The Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1

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1 The Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1

2 Outline Properties of the region of convergence (10.2) The inverse Z-transform (10.3) Definition Computational techniques Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 2

3 Observations Specification of the Z transform requires both algebraic expression and region of convergence Rational Z-transforms are obtained if x[n]=linear combination of exponentials for n>0 and n<0 Rational Z-transforms are completely characterized by their poles and zeros (except for the gain) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 3

4 Properties of the ROC of the Z-transform 1. The ROC of X(z) consists of a ring in the z- plane centered about the origin Convergence is dependent only on r, not on ω In some cases, the inner boundary can extend inward to the origin (ROC=disc) In other cases, the outer boundary can extend outward to infinity Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 4

5 Properties of the ROC of the Z- transform 2. The ROC does not contain any poles At a pole X(z) is infinite and therefore does not converge. 3. If x[n] is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and/or z=. X ( z) N = 2 n= N 1 x[ n] z n N1<0, N2>0 summation includes terms with both positive and negative powers of z. z=0 and z= are not in the ROC. N 1 0 summation includes only terms with negative powers. ROC includes z= N 2 0 summation includes only terms with positive powers. ROC includes z=0 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 5

Properties of the ROC of the Z-transform 4. If x[n] is a right-sided sequence, and if z =r 0 is in the ROC, then all finite values of z for which z >r 0 are also in the ROC. 5.

6 Properties of the ROC of the Z-transform 4. If x[n] is a right-sided sequence, and if z =r 0 is in the ROC, then all finite values of z for which z >r 0 are also in the ROC. 5. If x[n] is a left-sided sequence, and if the circle z =r 0 is in the ROC, tha all values of z for which 0< z <r 0 will also be in the ROC. 6. If x[n] is two-sided, and if z =r 0 is in the ROC, then the ROC consists of a ring in the z-plane including the circle z =r 0. Which type of signals do the following ROCs correspond to? right-sided left-sided two-sided Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 6

7 Properties of the ROC of the Z-transform 7. If the Z-transform X(z) of x[n] is rational, then its ROC is bounded by poles or extends to infinity. 8. If the Z-transform X(z) of x[n] is rational, and x[n] is right-sided, then the ROC is the region in the z-plane outside the outermost pole (outside the circle of radius equal to the largest magnitude of the poles of X(z)) If x[n] is causal then ROC includes z= 9. If the Z-transform X(z) of x[n] is rational, and x[n] is left-sided, then the ROC is the region in the z- plane inside the innermost nonzero pole (inside the circle of radius equal to the smallest magnitude of the non-zero poles of X(z) If x[n] anticausal then ROC includes z=0 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 7

8 Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 8

9 Example p. 798 Let x[n] be an absolutely summable signal with rational Z-transform X(z). If X(z) is known to have a pole at z=1/2, could x[n] be: a) a finite-duration signal? b) a left-sided signal? c) a right-sided signal? d) a two-sided signal? Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 9

10 The inverse Z-transform 1) by expressing the Z-transform as the Fourier Transform of an exponentially weighted sequence, we obtain x[ n] 1 2πj n 1 = X ( z) z dz The formal expression of the inverse Z-transform requires the use of contour integrals in the complex plane. For rational Z-transforms we can compute the invrese Z- transforms using alternative procedures: Partial fraction expansion Z transform pairs Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 10

11 The inverse Z-transform A. for rational Z-transforms: by expanding the algebraic expression into a partial fraction expansion Example: Compute the inverse Z-transform for: X ( z) 2 5 3z z = z z z 4 3 > 1 3 See also textbook examples 10.10, Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 11

12 The inverse Z-transform B. by power-series expansion of X(z) textbook examples 10.12, 10.13, Main idea: the expression of the Z-transform + n= X ( z) = x[ n] z can be interpreted as a power-series involving both positive and negative powers of z. n The coefficients in this power series are x[n] For right sided signals : power expansion in negative powers of z For left-sided signals: power expansion in positive powers of z Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 12

The Z transform (2) 1

The Z transform (2) 1 Today Properties of the region of convergence (3.2) Read examples 3.7, 3.8 Announcements: ELEC 310 FINAL EXAM: April 14 2010, 14:00 pm ECS 123 Assignment 2 due tomorrow by 4:00 pm