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
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
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
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 eac
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 an
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
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)
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 p
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]