Find the Laplace transform of the following
functions.
(a)
f (t) =
{
6
0 < t ≤ 4
8
t ≥ 4
(b)
f (t) =
{
t2
0 ≤ t < 3
0
t ≥ 3
(c)
f (t) =
{
0
0 ≤ t < π/4
cos[7(t − π/4)]
t ≥ π/4
1. Find the Laplace transform of each of the following
functions: (a). f(t) = t , (b). f(t) = t2 ,
(c) f(t) = tn where n is a positive
integer
Laplace transform of the given function
2. . f(t) = sin bt
3. f(t) = eat sin bt
Find the Laplace Transform of the functions
t , 0 ≤ t < 1
(a) f(x) = 2 − t , 1 ≤ t < 2
0 , t ≥ 2
(b) f(t) = 12 + 2 cos(5t) + t cos(5t)
(c) f(t) = t 2 e 2t + t 2 sin(2t)
Q1-Find all possible time domain signals corresponding
to the following z-transform:
X(z) = (z3 + z2 + 3/2 z + 1/2 ) /
(z3 + 3/2 z2 + 1/2 z)
Q2-A digital linear time invariant filter has the
following transfer function:
H(z) = (5 + 5z-1) / (1 - 3/8 z-1 + 1/16
z-2)
a) Find the impulse response if the filter is causal.
For each of the following functions, find the extreme value of z
(this can be either max or min), subject to a given constraint by
the method of direct substitution.
(a) z = xy, subject to the constraint x2 +
y2 = 16 (Note that there are four solutions.)
(b) z = 3x2 − 10xy + 12y2 , subject to the
constraint y = 20 − 1/2 x
(c) z = xy, subject to the constraint x − y =...
QUESTION 6. Z is a standard normal variable. Find the value of Z
in the following. (12 points)
2 Points each
a. The area to the left of Z is 0.8554
b. The area to the right of Z is 0.1112.
c. The area to the left of -Z is 0.0681.
d. The area to the right of -Z is 0.9803.
e. The area between 0 and Z is 0.4678.
f. The area between -Z and Z is 0.754.
1) Find the Laplace transform of
f(t)=−(2u(t−3)+4u(t−5)+u(t−8))
F(s)=
2) Find the Laplace transform of f(t)=−3+u(t−2)⋅(t+6)
F(s)=
3) Find the Laplace transform of f(t)=u(t−6)⋅t^2
F(s)=