Solved ECE113: DSP Homework 6

Description

Problem 1: Problem 3.14 ((d) and (g) only) in R1 (Proakis 4th Edition)

Problem 2: Problem 3.16 ((d) only) in R1

Problem 3: Problem 3.18 ((d) only) in R1

Problem 4: Problem 3.32 in R1

Problem 5: Problem 3.35 ((c) and (g) only) in R1
(Hint: In Part (c), a “,” is obviously missing between x(n-1) and x(n). For Part (g), note that x(n) does not have a Z-transform and you should instead use the fundamental property of Transfer Functions relating to how an LTI system responds to a sinusoidal sequence)

Problem 6: Problem 3.38 ((b) only) in R1

Problem 7: Problem 3.40 in R1

Problem 8: Problem 3.42 in R1

Problem 9: Problem 3.51 in R1

Problem 10: Problem 5.20 in R1

MATLAB Exercises:

P4.11

Determine the following inverse $z$-transforms using the partial fraction expansion method.

1. X1(z)=1−z−1−4z−2+4z−31−114z−1+138z−2−14z−3X1​(z)=1−411​z−1+813​z−2−41​z−31−z−1−4z−2+4z−3​
   The sequence is right-sided.

2. X4(z)=zz3+2z2+1.25z+0.25,∣z∣>1X4​(z)=z3+2z2+1.25z+0.25z​,∣z∣>1

Note: For PFE, you can use the residuez function in MATLAB.

P4.21

A digital filter is described by the frequency response function

H(ejω)=[1+2cos⁡(ω)+3cos⁡(2ω)]cos⁡(ω2)e−j5ω/2H(ejω)=[1+2cos(ω)+3cos(2ω)]cos(2ω​)e−j5ω/2

1. Determine the difference equation representation.

2. Using the freqz function, plot the magnitude and phase of the frequency response of the filter. Note the magnitude and phase at $\omega =\pi /2$ and at $\omega =\pi$.

3. Generate 200 samples of the signal $x(n)=\sin (\pi n/2)+5\cos (\pi n)$, and process through the filter to obtain $y(n)$. Compare the steady-state portion of $y(n)$ to $x(n)$. How are the amplitudes and phases of the two sinusoids affected by the filter?

Note:

• For Part (1), you can use Euler equation along with the relationship between DTFT and Z-transform (i.e., the relationship between z and $\omega$) to find the associated Transfer Function H(z) and from that, obtain the LCCDE.

• For Part (3), you can use the filter function in MATLAB.