Description
1. (28 points) Fourier Series
(a) (18 points) Find the Fourier series coefficients for each of the following periodic signals:
i. f(t) = cos(3πt) + 1
2
sin(4πt)
ii. f(t) is a periodic signal with period T = 1 s, where one period of the signal is
defined as e
−2t
for 0 < t < 1 s, as shown below.
iii. f(t) is the periodic signal shown below:
(b) (10 points) Suppose you have two periodic signals x(t) and y(t), of periods T1 and
T2 respectively. Let Xk and Yk be the Fourier series coefficients of x(t) and y(t),
respectively.
i. If T1 = T2, express the Fourier series coefficients of z(t) = x(t) + y(t) in terms of
Xk and Yk.
ii. If T1 = 2T2, express the Fourier series coefficients of w(t) = x(t) + y(t) in terms of
Xk and Yk.
2. (20 points) Fourier series of transformation of signals
Suppose that f(t) is a periodic signal with period T0, with the following Fourier series:
f(t) = X∞
k=−∞
cke
jkω0t
1
Determine the period of each of the following signals, then express its Fourier series coefficients
in terms of ck:
(a) g(t) = f(t) + 1
(b) g(t) = f(−t)
(c) g(t) = f(t − t0)
(d) g(t) = f(at), where a is positive real number
3. (10 points) Eigenfunctions and LTI systems
(a) (5 points) Show that f(t) = cos(ω0t) is not an eigenfunction of an LTI system.
(b) (5 points) Show that f(t) = t is not an eigenfunction of an LTI system.
4. (29 points) LTI systems
Consider the following system:
The system takes as input x(t), it first multiplies the input with e
t
, then sends it through an
LTI system. The output of the LTI system gets multiplied by e
−t
to form the output y(t).
(a) (3 points) Show that we can write y(t) as follows:
y(t) = e
tx(t)
∗ h(t)
e
−t
(1)
(b) (4 points) Use the definition of convolution to show that (1) can be equivalently written
as:
y(t) = Z ∞
−∞
h
0
(τ )x(t − τ )dτ (2)
where h
0
(t) is a function to define in terms of h(t).
(c) (12 points) Equation (2) represents a description of the equivalent system that maps
x(t) to y(t). Show using (2) that the equivalent system is LTI and determine its impulse
response heq(t) in terms of h(t).
(d) (10 points) Suppose that system S1 is given by its step response s(t) = r(t − 1). Find
the impulse response h(t) of S1. What can you say about the causality and stability of
system S1? What can you say about the causality and stability of the overall equivalent
system?
5. (13 points) MATLAB
You must submit all plots and code for each task to receive full credit
(a) (6 points) Task 1
Write an m-file that takes a set of Fourier series coefficients, a fundamental frequency,
and a vector of output times, and computes the truncated Fourier series evaluated at
these times. The declaration and help for the m-file might be:
function fn = myfs(Dn,omega0,t)
%
% fn = myfs(Dn,omega0,t)
% % Evaluates the truncated Fourier Series at times t
%
% Dn — vector of Fourier series coefficients
%
% omega0 — fundamental frequency
% t — vector of times for evaluation
%
% fn — truncated Fourier series evaluated at t
The output of the m-file should be
fN (t) = ΣN
n=−N Dne
jω0nt
The length of the vector Dn should be 2N + 1. You will need to calculate N from the
length of Dn.
(b) (7 points) Task 2
Verify the output of your routine by checking the Fourier series coefficients for the signal
from Problem 1-a-ii. Try for N = 10, N = 50 and N = 100. Use the MATLAB subplot
command to put multiple plots on a page.
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