EE341, Spring 2016
Problem Set #0 (Optional)
Due: 9PM, 1 April 16
1. These five expressions of discrete-time functions correspond to three signals. Sketch them
and determine which expression matches which signal.
x1 [n] = u[n + 1] u[n 3]
x2 [n] =
3
X
[n k
EE 341 Exam 2 Solution
Problem 1:
T
2
2
3
and T2 =
. Since 1 = is
7
3
T2 7
rational, x(t) is periodic with period T = 7T1 = 3T2 = 2 , or 0 = 1 . Using Euler identity,
Signal x(t) is a sum of two sinusoids with time periods: T1 =
j (3t )
6
e j 7t + e j 7t
EE341: Exam 1 Solution
1. (25 points)
3 cos 3
a. Determine whether the continuous-time signal
2 sin 4 is
periodic. If yes, find its time period ; otherwise, state why it is not periodic.
Solution:
is a sum of two sinusoids with time periods
is rational,
i
EE341: Exam 3 Solution
1. (30 points) Consider the conventional AM signal
cos
is the message signal and
cos
1
, where
is the carrier, as shown below.
15
AM wave s(t) in volt
10
5
0
-5
-10
-15
0
10
20
30
40
50
60
70
80
90
100
Time t in micro second
and .
EE341 Homework 12 Solution
Problem 1
1
1
+ (1 p ) log 2 (
) = [ p log 2 p + (1 p ) log 2 (1 p )]
p
1 p
(b) To maximize the entropy, take the derivative of H(X) with respect to p, set it to zero, and
solve for p:
1
1
p ( ) + (1) log 2 p (1 p)(
) (1) log
EE 341 Exam 3 Solutions
1.
(30 points) Consider the conventional AM signal
1
, where
the message signal and
cos
is the carrier, as shown below.
sin
is
0.18
0.2
5
4
AM wave s(t) in volt
3
2
1
0
-1
-2
-3
-4
-5
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
Time t
EE 341 Exam 3 Solutions
Problem 1
s = 10kHz . R p = 2dB = 1 10
(a) p = 2.5kHz
Rs = 40dB = 10
Rs
20
1
2N
2 log s
p
5
c s 2 1
7
j
= 0.21 and
= 3.5 N = 4 . This gives
1
9
j
j
s2 = 3e 8 = 3e
2N
2.67kHz c 3.16kHz , say c = 3kHz .
j
(
7
j
8
(b) Poles of H(
EE 341 Exam # 1
(September 27, 2006)
Name:
Score:
1 (25 points):
2 (25 points):
3 (25 points):
4 (25 points):
Total (100 points):
This is a closed book/notes exam.
You may use a calculator and one hand-written 8.500 1100 sheet of formulae.
Programming
EE 341 Exam 2 Solution
Problem 1:
Signal x(t) is a sum of two sinusoids with time periods:
. Since
and
is rational, x(t) is periodic with period 4
3 2, or 1.
Using Euler identity,
2cos 4 3 sin3
2
(
#
!"
$
#
"
')
(
#
" !"
#
3
%! & "%!
* ( ( '&)
EE 341 Exam # 2
(October 25, 2006)
Name:
Score:
1 (25 points):
2 (25 points):
3 (25 points):
4 (25 points):
Total (100 points):
This is a closed book/notes exam.
You may use a calculator and one 8.500 1100 sheet of formulae.
Maximum time allowed is 50
EE341: Exam 1 Solution
1.
(25 points)
a. Determine whether the continuous-time signal
3 cos 4
2 sin 6
periodic. If yes, find its time period ; otherwise, state why it is not periodic.
Solution:
is a sum of two sinusoids with time periods
3
is periodic wit
EE 341 Exam 2 Solution
1. (45 points) A signal
has spectrum
. Find the spectrum of the
following signals. It is not necessary to simplify the final expression for the spectrum - just
make sure you have applied the property(ies) completely and correctly.
a
EE 341 HW # 12
(due April 26, 2010)
1. A coin toss produces one of two possible outcomes: cfw_Head, Tail. Let p denote the probability of
getting a Head.
(a) Write an expression for the entropy as a function of p.
