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) Z Domain
(t=kT)
unit impulse      unit impulse
unit step   (note)
ramp
parabola
tn
(n is integer)
 
exponential
power   b to the k, Z
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
  Generic second order, Discrete Time

(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

© Copyright 2005 to 2014 Erik Cheever    This page may be freely used for educational purposes.

Erik Cheever       Department of Engineering         Swarthmore College