1、Chapter6 z-TransformDefinitionROC (Region of Converges)z-Transform PropertiesTransfer Function 1z-TransformIn continuous signal system, we use S-Transform and FT as the tools to process problems in the transform domain; so in discrete signal system, we use z-Transform and DFT.z-Transform can make th

2、e solution for discrete time systems very simple.2z-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 exist.36.1 Definition and PropertiesDTFT defined

3、by：leads to the z-transform。 z-transform may exist for many sequences for which the DTFT does not exist。4Definition and PropertiesFor a given sequence gn, its z-transform G(z) is defined as:where z= Re(z) + jIm(z) is a complex variable.5Definition and PropertiesIf we let z=rej, then the z-transform

4、reduces to:For r = 1 (i.e., |z| = 1), z-transform reduces to its DTFT, provided the latter exists。The contour |z| = 1 is a circle in the z-plane of unity radius and is called the unit circle。6Definition and PropertiesLike the DTFT, there are conditions on the convergence of the infinite series： For

5、a given sequence, the set R of values of z for which its z-transform converges is called the region of convergence (ROC)7Definition and PropertiesFrom our earlier discussion on the uniform convergence of the DTFT, it follows that the series：converges if gnr-n is absolutely summable, i.e., if：8Defini

6、tion and PropertiesIn general, the ROC of a z-transform of a sequence gn is an annular region of the z-plane:where Note: The z-transform is a form of a Laurent series and is an analytic function at every point in the ROC。9Definition and PropertiesExample - Determine the z-transform X(z) of the causa

7、l sequence xn=nn and its ROC.NowThe above power series converges to:ROC is the annular region |z| |.10Definition and PropertiesExample - The z-transform m(z) of the unit step sequence mn can be obtained from:mROC is the annular region .by setting a = 1,11Definition and PropertiesExample - Consider t

8、he anti-causal sequence:Its z-transform is given by:ROC is the annular region12Definition and PropertiesNote: The unit step sequence mn is not absolutely summable, and hence its DTFT does not converge uniformly.Note: Only way an unique sequence can be associated with a z-transform is by specifying i

9、ts ROC.13Commonly Used z-transform146.2 Rational z-TransformIn the case of LTI discrete-time systems we are concerned with in this course, all pertinent z-transforms are rational functions of z-1.That is, they are ratios of two polynomials in z-1 :15Rational z-TransformA rational z-transform can be

10、alternately written in factored form as:16Rational z-TransformAt a root z=l of the numerator polynomial G(l)=0 and as a result, these values of z are known as the zeros of G(z).At a root z= l of the denominator polynomial G(l), and as a result, these values of z are known as the poles of G(z).17Rati

11、onal z-TransformConsider:Note G(z) has M finite zeros and N finite poles:If N M there are additional N - M zeros at z = 0 (the origin in the z-plane)If N M there are additional M - N poles at z = 0.18Rational z-TransformExample - The z-transformmhas a zero at z = 0 and a pole at z = 1.19Rational z-T

12、ransformA physical interpretation of the concepts of poles and zeros can be given by plotting the log-magnitude 20log10|G(z)| as shown on next figure for:20Rational z-Transformpoles z=0.4j0.6928, zeros z=1.2j1.2.216.3 ROC of a Rational z-transformROC of a z-transform is an important concept.Without

13、the discussion of the ROC, there is no unique relationship between a sequence and its z-transform. Hence, the z-transform must always be specified with its ROC.22ROC of a Rational z-transformMoreover, if the ROC of a z-transform includes the unit circle, the DTFT of the sequence is obtained by simpl

14、y evaluating the z-transform on the unit circle.ROC of z-transform of the impulse sequence of a causal, stable LTI discrete time system.23ROC of a Rational z-transformThe ROC of a rational z-transform is bounded by the locations of its poles.To understand the relationship between the poles and the R

15、OC, it is instructive to examine the pole-zero plot of a z-transform.24ROC of a Rational z-transformConsider again the pole-zero plot of the z-transform m(z) .25A sequence can be one of the following types: finite-length, right-sided, left-sided and two-sided.In general, the ROC depends on the type

16、of the sequence of interest.ROC of a Rational z-transform261）Finite Length Sequence27 If n20 ，ROC doesnt include point If n10 ，ROC doesnt include point 0 If n10| and |z|:If | |, then there is an overlap and the ROC is an annular region |z| , then there is no overlap and V(z) does not exist. 67z-Tran

17、sform PropertiesUsing the conjugation property we obtain the z-transform of v*n as:Finally, using the linearity property we get:68z-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 by:69z-Transf

18、orm PropertiesOr:Example - Determine the z-transform Y(z) and the ROC of the sequence: We can write yn=nxn+xn where xn= n n 70z-Transform PropertiesNow, the z-transform X(z) of xn= n n is given by:Using the differentiation property, we arrive at the z-transform of nxn as:71z-Transform PropertiesUsin

19、g the linearity property we finally obtain:726.6 Computation of the Convolution Sum of Finite-Length SequencesTabular methods for the computation of the linear and circular convolution have been outlined.Now, we describe alternate methods for the computation of the linear and circular convolution th

20、at based on the multiplication of two polynomial.736.6.1 Linear ConvolutionLet xn and hn be two sequences of lengths L+1 and M+1, respectively. Their z-transforms, X(z) and H(z) be:So, the z-transform of linear convolution sequence yn is Y(z):And its coefficient is yn.746.6.2 Circular ConvolutionLet

21、 xn and hn both be two sequences of degree N-1, Their z-transforms, X(z) and H(z) be:So, the z-transform of linear convolution sequence yn is Y(z):75Circular ConvolutionLet Yc(z) denote the polynomial of degree N-1 whose coefficients is ycn , it can be shown that (Problem 6.17) :The modulo operation

22、 with respect to z-N=1 in above equation of YL(z).Please look at P331 about the process example.766.7 The Transfer FunctionThe z-transform of the impulse response of an LTI system, called the transfer function, is a polynomial in z-1.In most practical cases, the LTI digital filter of interest is cha

23、racterized by a linear different equation with constant and real coefficient, so its transfer function is rational z-transform.776.7.1 DefinitionOrigin:In time domain, use unit sample response hn to represent a LTI system:To do ZT for both sides, we get:Then:Its the transfer function of a LTI system

24、.786.7.2 Transfer Function ExpressionFIR Digital FilterIn the case of an LTI FIR digital filter, with its impulse response hn defined for N1 nN2, and thus, hn=0 for nN1 and nN2. Therefore, the transfer function is given by:79Transfer Function ExpressionFinite-Dimension Linear Time-Invariant IIR Disc

25、rete-Time SystemConsider an LTI discrete-time system characterized by a difference equation Its transfer function is obtained by taking the z-transform of both sides of the above equation, Thus:80Transfer Function Expression1. are the finite zeros, and are the finite poles of H(z).2.If NM, there are

26、 additional (N-M) zeros at z=0 ; if NM, there are additional (M-N) poles at z=0.81Transfer Function Expression3.For a causal IIR filter, the ROC of the transfer function H(z) is:826.7.3 Frequency Response from Transfer FunctionThe relationship between H(z) and H(ejw).Because:836.7.4 Geometric Interp

27、retation of FR ComputationUse zero-vectors and pole-vectors in z-plane to interpret the frequency response. (6.97) (6.98)84Geometric Interpretation of FR ComputationAs shown below in the z-plane the factor represents a vector starting at the point and ending on the unit circle at8586lowpasshighpass87bandpassbandstop88allpassmultiband896.7.5 Stability Condition in Term