ENEE 222 0201/2
HOMEWORK ASSIGNMENT 1
Due Tue 02/05/13
Problem 1A
Consider the complex numbers
z1 = 4 5 j
and
z2 = 2 + 7j
(i) (2 pts.) Plot both numbers on the complex plane.
(ii) (2 pts.) Evaluate |zi | and zi for both values of i (i = 1, 2).
2
(iii) (6
ENEE222: HW Assignment #2
Due Tue 9/25/2012
1. Answer the following questions.
(a) Consider the frequency f0 = 420 Hz and the sampling rate fs = 600 samples/sec. List all the
aliases of f0 with respect to fs in the frequency range 0.0 to 3.0 kHz.
(b) If w
ENEE222: HW Assignment #1 Solution
Due Tue 9/18/2012
1. Consider the complex numbers
z 1 = 4 5j
and
z 2 = 2 + 7j
(a) Plot both numbers on the complex plane.
(b) Evaluate |zi | and zi for both values of i (i = 1, 2).
2
(c) Express each of z1 + 3z2 , z1 + 2
Solved Example 10.1
We have x(t) = s(t) + s(t), and thus
Xk = Sk + Sk
Since s(t) is real-valued, Sk = Sk and thus also
Xk = 2 ecfw_Sk
The spectrum is real and even, as is x(t).
For y (t), we have
y (t) = x(t) + x(t T0 /2)
and thus the two sets of Fourier
ENEE222 Homework #6 solution
Problem 1
_
(a) Since s is real-valued, S has circular conjugate symmetry, i.e.,
S=[5
1
-2+j*3
j*3
4-j
-8
4+j
-j*3
-2-j*3
1 ].'
(b)
Sum of s[n] (where n = 0:9) equals S[0].
Sum of (-1)^n*s[n] equals S[5].
Therefore s[0]+s[2]+s
hw3solution
Problem 1
_
A counteclockwise rotation by q = theta on the plane is a linear
transformation with matrix
[cos(q) -sin(q) ; sin(q) cos(q)]
If a = sqrt(r^2 + s^2), then
a*[r/a -s/a ; s/a r/a] = [r -s ; s r]
is rotation matrix scaled by a, i.e., i
hw2solution
Problem 1
_
(a) The aliases of f0 = 420 Hz with respect to fs = 600 samples/sec are at
(all frequencies in Hz):
f = 420 + k*600, k = 0,1,.; i.e., 420, 1020, 1620, 2220, 2820,.
and
f = -420 + k*600, k = 1,2,.; i.e., 180, 780, 1380, 1980, 2580,.
S 2.1
_
The sinusoid x[n] = cos(w*n) will satisfy
x[n] = x[n+16]
provided
w = (k/16)*2*pi
for some integer k. Depending on the value of k,
the fundamental period may equal 16 or a submultiple
of it - namely 1,2,4 or 8. It will equal 16
if and only if k an
ENEE222: HW Assignment #1
Due Tue 9/18/2012
1. Consider the complex numbers
z1 = 4 5 j
z2 = 2 + 7j
and
(a) Plot both numbers on the complex plane.
(b) Evaluate |zi | and zi for both values of i (i = 1, 2).
2
(c) Express each of z1 + 3z2 , z1 + 2z2 and z1
ENEE 222 0201/2
HOMEWORK ASSIGNMENT 6
Due Tue 03/26/13
Problem 6A
Let
V=
v(0) v(1) v(2) v(3) v(4) v(5) v(6) v(7)
be the matrix of Fourier sinusoids of length N = 8.
(i) (6 pts.) If
4
x = 4 1 2 1
1
2
1
T
,
use projections to represent x in the form x = Vc.
LAB ASSIGNMENT 4 (due 02/20/13)
_
DATA:
The vector chirp04 contains a sinusoid of unit amplitude generated by
chirp04 = cos(2*pi*v) ;
where the angle 2*pi*v is a NONLINEAR function of the sample index.
As a result, the signal frequency varies with time.
ENEE 222 Signals and Systems
Spring 2013 Test 2 4/10/2013 Solutions
Closed book, no calculators. All problems count the same 25 points
Problem 1: (a 10 points) Explain the concept of a decibel, and (b 15 points) its application in the
Bode plot of the tra
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 3 due 2013-02-19
Problem 1: Find the convolution y[ n ] =
h[n k ]x[ k ] of the discrete time signals h and x
k =
given by
n
1
h[ n ] = , n = 0,1, 2, .
2
n
1
x[ n ] = , n = 0,1, 2, .
