Lecture 7: Z-Transform

1. Lecture 7: Z-Transform • Remember the Laplace transform? This is the same thing but for discrete-time signals! • Definition: • z is a complex variable: imaginary z r w real EE421, Lecture 7

2. Z-transform • What is z-n or zn? • rate of decay (or growth) is determined by r • frequency of oscillation is determined by w real part imaginary part real part imaginary part EE421, Lecture 7

3. Z-Transform imaginary plots of zn real unit circle r = 1 EE421, Lecture 7

4. Z-Transform • Transfer function: • Notation: • Properties: • linearity • delay • convolution system impulse response EE421, Lecture 7

5. Z-Transform • Some simple pairs: • finite-length sequence • impulse n=0 EE421, Lecture 7

6. Z-Transform • The geometric series is important for deriving many z-transforms: EE421, Lecture 7

7. Z-Transform • unit step function • reversed step function only if |Z|>1! only if |Z|<1! Do these different functions have the same z-transform? EE421, Lecture 7

8. Z-Transform • Region of ConvergenceIn general, the z-transform is an infinite sum! This means it (the z-transform) may not exist for all values of z. More specifically, it is the value of r = |z| that is important. If x(n) = (0.5)nu(n), then z-plane only if |Z|>0.5 ! 0.5 ROC EE421, Lecture 7

9. Z-Transform • Region of ConvergenceHere’s what the ROC can look like: |z|<a b<|z| b<|z|<a all |z| EE421, Lecture 7