EEL5701: HW#6

Due Thursday, July 9 at the beginning of class, late homework will not be accepted. Remember, the open-book Exam II will be held Thursday, July 16, 9:30am-12:30pm in NEB 102.

Do all of the following problems in Mitra. Justify all of your answers.

1.

5.19

2.

5.21

3.

7.2

4.

7.4

5.

7.8

6.

7.14

7.

7.16

8.

A first-order analog filter has a zero at s=-2, a pole at -2/3, and a DC gain of one. Bilinear transformation yields the following digital filter:

$$H(z)=\frac{k}{1-\alpha z^{-1}}$$

Find k, $\alpha$, and T.

9.

(O&S 7.8) The system function of a discrete-time system is

$$H(z)=\frac{2}{1-e^{-0.2}z^{-1}}-\frac{2}{1-e^{-0.4}z^{-1}}$$

(a)

Assume that this discrete-time filter was designed by the impulse invariance method with T=2. Find the system function H_a(s) that could have been the basis for the design. Is your answer unique? Why or why not? If not, find another H_a(s).

(b)

Assume that this discrete-time filter was designed by the bilinear transform with T=2. Find the system function H_a(s) that could have been the basis for the design. Is your answer unique? Why or why not? If not, find another H_a(s).

10.

(O&S 7.9b) If a continuous-time system is an all-pass system, its poles will be at location s_k in the left-half s-plane and its zeros will be at corresponding location -s_k in the right-half s-plane. Which design of the two design methods (impulse invariance and bilinear transform) will result in an all-pass discrete-time system? Explain.