18-290 Signals And Systems
Homework 6 solutions
Spring 2012
1. Let x(t) and y(t) both be continuous periodic signals having period T and with Fourier series representations given by:
ak ejk0 t
x(t) =
bk ejk0 t .
and y(t) =
k=
k=
(a) Show that Fourier seri
18-290 Signals And Systems
Homework 1
Issued: Friday, January 20, 2012
Spring 2012
Total Points: 20
Due: Thursday, January 26, 2012 (in class)
Note: Please solve the problems using only paper and pencil, without a computer. Also, please show all
steps to
18-290 Signals and Systems
Spring 2017
Homework 4 Solutions
1. Consider sketches of 3 signals below:
b(t)
1
W
1
2W
t
t
-W
r(t)
b(t)*b(t)
2W
-2W
2W
-2W
-1
Sketch (a) b(t) r(t) and (b) r(t) r(t).
(Solution:)
From the sketches, r(t) can be represented as the
18-290 Signals and Systems
Spring 2017
Homework 6 Solutions
1. You learned about Parsevals theorem in class:
Z
Z
1
2
|x(t)| dt =
|X(j)|2 d
2
This relationship is important because the left side represents the total energy of x(t).
Hence the integral of
18-290 Signals and Systems
Spring 2017
Homework 1 Solutions
1. Let a, b > 0. Let
x(t) = eat u(t),
h(t) = ebt u(t).
Derive an expression of x(t) h(t).
Solution:
Z
x(t) h(t) =
x( )h(t )d
Z =
=
ea u( )eb(t ) u(t )d
=
Observe that u( ) is non-zero only for 0
18-290
Signals and Systems
Profs. Aswin Sankaranarayanan and Richard Stern
Spring 2017
Homework 1 Solutions
1. A continuous-time signal x(t) is periodic if x(t) = x(t T ) for all t. The fundamental
period T is the smallest value of that parameter for whic
18-290 Signals and Systems
Spring 2017
Homework 1 Solutions
1. Derive, from the defining equations, the Fourier transform of the following signal.
x(t) = teat u(t),
Hint:
R
t (1+t)
tet dt = e
2
a > 0.
. This expression is valid even when is complex-valued
18-290 Signals and Systems
Spring 2017
Homework 2 Solutions
1. Plot the following equation, x(t), on the interval [0, 2] for the different values of the
parameters , , and specified below:
x(t) = et cos(t + )u(t)
Please do not use MATLAB for this question
18-290 Signals and Systems
Spring 2017
Midterm 1
1. (10 pts) Suppose that x(t) can be written as
X
x(t) =
ak ejk t ,
k
where ak are scalars. Let x(t) also be periodic with period T . Show that k must be
.
an integer multiple of 2
T
(Solution:)
Since x(t)
18-290 Signals And Systems
Homework 6
Issued: Friday, March 2, 2012
Spring 2012
Total Points: 100
Due: Thursday, March 8, 2012 (in class)
1. Let x(t) and y(t) both be continuous periodic signals having period T and with Fourier series representations give
18-290 Signals And Systems
Homework 2
Issued: Thursday, January 26, 2012
Spring 2012
Total Points: 100
Due: Thursday, February 2, 2012 (in class)
1. Let z0 be a complex number with polar coordinates (r0 , 0 ) and Cartesian coordinates (x0 , y0 ). Determin
18-290 Signals And Systems
Homework 4
Issued: Friday, February 10, 2012
Spring 2012
Total Points: 100
Due: Thursday, February 16, 2012 (in class)
1. For the system in the gure, the following is specied:
x(t) = u(t), h1 (t) = u(t), h2 (t) = (t 1) and h3 (t
18-290 Signals And Systems
Homework 8
Issued: Friday, March 23, 2012
Spring 2012
Total Points: 100
Due: Thursday, March 29, 2012 (in class)
1. Compute the DTFT of each of the following signals:
(a) x[n] = [n + 1] + 2[n] + [n 1]
(b) x[n] =
1 |n|
u[n
2
(c)
18-290 Signals And Systems
Homework 8
Issued: Friday, March 23, 2012
Spring 2012
Total Points: 100
Due: Thursday, March 29, 2012 (in class)
1. Compute the DTFT of each of the following signals:
(a) x[n] = [n + 1] + 2[n] + [n 1]
(5pt)
X(ej )
x[n]ejn =
=
n=
18-290 Signals And Systems
Homework 10
Issued: Friday, April 13, 2012
Spring 2012
Total Points: 100
Due: Thursday, April 19, 2012 (in class)
1. (a)
F
sin(1000t)
G(j)
=
( ( 1000) ( + 1000) = j ( ( + 1000) ( 1000)
j
1
X(j) F cfw_sin(1000t)
2
(4pt)
(b)
F
co
18-290 Signals And Systems
Spring 2012
Homework 10
Total Points: 100
Issued: Friday, April 13, 2012 Due: Thursday, April 19, 2012 (by noon at the course hub)
1. Let x(t) be a real-valued signal whose X(j) is shown on Figure 1a and with maximum frequency
0
18-290 Signals And Systems
Homework 7
Issued: Friday, March 9, 2012
Spring 2012
Total Points: 100
Due: Thursday, March 22, 2012 (in class)
1. A raised-cosine lter is an LTI system with the following frequency response:
1 + cos() [, ]
.
0
otherwise
H(j) =
18-290 Signals And Systems
Homework 3
Issued: Friday, February 3, 2012
Spring 2012
Total Points: 100
Due: Thursday, February 9, 2012 (in class)
1. Compute the convolution y[n] = x[n] h[n] analytically and sketch y[n], for the following pairs of signals
(a