4. Using the Laplace transform pairs of Table 2.1 and the Laplace transform theorems of Table 2.2, derive the Laplace transforms for the following time functions: For table 2.1 & 2.2, see appendix at the end. e-at sin otu(t) a b C е cos at u(t) Bu(t)
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![Appendix
TABLE 2.1 Laplace transform table
Item no.
f(1)
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
TABLE 2.2 Laplace transform theorems
Item no.
Theorem
7.
8.
9.
10.
11.
12.
L[f(t)]=F(s) = f(t)e-sdt
L[kf (1)]
L
8(1)
u(t)
=kF(s)
Lf11) +f2(1)] = F₁(s) + F2(s)
Le-atf(1)]
= F(s+a)
L[f(1-T)]
= e-sTF(s)
L[f(at)] ---F(²)
HESE
tu(t)
t"u(t)
e-at u(t)
sin cotu(t)
cos atu(t)
dt
di
[d"f
den
L[fo_f(t)dt]
f(xo)
ƒ(0+)
F(s)
S
=lim sF(s)
S-0
=lim sF(s)
F(s)
-
1
1
S
1
$2
n!
sh +1
1
s+a
=SF (s)-f(0-)
=s²F(s)- sf (0-) - f'(0-)
=s" F (s)-s-kpk-1 (0-)
k=1
(
s² + w²
S
s²+w²
Name
Definition
Linearity theorem
Linearity theorem
Frequency shift theorem
Time shift theorem
Scaling theorem
Differentiation theorem
Differentiation theorem
Differentiation theorem
Integration theorem
Final value theorem¹
Initial value theorem²
¹For this the orem to yield correct finite results, all roots of the denominator of F(s) must have negative real
parts, and no more than one can be at the origin.
2For this theorem to be valid, f(t) must be continuous or have a step discontinuity at t= 0 (that is, no
impulses or their derivatives at t= 0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5c85250b-6a52-47e3-86e8-89d741652c47%2F09c5c760-f9e7-467f-a6a9-75246d3f2ac3%2F0y5oykn_processed.jpeg&w=3840&q=75)
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