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1. (18 points) Fourier Series
(a) (7 points) When the periodic signal f(t) is real, you have seen in class some properties
of symmetry for the Fourier series coefficients of f(t) (handout 8, slide 41). How do
these properties of symmetry change when f(t) is pure imaginary?
(b) (7 points) A real and even signal x(t) has the following properties:
• it is a periodic signal with period 1 s;
• it has a DC component of 1 and one positive frequency component;
• it has a power of 9.
What is x(t)?
(c) (4 points) Consider the signal y(t) shown below and let Y (jω) denote its Fourier transform.
Let YT (t) denote its periodic extension:
How the Fourier series coefficients of yT (t) can be obtained from the Fourier transform
Y (jω) of y(t)? (Note that the figures given in this problem are for illustrative purposes,
the question is for any arbitrary y(t)).
1
2. (32 points) Symmetry properties of Fourier transform
(a) (16 points) Determine which of the signals, whose Fourier transforms are depicted in
Fig. 1, satisfy each of the following:
i. x(t) is even
ii. x(t) is odd
iii. x(t) is real
iv. x(t) is complex (neither real, nor pure imaginary)
v. x(t) is real and even
vi. x(t) is imaginary and odd
vii. x(t) is imaginary and even
viii. There exists a non-zero ω0 such that e
jω0tx(t) is real and even
Figure 1: P2.a
(b) (8 points) Using the properties of Fourier transform, determine whether the assertions
are true or false.
i. The convolution of a real and even signal and a real and odd signal is odd.
ii. The convolution of a signal and the same signal reversed is an even signal.
(c) (8 points) Show the following statements:
i. If x(t) = x
∗
(−t), then X(jω) is real.
2
ii. If x(t) is a real signal with X(jω) its Fourier transform, then the Fourier transforms
Xe(jω) and Xo(jω) of the even and odd components of x(t) satisfy the following:
Xe(jω) = Re{X(jω)}
and
Xo(jω) = jIm{X(jω)}
3. (15 points) Fourier transform properties
Let X(jω) denote the Fourier transform of the signal x(t) sketched below:
Evaluate the following quantities without explicitly finding X(jω):
(a) R ∞
0 X(jω)dω
(b) X(jω)|ω=0
(c) X(jω)
(d) R ∞
−∞ e
−jωX(jω)dω
(e) Plot the inverse Fourier transform of Re{e
−3jωX(jω)}
4. (35 points) Fourier transform and its inverse
(a) (20 points) Find the Fourier transform of each of the signals given below:
Hint: you may use Fourier Transforms derived in class.
i. x1(t) = 2rect −t−3
2
cos(10πt)
ii. x2(t) = e
(2+3j)tu(−t + 1)
iii. x3(t) = (
1 + cos(πt), |t| < 1
0, otherwise
iv. x4(t) = te−tu(t)
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