Homework #5 Signals & Systems 

ECE102 Homework #5
Signals & Systems 
100 points total.
This homework covers questions relate to Fourier series and Fourier transform.
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) (see the Lecture 11 slide
titled: Fourier Series Properties: Fourier Symmetry (cont.)). How do these properties
of symmetry change when f(t) is imaginary (with no real component)?
(b) (7 points) A real and odd signal x(t) has the following properties:
• it is a periodic signal with period 1 s;
• it has one positive frequency component (positive frequency component meaning
ck with k > 0);
• it has a power of 9 (hint: consider Parseval’s relation. The power of the signal in
the time domain is the same as the sum of the powers of its frequency components).
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 can 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)).
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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 the 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.
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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
Hint: Consider the properties of x(t).
(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)}
Hint: Consider the ’even and odd’ properties of the Fourier transform
4. (35 points) Fourier transform and its inverse
(a) (18 points) Find the Fourier transform of each of the signals given below:
Hint: you may use Fourier Transforms derived in class.
i. x1(t) = (
1 + cos(πt), |t| < 1
0, otherwise
ii. x2(t) = e
(1+3j)tu(−t + 1)
iii. x3(t) = 2te−2tu(t)
Hint: You can consider Fourier transform of the derivative and its dual.
(b) (7 points) Find the inverse Fourier transform of the signal shown below (note that
|X(0)| = 0.5 and |X(2.5)| = 1):
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(c) (10 points) Two signals f1(t) and f2(t) are defined as
f1(t) = sinc(2t)
f2(t) = sinc(t) cos(3πt)
Let the convolution of the two signals be
f(t) = (f1 ∗ f2)(t)
i. Find F(jω), the Fourier transform of f(t).
ii. Find f(t).
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