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
01-Oct, 98
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
01-Oct, 98
EE421, Lecture 7

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

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

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

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

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

8. 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

9. 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