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

unit impulse | unit impulse | ||

unit step | (Note)
u(t) is more commonly used to represent the step function, but u(t) is also used to represent other things. We choose gamma (γ(t)) to avoid confusion (and because in the Laplace domain (Γ(s)) it looks a little like a step input). |
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ramp | |||

parabola | |||

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

power | |||

time multiplied exponential |
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Asymptotic exponential |
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double exponential |
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asymptotic double exponential |
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asymptotic critically damped |
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differentiated critically damped |
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sine | |||

cosine | |||

decaying sine |
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decaying cosine |
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generic decaying oscillatory |
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generic decaying oscillatory (alternate) |
(Note) atan is the arctangent (tan |
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Z-domain generic decaying oscillatory |
(Note) |
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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

Replace