Time Domain Convolution
ECE 3100
Western Michigan University
John Stahl
Probabilistic Methods Signal & System Analysis , 3rd edition, George R. Cooper, 1999
Time Domain
x(t)
h(t)
y(t)
𝑡
∞
𝑦 𝑡 =
𝑥(𝑡 − 𝜆) ∙ ℎ 𝜆 ∙ 𝑑𝜆
or
𝜆=0
Laplace Domain
X(s)
H(s)
𝑌 𝑠 = 𝐻(𝑠) ∙ X(s)
Y(s)
𝑦 𝑡 =
𝑥(𝜆) ∙ ℎ 𝑡 − 𝜆 ∙ 𝑑𝜆
𝜆=−∞
Time Domain Convolution
𝑥 𝑡 = 𝑀 ∙ 𝑢(𝑡)
Where M is a DC value.
ℎ 𝑡 = 𝑡 ∙ exp⁡(−2𝑡) ∙ 𝑢(𝑡)
𝑡
𝑦 𝑡 =
𝑥(𝜆) ∙ ℎ 𝑡 − 𝜆 ∙ 𝑑𝜆
𝜆=−∞
𝑡
=
𝑀𝑢(𝜆) ∙ (𝑡 − 𝜆) ∙ exp⁡(−2 ∙ (𝑡 − 𝜆))𝑢(𝑡 − 𝜆) ∙ 𝑑𝜆
𝜆=−∞
𝑢 𝜆 = 0 when 𝜆 < 0. Likewise 𝑢 𝑡 − 𝜆 = 0 when 𝑡 < 𝜆. In this
case 𝑢 𝜆 will set the limits of integration.
𝑡
=
𝑀 ∙ (𝑡 − 𝜆) ∙ exp⁡(−2𝑡 + 2𝜆) ⁡ ∙ 𝑑𝜆
0
𝑡
= 𝑀 ∙ exp⁡(−2𝑡)
(𝑡 − 𝜆) ∙ exp⁡(2𝜆) ∙ 𝑑𝜆
0
𝑦(𝑡) = 𝑀 ∙
𝑡
𝑒 −2𝑡
⁡(𝑡𝑒 2𝜆 − 𝜆 ∙ 𝑒 2𝜆 ) ∙ 𝑑𝜆
0
=𝑀∙𝑒
−2𝑡
𝑡 2𝜆 𝑒 2𝜆
∙ 𝑒 − 2 2𝜆 − 1 |𝑡0
2
2
Slowly and carefully apply the
limits to each term. Be careful of
sign errors.
2𝑡
2𝑡
0
0
𝑡
𝑒
𝑒
𝑒
𝑒
= 𝑀 ∙ 𝑒 −2𝑡 ∙ (𝑒 2𝑡 −𝑒 0 ) − 2𝑡
−
+ 2∙0∙ −
2
4
4
4
4
=𝑀∙
𝑒 −2𝑡
𝑡 2𝑡
𝑒 2𝑡 𝑒 2𝑡
1
∙ (𝑒 −1) − 2𝑡
−
+ 0−
2
4
4
4
2𝑡
2𝑡
𝑡
𝑒
𝑒
1
= 𝑀 ∙ 𝑒 −2𝑡 ∙ (𝑒 2𝑡 −1) − 2𝑡
−
+ 0−
2
4
4
4
1
1
1
1
1
−2𝑡
= 𝑀𝑡 − 𝑀 ∙ 𝑡𝑒
− 𝑀𝑡 + 𝑀 + 0 − 𝑀 ∙ 𝑒 −2𝑡
2
2
2
4
4
𝑦 𝑡 =
1
1
1
𝑀 − 𝑀 ∙ 𝑒 −2𝑡 − 𝑀 ∙ 𝑡𝑒 −2𝑡
4
4
2
Laplace Domain
X(s)
H(s)
Convert the x(t) and
h(t) into X(s) and H(s).
Y(s)
𝑌 𝑠 = 𝐻(𝑠) ∙ X(s)
𝑥 𝑡 = 𝑀 ∙ 𝑢(𝑡)
ℎ 𝑡 = 𝑡 ∙ exp⁡(−2𝑡) ∙ 𝑢(𝑡)
Use the Laplace table.
𝑀
𝑠
𝑋 𝑠 =ℒ 𝑥 𝑡
=
𝐻 𝑠 =ℒ ℎ 𝑡
1
=
𝑠+2
2
𝑌 𝑠 = 𝐻(𝑠) ∙ X(s)
1
𝑌 𝑠 =
𝑠+2
𝑀
∙
2 𝑠
𝑟𝑜𝑜𝑡𝑠:
𝑠=0
𝑠 = −2
Partial Fraction Expansion with repeated roots
1
𝑠+2
2∙
𝑀 𝐴
𝐵
= +
𝑠
𝑠
𝑠+2
𝐴 𝑠+2 𝑠+2
2
𝐶
𝑠+2
1+
+ Bs 𝑠 + 2
@𝑠 = 0
2
2
+ Cs 𝑠 + 2 = 𝑀 𝑠 + 2
@𝑠 = −2
8𝐴 + 0 = 2𝑀
0+0= 0
𝐵=0
𝑀
𝐴=
4
Fails for repeated
roots
𝐶=0
With repeated roots:
Eq 1. @𝑠 = −1
𝐴 + B(−1) 1
Eq 2. @𝑠 = −3
𝐴 −1 −1
2
2
+ C(−1) 1 = 𝑀 1
+ B(−3) −1
2
+ C(−3) −1 = 𝑀 −1
Eq 1.
𝑀
−B−C=𝑀
4
3𝑀
−B − C =
4
Eq 2.
−
𝑀
− 3B + 3C = −𝑀
4
3𝑀
−3B + 3𝐶 = −
4
3𝑀 1
−1 −1 𝐵
∙
=
∙
−3 3
𝐶
4 −1
𝐴
𝐵
𝑌(𝑠) = +
𝑠
𝑠+2
𝑀
B=−
4
𝑀
C=−
2
𝐶
1+ 𝑠+2
2
Take Y(s) back into the time-domain.
𝐴
𝐵
𝑌(𝑠) = +
𝑠
𝑠+2
ℒ −1
𝐴
𝑠
ℒ −1
𝐴 ∙ 𝑢(𝑡)
𝐵
𝑠+2
𝐶
+
1
𝑠+2
1
𝐵𝑒 −2𝑡 ∙ 𝑢(𝑡)
2
ℒ −1
𝐶
𝑠+2
2
𝐶𝑡𝑒 −2𝑡 ∙ 𝑢(𝑡)
1
1
1 −2𝑡
−2𝑡
𝑦(𝑡) =
𝑀 + 𝑀𝑒
+ 𝑡𝑒
∙ 𝑢(𝑡)
4
2
4