6.003 Homework #6 Solutions
Problems
1. Maximum gain
For each of the following systems, nd the frequency m for which the magnitude of the
gain is greatest.
a.
1
1 + s + s2
w
m =
This system has poles at s = 1 j
2
1
2
3
2.
3
2
1
2
23
We must minimize the
6.003 Homework #4 Solutions
Problems
1. Laplace Transforms
Determine the Laplace transforms (including the regions of convergence) of each of the
following signals:
a. x1 (t) = e2(t3) u(t 3)
X1 =
e3s
s+2
ROC:
Re(s) > 2
X1 (s) =
x1 (t)est dt =
= e6
e(s+2)t
6.003 Homework #3 Solutions
Problems
1. Complex numbers
a. Evaluate the real and imaginary parts of j j .
e/2
Real part =
Imaginary part =
0
Eulers formula says that j = e j /2 , so
j
j j = e j/2
= e/2
.
Thus the real part is e/2 and the imaginary part i
6.003 Homework #2 Solutions
Problems
1. Finding outputs
Let hi [n] represent the nth sample of the unit-sample response of a system with system
functional Hi (R). Determine hi [2] and hi [119] for each of the following systems:
a. H1 (R) =
R
1 3
R
4
3
4
h
6.003 Homework #1 Solutions
Problems
1. Solving dierential equations
Solve the following dierential equation
y (t) + 3
dy (t)
d2 y (t)
=1
+2
dt
dt2
for t 0 assuming the initial conditions y (0) = 1 and dy(t)
dt t=0 = 2. Express the solution
in closed form
6.003 (Spring 2010)
April 7, 2010
Quiz #2
Name:
Kerberos Username:
Please circle your section number:
Section
1
2
3
4
Instructor
Time
Peter Hagelstein
Peter Hagelstein
Rahul Sarpeshkar
Rahul Sarpeshkar
10 am
11 am
1 pm
2 pm
Grades will b e determined by t
6.003 Homework #9 Solutions
Problems
1. Fourier varieties
a. Determine the Fourier series coecients of the following signal, which is periodic in
T = 10.
x1 (t)
1
t
10
3 1
1
3
10
a0 =
2
5
ak =
sin 3k sin 5k
5
k
1
1
ak =
10
=
e
e
k
j 210 t
3
k
j 3 210
e
j
6.003 Homework #8 Solutions
Problems
1. Fourier Series
Determine the Fourier series coecients ak for x1 (t) shown below.
x1 (t)= x1 (t + 10)
1
t
1
10
a0 =
1
10
ak =
1 jk/10
sin(k/10)
k e
1
ak =
T
x(t)e
T
j 2 kt
T
1
dt =
10
for k = 0
1
1e
j 2 kt
10
0
1 ej
6.003 Homework #7 Solutions
Problems
1. Second-order systems
The impulse response of a second-order CT system has the form
h(t) = et cos(d t + )u(t)
where the parameters , d , and are related to the parameters of the characteristic
polynomial for the syst
6.003 Homework #5 Solutions
Problems
1. DT convolution
Let y represent the DT signal that results when f is convolved with g , i.e.,
y [n] = (f g )[n]
which is sometimes written as y [n] = f [n] g [n].
Determine closed-form expressions for each of the fol