2
ENEE 222 Signals and Systems
Spring 2013 Test 1 2013-02-27 - Solutions
Problem 1: (33 points) (a 20 points) Show that the Fourier transform of the convolution of two
time signals is the product of their Fourier transforms. (b 13 points) Compute the Fourie
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 1 due 2013-01-31 (at end of the lab session)
Problem 1: Start MATLAB both from the university computer system and remotely using the
Virtual Computer Lab website: virtlab.eng.umd.edu. For th
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 2 Due 2013-02-10
Problem 1: (a) Write a MATLAB program to compute the sum S N =
N
1
m
m =1
2
. Prove that the
sum converges as N . (b) Write a MATLAB program to show that the sum QN =
N
1
m
The graphs below show one period of the sinusoid cos(t+) starting
at t = 0. The dotted lines are at levels 1/2 and -1/2.
Which graph corresponds to:
=0:
0 < < /3 :
/3 < < /2 :
= /2 :
/2 < < 2/3 :
2/3 < < :
A
=:
< < 2/3 :
2/3 < < /2 :
= /2 :
/2 < < /3 :
ENEE 222 Discrete Signal Processing
Fall 2013
Problem Set 10 - Due 12/12/2013
1. Write a MATLAB program to directly implement the Discrete Fourier Transform and its
inverse defined by
N 1
xn = xk e j 2 kn/ N , DFT
k=0
xk =
1 N 1 j 2 kn/ N
, IDFT
x e
N n=
PROBLEM 1 (15 pts.)
(i) (4 pts.) What do the equations
|z| = |z + 2 j 2|
|z| = 1
where z is a variable point, represent on the complex plane?
(ii) (4 pts.) Sketch the two lines/curves represented by the two equations in part (i). Show any
points of tang
ENEE 222 Signals and Systems
Spring 2012 Test 2 4/11/2012 Solutions
Closed book, no calculators. All problems count the same 25 points
cfw_
Problem 1: (a) Calculate the DFT of the signal x[n] = ,n = 0,1,., N 1 . (b) Find the
n
1
k=0
0 k = 1,2,., N 1
si
ENEE 222 Signals and Systems
Fall 2011 Final Exam 2011-12-19
All problems count the same
Problem 1: Suppose x(t) is the causal signal shown in the figure. (The sequence continues to
t.)
Let X(f) be the Fourier transform of x(t). Find
2
X ( f ) df . Draw a
ENEE 222 Signals and Systems
Fall 2011 Sample Test 1 2011-09-23 - Solutions
Problem 1. Prove that the following statements are equivalent: (1) The equation Ax=b has a solution for any vector b.
(2) The equation Av=0 has the unique solution x=0. Give a car
ENEE 222 Discrete Time Signal Processing
Fall 2013
Problem Set 8 Due 11/19 - Solutions
Problem 1: Consider the function
x ( t ) = cos(t )e 3 t , < t <
(a) Compute and plot the power spectral density of x as a function of frequency. Recall the power
spect
CMSC 351:Summer 2013
Clyde Kruskal
Information CMSC351
Asymptotic Notations.
(g(n) = cfw_f (n): there exist positive constants c1 , c2 , and n0 such that 0 c1 g(n) f (n) c2 g(n) for
all n n0 .
O(g(n) = cfw_f (n): there exist positive constants c and n0 su
ENEE 222 Discrete Time Signal Processing
Fall 2013
Problem Set 7 Due 11/5
Problem 1: The periodic impulse train and its Fourier series are
s(t ) =
(t nT ) =
n =
1 j 2 nt /T
e
T n =
Verify this by computing the finite sum (in MATLAB) and plotting
sN (t )
Fall 2013
Clyde Kruskal and Vibha Sazawal
CMSC 351: Practice Questions for Final Exam
These are practice problems for the upcoming final exam. You will be given a sheet of notes for
the exam. Also, go over your homework assignments. Warning: This does not
4.2 DTFT Examples
4.2 DTFT Examples
Example 4.1 Find the DTFT of a unit-sample x[ n] = [ n].
DTFT of a unit-sample x[ n] = [ n].
jwn Processing
jwn
ENEE
222
Discrete
Time
Signal
X
(
w
)
=
x
[
n
]
e
=
= e j0 =1
[ n]e
x[ n]e jwn = [ n]e jwn = e j 0 = 1
(
ENEE 222 Discrete Time Signal Processing
Fall 2013
Problem Set 8 Due 11/19
Problem 1: Consider the function
x ( t ) = cos(t )e 3 t , < t <
(a) Compute and plot the power spectral density of x as a function of frequency. Recall the power
spectral density
4.2 DTFT Examples
Example 4.1 Find the DTFT of a unit-sample x[ n] = [ n].
n =
n =
jwn
ENEE 222 Discrete
X ( w) =Time
]e jwn =Processing
= e j0 =1
x[ nSignal
[ n]e
Fall 2013
Similarly, the DTFT of a generic unit-sample is given by:
Problem Set 9 Due