Z Transform Primer

Basic Concepts• Consider a sequence of values: {xk : k = 0,1,2,... }• These may be samples of a function x(t), sampled at

instants t = kT; thus xk = x(kT).• The Z transform is simply a polynomial in z having the xk

as coefficients:

0

)(k

kkk zxxZzX

Fundamental Functions

• Define the impulse function: {k} = {1, 0, 0, 0,....}

1)( kZz

• Define the unit step function: {uk} = {1, 1, 1, 1,....}

11

11

0

z

z

zzuZzU

k

kk

(Convergent for |z| < 1)

Delay/Shift Property• Let y(t) = x(t-T) (delayed by T and truncated at t = T)

yk = y(kT) = x(kT-T) = x((k-1)T) = xk-1 ; y0 = 0

1

11

)(k

kk

k

kkk zxzyyZzY

• Let j = k-1 ; k = j + 1

)()( 1

0

1

0

1 zXzzxzzxzYj

jj

j

jj

• The values in the sequence, the coefficients of the polynomial, slide one position to the right, shifting in a zero.

The Laplace Connection

• Consider the Laplace Transforms of x(t) and y(t):

sXeTtxLtyLsY Ts

• Equate the transform domain delay operators:Tsez 1

Tsez

• Examine s-plane to z-plane mapping . . .

S-Plane to Z-Plane Mapping

Tsez

Anything in the Alias/Overlay region in the S-Plane will be overlaid on the Z-Plane along with the contents of the strip between +/- j/T. In order to avoid aliasing, there must be nothing in this region, i.e. there must be no signals present with radian frequencies higher than /T, or cyclic frequencies higher than f = 1/2T. Stated another way, the sampling frequency must be at least twice the highest frequency present (Nyquist rate).

Mapping Poles and Zeros

A point in the Z-plane rejwill map to a point in the S-plane according to:

T

rs

lnRe

Ts

Im

Conjugate roots will generate a real valued polynomial in s of the form:

22 2 nnss

2ln

1

r

Tn

T

r

n ln

Example 1: Running Average Algorithm

Block Diagram Transfer Function

4

23321

4

1

4

1

z

zzzzX

zzzzXzY

4

23

4

1

z

zzzz

X

Y

4321

kkkkk

xxxxy

Note: Each [Z-1] block can be thought of as a memory cell, storing the previously applied value.

(Non-Recursive)

Z Transform

Example 2: Trapezoidal Integrator

211

Txxyy kkkk

2

11 TzXzzXzYzzY

21

1

21

11

1 T

z

zzX

T

z

zzXzY

(Recursive)

Z Transform

Block Diagram

1

1

2 z

zT

zX

zY

Transfer Function

Ex. 2 (cont) Block Diagram Manipulation

Intuitive Structure

Equivalent Structure Explicit representation of xk-1 and yk-1 has been lost, but memory element usage has been reduced from two to one.

Ex. 2 (cont) More Block Diagram Manipulation

1

1

2 z

zT

zX

zY Note that the final form is equivalent to a rectangular integrator with an additive forward path. In a PI compensator, this path can be absorbed by the proportional term, so there is no advantage to be gained by implementing a trapezoidal integrator.