Q4. Determine the inverse LaPlace transform of each of the functions that follow. Compute the partial fraction expansion analytically for each case. (a) X (s) 8+2 (b) X (s) = s²+7s+12 = ste-s s²+8+1
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.
![TABLE 3.1 Properties of the Fourier Transform
Property
Linearity
Right or left shift in time
Time scaling
Time reversal
Multiplication by a power of t
Multiplication by a complex exponential
Multiplication by sin (wot)
Multiplication by cos(wot)
Differentiation in the time domain
Integration in the time domain
Convolution in the time domain
Multiplication in the time domain
Parseval's theorem
Special case of Parseval's theorem
Duality
Transform Pair/Property
ax(t) + bv(t)→aX(w) + bV(w)
x(tc)X(w)e-jose
x (at) < > ² x ( =) a >
X
a
x(−1)→X(-w) = X(w)
a>0
d"
t"x(t) → j" don X (w) n = 1, 2, ...
dw"
x(t)ejat →X(w - wo) wo real
x(t) sin(wt) ↔ { [X(w + w) − X(w − w)]
x(t) cos(wit) → [X(w + wo) + X(w − w)]
din X(t) → (jw)"X(w) n = 1, 2,...
1
[x(x) dx → — X (w) + 7X (0)8(w)
jw
x(t) *v(t) →X(w)V(w)
x(t)v(t) → __X(w)*V(w)
[x(1)v(1) dt =
1
[~_* x²(1) dt = 2 /
2TT
X(t)→2πx(-w)
[XV (oo) das
1X (w) ² do
-xx
TABLE 3.2 Common Fourier Transform Pairs
1, -∞ < t < ∞ → 2π8 (W)
−0.5 + u(t) ↔ -
jw
1
jw
u(t) <> πδ(ω) +
8(t) →1
8(tc)e-jwc, c any real number
1
jw + b²
e-btu(t).
ejut → 2πd (w - wo), wo any real number
τω
2π
P:(t)→7 sinc-
b>0
7 sinc→2πp, (w)
27
2|t|
τω
(1 - 2!!!) p.,(1)→ sinc²(e)
T
sinc²(# )+2π(1 - 2!!)P, (c)
2
T
cos(wot) →T[8(w + wo) + d(w = wo)]
cos(wat + 0)→π[e¯jªs(w + wo) + ejªs(w - wo)]
sin(wot) → jπ[8(w + wo) − 8(w - wo)]
sin(wat + 0) → jπ[e¯jªs(w + wo) — ejªts (w - wo)]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca6b5de9-d666-4be4-bec5-372f49facd74%2F7d851fd3-371e-426e-871e-f503bc9005fe%2F2bgu6em_processed.png&w=3840&q=75)
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