測試信號英文版課件：Chapter3_Lec3 z-Transform

1、1z-TransformThe DTFT provides a frequency-domain representation of discrete-time signals and LTI discrete-time systemsBecause of the convergence condition, in many cases, the DTFT of a sequence may not existAs a result, it is not possible to make use of such frequency-domain characterization in thes

2、e cases z-TransformFor discrete-time systems, ZT plays the same role of Laplace-transform does in continuous time systems. ZT characterizes signals or LTI systems in complex frequency domain.2Laplace-transform3z-TransformConsequently, z-transform has become an important tool in the analysis and desi

3、gn of digital filtersFor a given sequence gn, its z-transform G(z) is defined aswhere is a complex variableLaplace-transform4z-TransformIf we let , then the z-transform reduces toThe above can be interpreted as the DTFT of the modified sequenceFor r = 1 (i.e., |z| = 1), z-transform reduces to its DT

4、FT, provided the latter exists5z-TransformThe contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circleLike the DTFT, there are conditions on the convergence of the infinite seriesFor a given sequence, the set R of values of z for which its z-transform converges is cal

5、led the region of convergence (ROC)6z-TransformFrom our earlier discussion on the uniform convergence of the DTFT, it follows that the seriesconverges if is absolutely summable, i.e., if7z-TransformIn general, the ROC of a z-transform of a sequence gn is an annular region of the z-plane:where891011z

6、-TransformExample - Determine the z-transform X(z) of the causal sequence and its ROCNow The above power series converges toROC is the annular region |z| |a|12z-TransformExample - The z-transform m(z) of the unit step sequence mn can be obtained fromby setting a = 1:ROC is the annular regionm13z-Tra

7、nsformNote: The unit step sequence mn is not absolutely summable, and hence its DTFT does not converge uniformlyExample - Consider the anti-causal sequence14z-TransformIts z-transform is given byROC is the annular region15z-TransformNote: The z-transforms of the two sequences and are identical even

8、though the two parent sequences are differentOnly way a unique sequence can be associated with a z-transform is by specifying its ROCHence, z-transform must always be specified with its ROC16z-TransformThe DTFT of a sequence gn converges uniformly if and only if the ROC of the z-transform G(z) of gn

9、 includes the unit circleDTFT exists ROC of G(z) includes |z|=1 17Table : Commonly Used z-Transform Pairs18Rational z-TransformsIn the case of LTI discrete-time systems we are concerned with in this course, all pertinent z-transforms are rational functions of That is, they are ratios of two polynomi

10、als in :19Rational z-TransformsThe degree of the numerator polynomial P(z) is M and the degree of the denominator polynomial D(z) is NAn alternate representation of a rational z-transform is as a ratio of two polynomials in z:20Rational z-TransformsA rational z-transform can be alternately written i

11、n factored form as21Rational z-TransformsAt a root of the numerator polynomial , and as a result, these values of z are known as the zeros of G(z)At a root of the denominator polynomial , and as a result, these values of z are known as the poles of G(z)22Rational z-TransformsConsiderNote G(z) has M

12、finite zeros and N finite polesIf N M there are additional zeros at z = 0 (the origin in the z-plane)If N num=1 2; den=1 0.4 -0.12; h,t=impz(num,den); figure(1) stem(t,h) xlabel(n) ylabel(hn)hn=1.0000 1.6000 -0.5200 0.4000 -0.2224 6162z-Transform PropertiesExample - Consider the two-sided sequenceLe

13、t and with X(z) and Y(z) denoting, respectively, their z-transformsNowand63z-Transform PropertiesUsing the linearity property we arrive atThe ROC of V(z) is given by the overlap regions of and If , then there is an overlap and the ROC is an annular regionIf , then there is no overlap and V(z) does n

14、ot exist 64z-Transform PropertiesExample - Determine the z-transform and its ROC of the causal sequenceWe can express xn = vn + v*n whereThe z-transform of vn is given by65z-Transform PropertiesUsing the conjugation property we obtain the z-transform of v*n asFinally, using the linearity property we get66z-Transform Propertiesor,Example - Determine the z-transform Y(z) and the ROC of the sequenceWe can write where 67z-Transform PropertiesNow, the z-transform X(z) of is given byUsing the d