# Z Transform, Ramp and Parabola.

Cuthbert Nyack

The equation for a ramp and its samples are shown below:-
The Z transform of the ramp is given by:-
Multiplying by z^{-1} gives:-
Subtracting the last 2 equations give:-
Rearranging the expression for the Z transform gives the final expression
for the Z transform of the ramp as:-

The equation of a parabola and its samples are given below:-
Its Z transform is formally given as:-
Multipliing by z^{-1} gives:-
Subtracting the above 2 equations results in:-
The first expression in curly brackets can be summed using the result
from the ramp and second expression in curly brackets is a delayed step
which can also be readily summed. Summing and rearranging gives the following expression
for the Z transform of the parabola.

Both of these results can also be obtained from the following property of
the Z transform.

The following binomial expansion can also be used.

*Return to main page*

*Return to page index*

© 1999 Cuthbert A. Nyack.