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 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

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