(b) Analytically find the maximum possibl
Spring 2016
EE 341
Lab 1
Lab 1: Elementary Sound Synthesis
1. Purpose
The purpose of this lab is to: i) implement simple synthesis methods, and ii)
develop intuitions for the audio impact of different signal transformations.
2. Background
Sound effects, m
EE 341: Discrete-Time Linear Systems
Lab 5: Sound/Image Synthesis
Due Date: Saturday, 4 June 2016, noon
In this lab, you will synthesize a sound or an image digitally, from scratch or by modifying
and combining existing signals, using at least three of th
EE341, Spring 2016
Problem Set #6
Due: 20 May 16
1. Two bandpass filters have impulse responses h1 [n] = (0.4)n cos(n/3)u[n] and h2 [n] = (0.8)n cos(3n/4)u[n].
Without computing the DTFT, find the center frequency of each, and determine which one has the
EE341, Spring 2016
Problem Set #2
Due: 15 April 16
1. Pick three of the systems below, and determine whether each system is or is not: memoryless,
time-invariant, linear, causal, stable and invertible. Provide formal proofs for each answer:
a) y[n] = x[n
EE341, Spring 2016
Problem Set #5
Due: 13 May 16
1. Use the analysis equation to derive the Fourier transform of
x[n] = (0.6)n sin(
3n
)u[n].
5
Sketch the magnitude |X(ej )| over the [, ] range. (You can use Matlab or just do a
rough sketch leveraging the
Spring 2016
EE 341
Lab 0
Lab 0: Working with Discrete-Time Signals
1. Purpose
The purpose of this lab is to: i) learn how to read, display, play and write sound
files, and ii) choose appropriate signal representations for a specified sampling
frequency. I
EE341, Spring 2016
Problem Set #3
Due: 22 April 16
1. Find and sketch the convolution of x[n] = u[n]u[n5] and h[n] = u[n+3]2u[n]+u[n5],.
2. Consider y[n] = x[n] h[n] where x[n] = u[n + 3] u[n 3] and h[n] = 3n u[2 n].
(a) Find the values of n such that y[n
EE341, Spring 2016
Problem Set #4
Due: 29 April 16
1. Each system on the left side is a causal LTI system. Indicate which of the four system
descriptions on the right side goes with each system on the left side. (H(z) is the transfer
function, h[n] is the
EE341, Spring 2016
Problem Set #7
Due: 27 May 16
This assignment has two parts (each worth 2 points), which should be turned in separately.
Problem Set:
1. The input to a DT system is x[n] = 1 + cos(2n/5). Which of the following 5-point DFTs could
corresp
EE 341
Lab 4: Using the FFT for Frequency Analysis and Sound Transformations
In this lab, we will learn how to do frequency analysis of a discrete-time signal on a
computer, and investigate the effect of different frequency-domain transformations.
When us
EE 341
Lab 3: Digital Filtering
In this lab, we will consider different types of digital filters and look at their characterization
in time and frequency. This will give you some insight into how digital filters are
implemented and designed and into the p
Useful Summations
Geometnc progressions [for |a|<1. finite N
SCI 1 CD EN
ir_ k_
;a_la ;G_1a
Taking the derivative of the first sum above: again for laidi, we get
For general a and finite N
N E a = '3 N3 i a = i]
as: N+1 [1:1 23*: NgN1+1E1=1
k 1! oth
Spring 2016
EE 341
Lab 2
Lab 2: Introduction to Image Processing
(Based on an early version by Dr. A. Miguel of Seattle University)
This lab will involve signal processing on images, which is a two-dimensional version of what
we have been doing with audio
EE341, Winter 2016
Problem Set #1
Due: 8 April 16
Note: OWN is short for the Oppenheim, Willsky and Nawab text.
For problems asking for a sketch, be sure to label heights and times. Show intermediate steps
of your work.
1. For the signal x[n] shown below,
l.
. An analog signal with a bandwidth of 4 kHz is sampled with asampling frequency f,
10 kHz (assume
ideal impulse sampling). A lowpase filter is to he dmigned for remastructing the sampled signal. What
are the paneth edge or, and stopband edge a), frequ