-Transform,Note: G(z) has M poles at and N poles at z = 0 As can be seen from the expression for G(z), the z-transform of a finite-length bounded sequence converges everywhere in the z-plane except possibly at z = 0 and/or at,2020/7
26、/17,38,ROC of a Rational z-Transform,Example - A right-sided sequence with nonzero sample values for is sometimes called a causal sequence Consider a causal sequence Its z-transform is given by,2020/7/17,39,ROC of a Rational z-Transform,It can be shown that converges exterior to a circle , including
27、 the point On the other hand, a right-sided sequence with nonzero sample values only for with M nonnegative has a z-transform with M poles at The ROC of is exterior to a circle , excluding the point,2020/7/17,40,ROC of a Rational z-Transform,Example - A left-sided sequence with nonzero sample values
28、 for is sometimes called a anti-causal sequence Consider an anti-causal sequence Its z-transform is given by,2020/7/17,41,ROC of a Rational z-Transform,It can be shown that converges interior to a circle , including the point z = 0 On the other hand, a left-sided sequence with nonzero sample values
29、only for with N nonnegative has a z-transform with N poles at z = 0 The ROC of is interior to a circle , excluding the point z = 0,2020/7/17,42,ROC of a Rational z-Transform,Example - The z-transform of a two-sided sequence wn can be expressed as The first term on the RHS, , can be interpreted as th
30、e z-transform of a right-sided sequence and it thus converges exterior to the circle,2020/7/17,43,ROC of a Rational z-Transform,The second term on the RHS, , can be interpreted as the z-transform of a left-sided sequence and it thus converges interior to the circle If , there is an overlapping ROC g
31、iven by If , there is no overlap and the z-transform does not exist,2020/7/17,44,ROC of a Rational z-Transform,Example - Consider the two-sided sequence where a can be either real or complex Its z-transform is given by The first term on the RHS converges for , whereas the second term converges for,2
32、020/7/17,45,ROC of a Rational z-Transform,There is no overlap between these two regions Hence, the z-transform of does not exist!,2020/7/17,46,ROC of a Rational z-Transform,The ROC of a rational z-transform cannot contain any poles and is bounded by the poles As an example, assume that a rational z-
33、transform X(z) has two simple poles at z = a and z = b with There are three possible ROCs associated with X(z),2020/7/17,47,ROC of a Rational z-Transform,Right-sided,Left-sided,Two-sided,2020/7/17,48,ROC of a Rational z-Transform,In general, if the rational z-transform has N poles with R distinct ma
34、gnitudes, then it has ROCs Thus, there are distinct sequences with the same z-transform Hence, a rational z-transform with a specified ROC has a unique sequence as its inverse z-transform,2020/7/17,49,ROC of a Rational z-Transform,The ROC of a rational z-transform can be easily determined using MATL
35、AB determines the zeros, poles, and the gain constant of a rational z-transform with the numerator coefficients specified by the vector num and the denominator coefficients specified by the vector den,z,p,k = tf2zp(num,den),2020/7/17,50,ROC of a Rational z-Transform,num,den = zp2tf(z,p,k) implements
36、 the reverse process The factored form of the z-transform can be obtained using sos = zp2sos(z,p,k) The above statement computes the coefficients of each second-order factor given as an matrix sos,2020/7/17,51,ROC of a Rational z-Transform,where,2020/7/17,52,ROC of a Rational z-Transform,The pole-ze
37、ro plot零极点图is determined using the function zplane The z-transform can be either described in terms of its zeros and poles: zplane(zeros,poles) or, it can be described in terms of its numerator and denominator coefficients: zplane(num,den),2020/7/17,53,ROC of a Rational z-Transform,Example - The pol
38、e-zero plot of obtained using MATLAB is shown below,2020/7/17,54,Inverse z-Transform Z逆变换,General Expression: Recall that, for , the z-transform G(z) given by is merely the DTFT of the modified sequence Accordingly, the inverse DTFT is thus given by,2020/7/17,55,Inverse z-Transform,By making a chang
39、e of variable , the previous equation can be converted into a contour integral given by where is a counterclockwise contour of integration defined by |z| = r,2020/7/17,56,z-Transform,z-Transform: analysis equation,Inverse z-Transform: synthesis equation,time domain,z-domain,2020/7/17,57,Inverse z-Tr
40、ansform,But the integral remains unchanged when it is replaced with any contour C encircling the point z = 0 in the ROC of G(z) The contour integral can be evaluated using the Cauchys residue theorem resulting in The above equation needs to be evaluated at all values of n and is not pursued here,202
41、0/7/17,58,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,A rational z-transform G(z) with a causal inverse transform gn has a ROC that is exterior to a circle Here it is more convenient to express G(z) in a partial-fraction expansion form and then determine gn by summing the inverse tran
42、sform of the individual simpler terms in the expansion,2020/7/17,59,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,A rational G(z) can be expressed as If then G(z) can be re-expressed as where the degree of is less than N,2020/7/17,60,Inverse Transform by Partial-Fraction Expansion逆变换通过部
43、分分式展开,The rational function is called a proper fraction真分数 Example - Consider By long division we arrive at,2020/7/17,61,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Simple Poles单极点: In most practical cases, the rational z-transform of interest G(z) is a proper fraction with simple pol
44、es Let the poles of G(z) be at , A partial-fraction expansion of G(z) is then of the form,2020/7/17,62,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,The constants in the partial-fraction expansion are called the residues留数and are given by Each term of the sum in partial-fraction expansi
45、on has a ROC given by and, thus, has an inverse transform of the form,2020/7/17,63,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Therefore, the inverse transform gn of G(z) is given by Note: The above approach with a slight modification can also be used to determine the inverse of a rat
46、ional z-transform of a noncausal sequence,2020/7/17,64,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Example - Let the z-transform H(z) of a causal sequence hn be given by A partial-fraction expansion of H(z) is then of the form,2020/7/17,65,Inverse Transform by Partial-Fraction Expansi
47、on逆变换通过部分分式展开,Now and,2020/7/17,66,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Hence The inverse transform of the above is therefore given by,2020/7/17,67,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Multiple Poles多极点: If G(z) has multiple poles, the partial-fraction exp
48、ansion is of slightly different form Let the pole at z = n be of multiplicity L and the remaining poles be simple and at ,2020/7/17,68,Inverse Transform by Partial-Fraction Expansion逆变换通过部分分式展开,Then the partial-fraction expansion of G(z) is of the form where the constants are computed using The resi
49、dues are calculated as before,2020/7/17,69,Partial-Fraction Expansion Using MATLAB,r,p,k= residuez(num,den) develops the partial-fraction expansion of a rational z-transform with numerator and denominator coefficients given by vectors num and den Vector r contains the residues Vector p contains the poles Vector k contains the constants,2020/7/17,70,Partial-Fraction Expansion Using MA