For the three filters of assignment 1, find the transfer function of *H*(*z*). Write *H*(*z*) as a ratio of polynomials and identify the poles and zeros. Choose a *a* = 0.9.

*Solution.
*

(a) For the impulse response

its

The above equation is equivalent to

where

(b) For the impulse response

its

The above equation is equivalent to

where

(c) For the output sequence

its

Using the linearity and time-shifting properties of the

The above equation is equivalent to (1-

Thus

where