z transform primer. basic concepts consider a sequence of values: {x k : k = 0,1,2,... } these may...
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![Page 1: Z Transform Primer. Basic Concepts Consider a sequence of values: {x k : k = 0,1,2,... } These may be samples of a function x(t), sampled at instants](https://reader036.vdocuments.us/reader036/viewer/2022082820/5697bfa51a28abf838c97842/html5/thumbnails/1.jpg)
Z Transform Primer
![Page 2: Z Transform Primer. Basic Concepts Consider a sequence of values: {x k : k = 0,1,2,... } These may be samples of a function x(t), sampled at instants](https://reader036.vdocuments.us/reader036/viewer/2022082820/5697bfa51a28abf838c97842/html5/thumbnails/2.jpg)
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
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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)
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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
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.
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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 . . .
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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).
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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
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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
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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
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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.
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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.