Table of Laplace and Z Transforms

• Laplace and Z Transforms
• Laplace Properties
• Z Xform Properties

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
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)
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