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. 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. 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
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)
Problem Set 5 Solutions
1. (15 Pts) The impulse response of a continuous-time system is given as h(t) = e3t u(t +
2)
(a) (5 pts) Find the frequency response H (j )
18-290
Signals and Systems
Profs. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
This problem set is due in class on Thursday, October 17th, at 9 AM.
Problem Set 6
1. (20 Pts) Consider the following continuous time periodic signals. Determine th
18-290
Signals and Systems
Profs. Aswin Sankaranarayanan and Byron Yu
Fall 2013 (100 Points)
Midterm Exam 2
Name:
Andrew ID:
Problem
1
Score
Max.
12
2
18
3
12
4
15
5
12
6
16
7
15
Total
100
Use of tables in encouraged unless otherwise specied.
Midterm Exam
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. 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 & 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. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
This problem set is due in class on Thursday, November 12st 2013, at 9 AM.
Problem Set 10
Please solve all these problems using only paper and pencil, without resor
18-290
Signals and Systems
Profs. Aswin Sankaranarayan and Byron Yu
Fall 2013 (100 Points)
This problem set is due in class on Thursday, November 14th 2013, at 9 AM.
Problem Set 9
Please solve all these problems using only paper and pencil, without resort
18-290 Signals & Systems
Spring 2013
Homework #2 Solution
Total Points: 100
Due: Thursday, Jan. 31, 2013
1. (20 pts) Periodicity after time scaling, time shift, and summation.
Let x(t) be a continuous time signal, and let y1 (t) = x(3t) and y2 (t) = x(t/3
18-290 Signals & Systems
Spring 2013
Homework #3 Solution
Total Points: 100
Due: Thursday, Feb. 7, 2013
1. (20 pts) Memory, causality and invertibility in systems.
For the following systems, determine if they are memoryless, causal, and invertible. If no,
18-290 Signals & Systems
Spring 2013
Homework #4 Solutions
Total Points: 100
Due: Thursday, Feb. 14, 2013
1. (35 pts) Evaluating convolutions, causality and stability for CT and DT LTI systems.
(a) (7 pts) Consider the following CT convolution example:
Le
18-290 Signals & Systems
Spring 2013
Homework #8 Solutions
Total Points: 100
Due: Thursday, March 21, 2013
1. (15 pts) Fourier transform of an impulse train.
Justify your answers by showing your work or reasoning.
(t kT ), an impulse train with period T
18-290 Signals & Systems
Spring 2013
Homework #9 Solutions
Total Points: 100
Due: Thursday, March 28, 2013
1. (25 pts)
Compute the Fourier transform of these DT signals:
(a) (5 pts) x[n] = n, f or 2 n 2; x[n] = 0, otherwise
[Soln]:
X (ej ) = x[n]ejn = nej
18-290 Signals & Systems
Spring 2013
Homework #10 Solutions
Total Points: 100
Due: Monday, April 22, 2013
1. (15 pts)
According to the sampling theorem, the frequency that must be exceeded by the sampling
frequency is called the Nyquist rate. Determine th
18-290 Signals & Systems
Spring 2013
Homework #11 Solutions
Total Points: 100
Due: Thursday, April 25, 2013
1. (25 pts)
Please use the denition of Laplace transform (X (s) = x(t)est dt) to evaluate the Laplace
transform of these functions. Specify the reg