Signals and Systems
(40242)
Department of Computer Engineering
Sharif University
of Technology
Instructor: Dr Manzuri
Deadline: 14 Aban
HW 2 – Theory
Notes:
.
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
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Deliver your answers to Dr Manzuri at the end of class session at 14 Aban.
Each day delay in delivering will result into 25% decreasing of grade.
If there is any question about this homework, just contact Ms. Ebrahimi or Ms. Salimi .
1. We have information about signal x[n] as below: (8 points)
I.
It is real and even.
II.
Period of the signal equal to 10 (N=10) with Fourier series coefficients of
III.
=5
[ ] = 50
IV.
∑
(
). Next, find A,B and C.
Prove that x[n] =
2. Let ( ) be a periodic signal with period T = 5 and Fourier series coefficient
as below. (12 points)
a. Find ( ) without calculating ( ).
b. Evaluate the following sums without calculating x(t).
∫
3. Let ( )
( )
( )
(
a. Compute ( )
b. Compute ( )
1
, ∫
)
( )
( )
( )
( )
( )
(
)
, ∫
( ) (10 points)
( )
.
depicted
4. Let x[n] be a periodic sequence with period N and Fourier series representation
[ ]
∑
The Fourier series coefficients of each of the following signals can be expressed in terms
of .Derive the expressions. (27 points)
a.
b.
c.
[
[ ]
[ ]
]
[
[
]
] (Assume N is even)
d.
[ ]
[
] (Assume N is even; note that this signal is periodic with
period )
e.
[ ]
f. ( ) [ ] (Assume N is even)
g. ( ) [ ] (Assume N is odd; note that this signal is periodic with period 2N)
[ ]
h. [ ] {
[
i.
]
5. Let ( ) be a continuous-time periodic signal with fundamental frequency
and and
Fourier coefficients
Given that
( )
(
)
(
),
how is the fundamental frequency
of ( ) related to
? And find a relationship
between the Fourier coefficients
of ( ) and the coefficients . (10 points)
6. Find the Fourier series of the following signals: (16 points)
a)
b)
c)
d)
2
( )
( )
(
∑
)
(
(
)
)
7. A midterm sample from previous semesters. (10 points)
8. Problem 3.41 of textbook p. 261. (7 points)
Good Luck
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