Description

• Grouping: You are allowed to work in pairs.
• Submission: We provide a latex template for your solutions. Use that template and create a hw3.tar.gz file that includes hw3.tex and all other related files. Tar.gz file should not contain any directories and should create a hw3.pdf file with the following commands, otherwise you will get zero; tar xvzf hw3.tar.gz pdflatex hw3.tex
Submit hw3.tar.gz to the COW page of the course.
• Late Submission: Not allowed.

1. (25 pts)
(a) (10 pts) Find and plot the spectral coefficients of Fourier series for the following discrete time signal, x[n]:

(b) Consider the following discrete time signal, y[n]:

i. (5 pts) Define y[n] in terms of x[n]. ii. (10 pts) Find and plot the spectral coefficients of Fourier series for y[n].
2. (20 pts) Determine and plot a discrete-time signal x[n] satisfying the following conditions:
(a) x[n] is a real and periodic signal with N = 4 and has Fourier Series coefficients ak, which is complex for some k.
(b) P4k=−3x[k] = 8.
and |a1 − a11| = 1.
(d) One of the coefficients is zero.

3. (20 pts) Consider a periodic signal x(t) which can be represented by the first K Fourier Series coefficients. Determine the impulse response of the system that can yield x(t) when it is contaminated by a noise r(t) (i.e., the input to the system is x(t) + r(t) and the output is x(t)), assuming that r(t) is composed of only very high-frequency components (namely, F{r(t)} = R(jω) = 0 for |w| ≤ K2π/T, where T is the period of x(t)).
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4. (35 pts) Consider an LTI system given by the following block diagram:

(a) (15 pts) Find the frequency response of this system.
(b) (10 pts) Find the impulse response of this system from its frequency response.
(c) (10 pts) Find the output y(t) for the input ) using the frequency response.
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