1.
X ( jw )=
sin (3 w )
d
4 sin ( 4 w )
dw
w
(
)
, determine the time domain signal
2.
3. Use the properties of DTFT to determine the discrete time signal x[n]
X ( e j )=
1
j
4e
( )
( 32 ) +5 ( ) , < <
sin ( )
2
sin
18-290 Signals & Systems
Spring 2014
Homework #5
Total Points: 100
Due: Thursday, Feb. 20, 2013
1. (25 pts) Analysis Equation
Find the fourier series for each signal
(a) (5 pts) x(t) = t for 0 < t < 2 and x(t) has a period of 2.
If k = 0
ak =
=
=
1
2
2
te
18-290: Homework #3 Solution
Signals and Systems, Spring 2014
Prof. Pulkit Grover and Bruno Sinopoli
February 7, 2014
Problem 1. [True or False. [12 pts]
A system is described by y[n] = x[n]2 + 2x[n 1]. If the following statements are true,
prove them. If
18-290 Signals & Systems
Spring 2014
Homework #9
Total Points: 100
1
Due: Tuesday, Apr. 8, 2013
Problem Set 9 - Part 1
1. Signal Reconstruction ( 25 pts)
Let x(t) = et u(t). We can sample this signal at Ts = 105 seconds meaning xs [n] =
x(nTs ) = enTs u[n
18-290 Signals & Systems
Spring 2014
Homework #11
Total Points: 100
1
Due: May 1, 2014
Problem Set 11 - Part 1
1. Laplace
As we have learned in class, Laplace Transform is more generalized version of Fourier Transform. If so, can we get the Fourier Transf
18-290 Signals & Systems
Spring 2014
Homework #8
Total Points: 100
1
Due: Thursday, Mar. 27, 2013
Problem Set 8 - Part 1
1. (20 pts) Discrete Time Fourier Transform
Compute the Discrete Time Fourier Transform (DTFT) for each of the following signals:
(a)
18-290
Signals and Systems
Profs. Pulkit Grover and Bruno Sinopoli
Spring 2014
Problem Set 7
This problem set is due in class on Thursday, March 20, at 10:30 AM.
Please show all work using pencil and paper, but feel free to use MATLAB to produce any plots
18-290 Signals & Systems
Spring 2014
Homework #2
Total Points: 100
Due: Thursday, Jan. 30, 2013
1. (25 pts) SIFT property of the Dirac Delta
1
For any real number k dene k (t) = 2k for t [k, k] and 0 everywhere else
(a) (10 pts) Dene (t) = limk0 k (t). Sh
18-290
Signals and Systems
Profs. Pulkit Grover and Bruno Sinopoli
Spring 2014
Problem Set 1 Solutions
1.a.
1.b.
This is not a periodic function as such, but a rectangular pulse train starting at t=0 and continuing
for all time afterwards:
Total energy is
18-290
Signals and Systems
Profs. Aswin Sankaranarayanan and Byron Yu
Fall 2013 (100 Points)
Final Exam
Name:
Andrew ID:
Problem
1
Score
Max.
6
2
8
3
6
4
8
5
9
6
6
7
12
8
7
9
8
10
8
11
12
12
10
Total
100
Use of tables in encouraged unless otherwise specie
18-290
Signals and Systems
Profs. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
This problem set is due in class on Thursday, September 26th, at 9 AM.
Problem Set 4 Solutions
1. (30 pts) Compute the following convolution integrals:
(a) (6 pts)
18-290
Signals and Systems
Profs. Pulkit Grover and Bruno Sinopoli
Spring 2014
Problem Set 4 Solutions
This problem set is due in class on Thursday, February 13, at 10:30 AM.
Please show all work using pencil and paper, but feel free to use MATLAB to prod
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 12 Solutions
1. Let x(t) be a real valued signal, band-limited to m , i.e, X(j) = 0 for | > m . Let
y(t) = x(t)ejc t .
(a) What constraints, if any, should be placed on c to e
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Midterm 1 Solutions
Name:
Andrew ID:
Problem Score Max
1
10
2
10
3
4
4
10
5
8
6
10
7
8
8
12
9
10
10
10
11
8
Total
100
Fall 2015
2
Midterm 1 Solutions
1. (10 points) Determine if the following si
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 11 Solutions
1. (16 points) Let x(t) and y(t) be two signals that have FT X(j) = 0 for | > x
and Y (j) = 0 for | > y , respectively. For each of the signals listed below, find
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 3 Solutions
1. (8 points) Compute:
(a) (4 points) y(t) =
R1
(b) (4 points) y(t) =
gers.
R1
cos(m )d , where m is a non-zero integer.
3
0
2
0
sin(2m ) sin(2n )d , where m and n
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 5 Solutions
1. (21 points Doru) For each of the following impulse responses, determine whether the
corresponding system is (i) stable, (ii) causal and (iii) memoryless.
(a) (7
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 2 Solutions
1. (20 points) Consider the complex exponential signal x(t) = ej 3 t .
(a) (5 points) Is x(t) periodic? If so, what is the period of the signal?
(b) (5 points) Cal
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 7 Solutions
1. (21 points) Determine the Fourier transform (FT) for the continuous time signal in
part (a) and the discrete-time Fourier transform (DTFT) for the discrete time
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 1 Solutions
1. (15 points) Determine which of the following signals are periodic (show your reasoning). For the periodic signals, compute the fundamental period.
(a) (5 points
18-290
Signals and Systems
Profs. Byron Yu and Pulkit Grover
Fall 2015
Homework 6 Solutions
1. (24 points) Determine the Fourier series (FS) or discrete-time Fourier series (DTFS)
coefficients for the following periodic signals.
(a) (8 points) x(t) = 4 co
18-290
Signals and Systems
Profs. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
Problem Set 3 Solutions
1. (10 Pts) Consider the system y (t) = A cos(t + )x(t). Is the system linear or nonlinear? Is it time-invariant? (Hint: what happens to the
18-290
Signals and Systems
Profs. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
This problem set is due in class on Thursday, September 12th, at 9 AM.
Problem Set 2
Please solve all these problems using only paper and pencil, without resorting