EE301 Homework 3 Solution

1) Problem 1: Let 𝑥[𝑛] = 𝛿[𝑛] + 3𝛿[𝑛 − 1] + 𝛿[𝑛 − 2].
a) Plot 𝑥[𝑛] and its periodic extension, 𝑥̃[𝑛], for 𝑁 = 3 .
𝑥̃𝑁
𝑘=−∞
b) Find the Discrete-time Fourier Series (DTFS) coefficients, 𝑋̃3[𝑘], of 𝑥̃3[𝑛]. Write the DTFS representation of 𝑥̃3[𝑛].
c) Find the Discrete-time Fourier Series (DFS) coefficients for N=5, 𝑋̃5[𝑘], of 𝑥̃5[𝑛]. Write the DTFS representation of 𝑥̃5[𝑛].
d) Find the DTFT, 𝑋(𝑒𝑗), of 𝑥[𝑛]. Plot its magnitude and phase.
3[𝑘] = 𝑋(𝑒𝑗)|=𝑘2𝜋 and 𝑋̃5[𝑘] = 𝑋(𝑒𝑗)|=𝑘2𝜋, i.e., uniformly spaced samples e) Verify that 𝑋̃
3 5
of DTFT of 𝑥[𝑛]. Show these samples on the magnitude and phase plot of 𝑋(𝑒𝑗).

Problem 2: The following difference equation for a LTI system is given, yn− yn−1=xn−xn−1+xn−2
a) Find the frequency response, H(ej).
b) Plot the magnitude, |H(ej)| and phase response, H(ej), for H(ej) in MATLAB using freqz command.
c) Let xn= cos n+sinn+ be the system input. Find the output y[n].
 3   2 4 
d) Prove that H(ej)= H*(ej(2π - )).

Problem 3:
For the given system, Xc(jω) and H(ej), sketch and label the Fourier Transform of yc(t) for the following cases
a) 1/T1=1/T2=104 b) 1/T1=1/T2= 2.104 c) 1/T1=2.104, 1/T2= 104 d) 1/T1= 104, 1/T2=2.104
Xc(jω)
xc(t) c


ω (rad/s)

Problem 4:
Assume the following system is provided:


Determine the maximum value for T and appropriate H(ej) of the DT system, such that Yr(jω)=Yc(jω).