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 01-Oct, 98 EE421, Lecture 7

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 01-Oct, 98 EE421, Lecture 7

Z-Transform imaginary real unit circle r = 1 plots of zn 01-Oct, 98 EE421, Lecture 7

Z-Transform Transfer function: Notation: Properties: linearity delay convolution system impulse response 01-Oct, 98 EE421, Lecture 7

Z-Transform Some simple pairs: finite-length sequence impulse n=0 01-Oct, 98 EE421, Lecture 7

Z-Transform The geometric series is important for deriving many z-transforms: 01-Oct, 98 EE421, Lecture 7

Z-Transform only if |Z|>1! only if |Z|<1! unit step function reversed step function only if |Z|>1! only if |Z|<1! Do these different functions have the same z-transform? 01-Oct, 98 EE421, Lecture 7

Z-Transform Region of Convergence In 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 01-Oct, 98 EE421, Lecture 7

Z-Transform Region of Convergence Here’s what the ROC can look like: |z|<a b<|z| b<|z|<a all |z| 01-Oct, 98 EE421, Lecture 7