EEL5701: HW#7

Due Thursday, July 30 at the beginning of class, late homework will not be accepted. Remember, the final exam will be held Thursday, August 6, 9:30am-12:30pm in NEB 102.

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

1.

6.16 in Mitra

2.

7.18 in Mitra

3.

7.23 in Mitra

4.

Derive an Nth order FIR filter using a rectangular window for a linear-phase filter that passes frequencies between $.3\pi$ and $.7\pi$.

5.

We are given a low-pass type-I FIR filter h[n] with given parameters $\omega_p$, omega_s, $\delta_p$, and $\delta_s$. Define:

$$g[n]=(-1)^{N/2} \delta[n-0.5N] - (-1)^nh[n]$$

• What type of filter is g[n] and what is the nature of its frequency response?
• Express the tolerance and band-edge parameters of the filter g[n] in terms of the parameters of h[n].

6.

Does the transition region have to be monotonic for the Parks-McClellan algorithm? Why or why not? Hint: think about the alternation theorem discussed in class.

7.

An optimal equiripple filter was designed using the Parks-McClellan algorithm. The magnitude of its frequency response is shown below. What type of linear-phase filter is this? How can you tell?

figure=../hw7/pm.eps,height=2.2in,angle=0

turn over

8.

What is the length of the impulse response of the system in problem 7? If the system is causal, what is the shortest delay it can have?

9.

Plot the zeros of the system function H(z) in problem 7 as accurately as you can in the z-plane.

10.

Construct a flow graph for a 16-point radix-2 decimation-in-time FFT algorithm. Label all operators in terms of powers of W₁₆ and also label any branch transmittances that are equal to -1. Label the input and out sequences properly.