Description

• Grouping: You are allowed to work in pairs.
• Submission: We provide a latex template for your solutions. Use that template and create a hw4.tar.gz file that includes hw4.tex and all other related files. Tar.gz file should not contain any directories and should create a hw4.pdf file with the following commands, otherwise you will get zero; tar xvzf hw4.tar.gz pdflatex hw4.tex
Submit hw4.tar.gz to the COW page of the course.

where D is the unit-delay operator.
(a) (5 pts) Find the difference equation which represents this system.
(b) (5 pts) Find the frequency response of this system.
(c) (10 pts) Find the impulse response of this system from its frequency response.
(d) (10 pts) Find the output y[n] for the input ] using the frequency response.
2. (10 pts) Suppose that two discrete-time LTI systems with the impulse responses h1[n] and h2[n] are connected in parallel. We have the following information about this combined system:
i. The frequency response of the combined system is; .
ii. The impulse response of the first system is:
Find h2[n], the impulse response of the second system.
3. (30 pts) Consider the following sampling system where the input is
x(t) = sin2πtπt + cos3πt

(a) (10 pts) Find and plot the Fourier Transform of x(t).
(b) (10 pts) What is the Nyquist frequency and period for sampling?
(c) (10 pts) Find and plot the Fourier Transform of xp(t) using the Nyquist rate.
1
4. (30 pts) Consider the following system with a C/D converter:
h[n] = cosπn
where the sampling frequency is ωs = π and
if
0, otherwise
(a) (10 pts) Find Xd(ejω).
(b) (10 pts) Find H(ejω).
(c) (10 pts) Find Yd(ejω).
2