Department of Electrical and Computer Engineering & Computer
Science
ELEC-6603: Discrete & Continuous Systems-I
Homework-1: solutions
Directions:
o Please go through the solutions thoroughlyo Compare the
suggested solutions to your own, and make the necessary
correctionso Make sure you understand each problem (do not
memorize!)o If you have doubts, feel free to ask questionso Please
let me know of potential errorsProblem-1:
Part-1a:
( ) ( ) ( )( ) ( ) ( )( ) ( )1 3 2 1 1 1 2 3 3x t tu t t u t t u
t= + +
Part-1b:
Part-2a:
( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 3 3 5x t u t t u t t u t u t= +
+
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Part-3a:
( ) ( ) ( ) ( ) ( ) ( )1 2 2 2 3 6 4 5x t u t u t u t u t u t= +
+ + +
Part-3b:
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Problem-2:
Part-1:
Part-2:
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Part-3:
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Problem-3:
Part-1:
Part-2:
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Part-3:
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Problem-4:
Part-1:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
22 3 1 3 2 3 10 3
1 2 1cos 4 sin cos 4
4 2 4
dxtu t t t tu t t
dt
dyt u t t t t u t t
dt
= + + = +
= + = +
Part-2:
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Problem-5:
Part-1:
Part-11:
( )1x t is simply equal to ( )x t right shifted by one.
Part-12:
Write ( ) ( )( )2 1 2x t x t = . Get the mirror image of ( )x t
then shift the result right by 2
Part-13:
( ) 1
2 1 22
x t x t
+ = +
. Scale ( )x t by 2 (compression), then shift left by
one-half
Part-14:
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( )1
4 82 2
tx x t
=
. Scale the signal ( )x t (expansion), then take a mirror image
of the
resulting signal. Lastly shift right by 8.
Part-2:
Part-11
[ ]4x n is simply [ ]x n shifted right by 4
Part-12:
[ ] ( )3 3x n x n = . Get the mirror image of [ ]x n , then
shift the result right by 3.
Part-13:
[ ]
[ ] [ ] [ ]
1, 33
0, 3
, 33
0, 3
nu n
n
x n nx n u n
n
=
>
=
>
Problem-6:
Part-1:
( )sin sin
2 2 2 2t t t dt
= =
Part-2:
( )4 4
3 3
5 5
174 4 17 0
4t t dt t t dt
+ = =
Part-3:
( ) ( ) ( ) ( )2 3 0 because 2 3 0 (no overlap)t t dt t t
