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).
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 step | ![]() | ![]() | ![]() |
ramp | ![]() | ![]() | ![]() |
parabola | ![]() | ![]() | ![]() |
tn (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) | ![]() | ![]() ![]() ![]() | |
Z-domain generic decaying oscillatory | ![]() ![]() (note) | ![]() | |
Prototype Second Order System (ζ<1, underdampded) | |||
Prototype 2nd order lopass step response | ![]() | ![]() | ![]() |
Prototype 2nd order lopass impulse response | ![]() | ![]() | ![]() |
Prototype 2nd order bandpass impulse response | ![]() | ![]() | ![]() |
Using this table for Z Transforms with discrete indices
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

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