# Table of Laplace and Z Transforms

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).

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)

(Note)

atan is the arctangent (tan-1) function. The atan function can give incorrect results (typically the function is written so that the result is always in quadrants I or IV). To ensure accuracy, use a function that corrects for this. In most programming languages the function is atan2. Also be careful about using degrees and radians as appropriate.

Z-domain
generic
decaying
oscillatory

(Note)
Prototype Second Order System (ζ<1, underdampded)
Prototype
2nd order
lowpass
step
response

Prototype
2nd order
lowpass
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

References

Replace