Mathematics - Signals and Systems

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Signals and Systems
Student’s Name
Institutional Affiliation
Date

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Question 4
a)
∫
−∞
∞
cos ( 2 πt + π
3 ) δ ( t−2 ) dt
Using the property,
∫
−∞
∞
f ( x ) δ ( x−a ) dx=f ( a )
∫
−∞
∞
cos ( 2 πt + π
3 ) δ ( t−2 ) dt
¿ cos (2 πt + π
3 ), t=2
¿ cos ( 4 π + π
3 )
¿ cos ( 13 π
3 ) =0.5
∫
−∞
∞
cos (2 πt + π
3 )δ ( t−2 ) dt=f ( a )=2
b)
∫
0
4
( t2 +1 ) u ( t+ 4 ) δ ( t−2 ) dt=∫
0
4
( t2 +1 ) δ ( t+ 2 ) dt
¿∫
0
4
(t2+ 1 ) δ ( t−(−2) ) dt
¿(−2)2 +1=5
c)
∫
0
4
( t2 +1 ) u ( t−4 ) δ ( t−2 ) dt =∫
0
4
( t2 +1 ) δ ( t−6 ) dt
¿ 0
This is because 6 is not in [0,4]
d)
∫
−∞
∞
( t2 +1 ) δ ( 2 t−4 ) dt =∫
−∞
∞
( t2 +1 ) δ {2 ( t−3 ) +2 }
¿ t 2+1 , t=3

3
¿ 32 +1=10
Question 5
Question 6
a)
From −1 to 0 the signal has a slope of 1 and can be represented by,
(t +1) u ( t +1 )
From 0 to 1 the signal can be represented as,
u(t )
From 1 to 2 the signal can be represented as,
(−t +2)u(t−2)