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
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
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,.
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
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
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
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
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.
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
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
PROBLEM 1 (15 pts.)
(i) (4 pts.) What do the equations
|z| = |z 6 8j|
|z| = 5
where z is a variable point, represent on the complex plane? Sketch the corresponding lines.
(ii) (3 pts.) Do the two lines in (i) intersect, and if so, at which point(s)?
Par
PROBLEM 1 (15 pts.)
(i) (6 pts.) Sketch the curve on the complex plane given by
|z 4 + 2j| = 5
Find the maximum values of
here.)
(ii) (9 pts.) Let
ecfw_z and
mcfw_z as z ranges over this curve. (No calculus is needed
x(t) = A cos(t + ) + 3 2 sin(t + /4) ,
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
PROBLEM 1 (15 pts.)
(i) (3 pts.) Sketch the curve |z 2 2j| = 1.
(ii) (5 pts.) Verify that z(t) = 2 + 2j + ej2t lies on the curve of part (i). Hence determine all positive
values of t for which |z(t)| equals its minimum value.
Part (iii) is unrelated to pa
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
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 8 Due 4/23
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
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 4 Due 2/28/2013
Problem 1: (a) Find the Fourier transform of the impulse train shown in the figure below.
(b) Find the Fourier transforms of each of the following signals and sketch the magn
ENEE 222 Discrete Signal Processing
Spring 2013
Problem Set 9 - Due 05/09/2013
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
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 7 Due 4/11
cfw_
cfw_
Problem 1: Suppose that x (t ) , < t < X ( f ) , < f < is a Fourier transform pair, and
that
x (t )
2
dt < . Prove Parsevals Theorem
2
x (t ) dt =
2
X ( f ) df .
Prob
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 6 Due 4/2 Solutions
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 plot
PROBLEM 1 (15 pts.)
The two signals r(t) and y (t) shown below are periodic and have complex Fourier series expansions of
the form
Sk ejk0 t ,
s(t) =
k=
where 0 is the fundamental angular frequency. The curved segments are sinusoidal.
r(t)
1
.
-9
.
-6
-3
PROBLEM 1 (15 pts.)
Both signals shown below are periodic and have complex Fourier series expansions of the form
Sk ejk0 t ,
s(t) =
k=
where 0 is the fundamental angular frequency.
x(t)
1
.
.
-10
-8
-6
-4
-2
2
4
6
8
10
2
4
6
8
10
t
-1
y(t)
1
.
.
-10
-8
-6
ENEE 222 Discrete Time Signal Processing
Spring 2013
Problem Set 5 Due 2013-03-12
Problem 1: There is a signal on the course web site called signal1.txt. It is made up of several
sinusoids. Load the signal into MATLAB and plot it. See if you can determine