Using this table
for Z Transforms with Discrete Indices

Shortened 2-page pdf of Laplace
Transforms and Properties

Shortened 2-page pdf of Z
Transforms and Properties

All time domain functions are implicitly=0 for
t<0 (i.e. they are multiplied by unit step).

Entry |
Laplace Domain | Time Domain (note) | Z Domain (t=kT) |
---|---|---|---|

unit impulse | unit impulse | ||

unit step | (note) | ||

ramp | |||

parabola | |||

t^{n}(n is integer) |
|||

exponential | |||

power | |||

time multiplied exponential |
|||

Asymptotic exponential |
|||

double exponential |
|||

asymptotic double exponential |
|||

asymptotic critically damped |
|||

differentiated critically damped |
|||

sine | |||

cosine | |||

decaying sine |
|||

decaying cosine |
|||

generic decaying oscillatory |
|||

generic decaying oscillatory (alternate) |
(note) |
||

Z-domain generic decaying oscillatory |
(note) |
||

Prototype Second Order System (ζ<1, underdampded) | |||

Prototype 2 ^{nd} orderlowpass step response |
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Prototype 2 ^{nd} orderlowpass impulse response |
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Prototype 2 ^{nd} orderbandpass impulse response |

Commonly the "time domain" function is given in terms of a discrete index, k, rather than time. This is easily accommodated by the table. For example if you are given a function:

Since t=kT, simply replace k in the function definition by k=t/T. So, in this case,

and we can use the table entry for the ramp

The answer is then easily obtained

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College