## Description

This homework assignment will look at some simple examples of discrete systems.

1. Z

z

z

This question involves a digital filter with the following transfer function:

H(z) =

1

2 +

2

3

z +

3

7

z

2

2 + 1

3

z +

1

2

z

3

(a) Find the poles and zeros of H using tf2zp.

(b) Plot the poles and zeros using zplane.

(c) Obtain the first 50 points of the impulse response using impz. Plot them using the appropriate

plotting function.

(d) Let x = (−

1

2

)

n where n ranges from 0 to 49. Apply the digital filter to x using filter. Plot the

result using the appropriate plotting function.

(e) Now apply the digital filter to x using conv. You will have to throw out some values to get the

same result. Make another plot to show that the results are the same.

2. You Gotta be Fibbin Me!

The Fibonacci sequence is the sequence of numbers such that every number after the first two is

the sum of the two preceding numbers. The first two numbers are both 1.

(a) We are interested in thinking of the n

th number (starting at 0) in the sequence as the output

of a discrete system at time n. Assume the input to the said system is an impulse at n = 0,

that is, the Fibonacci sequence is the impulse response of this system. what is the difference

equation governing the system? Use a for loop to generate the first 100 values of the impulse

response of this system. Plot each point with semilogy, make the linetype dots (i.e. ’.’ is the

third argument).

(b) Use the Z-transform to determine the transfer function of the system–Don’t use MATLAB to

do this, just right it down. Calculate the exact poles and zeros by hand. Calculate the poles

and zeros with MATLAB and generate a pole zero plot. Is the system causal? Is it stable?

Explain why using the pole zero plot and the difference equation.

1