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?