Z Transform Primer

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Z Transform Primer
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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

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Fundamental Functions

• Define the impulse function dk 1, 0, 0,
  0,....

• Define the unit step function uk 1, 1, 1,
  1,....

(Convergent for z lt 1)
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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

• Let j k-1 k j 1

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

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The Laplace Connection

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

• Equate the transform domain delay operators

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

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S-Plane to Z-Plane Mapping
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 /- jp/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 w
p/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).
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Mapping Poles and Zeros

• A point in the Z-plane rejq will map to a point
  in the S-plane according to

Conjugate roots will generate a real valued
polynomial in s of the form
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Example 1 Running Average Algorithm
(Non-Recursive)
Z Transform
Block Diagram Transfer
Function
Note Each Z-1 block can be thought of as a
memory cell, storing the previously applied value.
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Example 2 Trapezoidal Integrator
(Recursive)
Z Transform
Block Diagram
Transfer Function
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Ex. 2 (cont) Block Diagram Manipulation
Intuitive Structure
Explicit representation of xk-1 and yk-1 has been
lost, but memory element usage has been reduced
from two to one.
Equivalent Structure
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Ex. 2 (cont) More Block Diagram Manipulation
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.