01-Oct, 98 EE421, Lecture 7 1

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Lecture 7: Z-Transform

Remember the Laplace transform? This is the same thing but for discrete-time signals!

Definition:

– z is a complex variable:

n

nznxzX )()(

sincos jrr

rez j

rz

real

imaginary

01-Oct, 98 EE421, Lecture 7 2

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-transform

What is z-n or zn?

– rate of decay (or growth) is determined by r– frequency of oscillation is determined by

njrnr

erznn

njnn

sincos

njrnr

erznn

njnn

sincos

real part imaginary part real part imaginary part

01-Oct, 98 EE421, Lecture 7 3

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

real

imaginary

unit circler = 1

plots of zn

01-Oct, 98 EE421, Lecture 7 4

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

Transfer function:

Notation:

Properties:– linearity– delay

– convolution

n

nznhzH )()(

system impulse response

)()( zXznnx 0nZ0

)()( zXnx Z

)()()()()()( zXzHzYnxnhny Z

01-Oct, 98 EE421, Lecture 7 5

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

Some simple pairs:– finite-length sequence

– impulse

321Z z3z1z213121

n=0

1zznn 0

n

nZ

)()(

01-Oct, 98 EE421, Lecture 7 6

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

The geometric series is important for deriving many z-transforms:

a1aa

a1NNN

Nn

n212

1

01-Oct, 98 EE421, Lecture 7 7

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

– unit step function

– reversed step function

10n

n

n

nZ

z11

zznunu

)()(

only if |Z|>1!

1

1

n

n

n

nZ

z11

zz1nu1nu

)()(

only if |Z|<1!

Do these different functions have the same z-transform?

01-Oct, 98 EE421, Lecture 7 8

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

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

1

0n

n1

n

0n

n

z5011

z50

z50zX

.

).(

).()(

only if |Z|>0.5 !

0.5

z-plane

ROC

01-Oct, 98 EE421, Lecture 7 9

EE421, Fall 1998Michigan Technological University

Timothy J. Schulz

Z-Transform

Region of Convergence

Here’s what the ROC can look like:

|z|<a b<|z| b<|z|<a all |z|