ENEE 241 02
HOMEWORK ASSIGNMENT 3
Due Tue 02/22
Problem 3A
(i) (4 pts.) Consider the frequency f0 = 180 Hz and the sampling rate fs = 640 samples/sec. List
all the aliases of f0 with respect to fs in the frequency range 0.0 to 2.5 kHz.
(ii) (2 pts.) If we

ENEE 241 02*
HOMEWORK ASSIGNMENT 18
Due Tue 04/28
(i) (2 pts.) In the lecture notes, you will nd the Fourier series for the symmetric (even) rectangular
pulse train of unit height and duty factor . Write down both the complex and real (cosines-only)
form

S 13.1 (P 3.7)
_
X = [4 1+j 3-j z1 z2].'
(i) From the analysis equation (k=0), we have that
x[1] + x[2] + x[3] + x[4] + x[5] = X[0] = 4
(ii) Since x is real-valued, X has conjugate symmetry about
index k=5/2. Thus (with ' denoting complex conjugate, as

ENEE 241 02*
HOMEWORK ASSIGNMENT 11
Due Thu 03/26
Exam 2 (Tue 03/31) will cover Assignments 611.
Let
V=
v(0) v(1) v(2) v(3) v(4) v(5)
be the matrix of Fourier sinusoids of length N = 6.
T
6 2
6 2
6 2 , use projections to represent x in the form
(i) (7 pts

ENEE 241 02
HOMEWORK ASSIGNMENT 12
Due Thu 04/02
Solve by hand (no inner products are needed). You may want to verify your answers on MATLAB.
All vectors have length N = 12.
(i) (4 pts.) The time-domain vector
x(1) =
1
1
1
1
1
1
1
1
1
1
1
1
T
is a Fourier

ENEE 241 02*
HOMEWORK ASSIGNMENT 13
Due Tue 04/07
Problem 13A
The real-valued signal vector s has DFT
12
S=
z1
2j
z2
4j
z3
3 + 2j
z4
T
2 5j
(i) (2 pts.) What are the values of z1 , z2 , z3 and z4 ?
(ii) (3 pts.) Without using complex algebra (or MATLAB),

ENEE 241 02
HOMEWORK ASSIGNMENT 14
Due Thu 04/09
Let the signal
s=
abcdef
T
gh
of Homework Problem 13A have DFT
S=
3
2
4
1
2
1
5
3
T
Without computing any DFTs or inverse DFTs, determine (in numerical form) the DFTs of the
signal vectors s(1) through s(7)

ENEE 241 02*
HOMEWORK ASSIGNMENT 17
Due Tue 04/21
Problem 17A (Submit both plots.)
The data set data17A.txt contains fty noisy samples of a continuous-time signal which is a sum
of two sinusoids at frequencies f1 and f2 (Hz), i.e.,
s(t) = A1 cos(2f1 t + 1

ENEE 241 02*
READING ASSIGNMENT 1
Tue 02/03 Lecture
Topic: complex multiplication; complex exponentials
Textbook References: section 1.3
Key Points:
Two complex numbers can be multiplied by expressing each number in the form z = x + jy ,
then using distr

ENEE 241 02*
READING ASSIGNMENT 2
Thu 02/05 Lecture
Topic: nth root of a complex number; continuous-time sinusoids; phasors
Textbook References: sections 1.3 and 1.4
Key Points:
The equation z n = ej , where is a given angle, has n roots of the form z =

ENEE 241 02*
READING ASSIGNMENT 3
Tue 02/10 Lecture
Topic: discrete-time sinusoids; sampling of continuous-time sinusoids
Textbook References: sections 1.5, 1.6
Key Points:
The discrete time parameter n counts samples. The (angular) frequency parameter i

ENEE 241 02*
READING ASSIGNMENT 4
Thu 02/12 Lecture
Topic: aliasing; introduction to matrices and vectors
Textbook References: sections 1.6, 2.1, 2.2.1, 2.2.2
Key Points:
Two continuous-time sinusoids having dierent frequencies f and f (Hz) may, when sam

ENEE 241 02*
READING ASSIGNMENT 5
Tue 02/17 Lecture
Topic: matrix-vector product; matrix of a linear transformation; matrix-matrix product
Textbook References: sections 2.1, 2.2.1, 2.2.2
Key Points:
The matrix-vector product Ax, where A is a m n matrix a

ENEE 241 02*
READING ASSIGNMENT 6
Thu 02/19 Lecture
Topic: cascaded linear transformations; row and column selection; permutation matrices; matrix
transpose
Textbook References: sections 2.2.3, 2.3
Key Points:
AB represents two linear systems connected i

ENEE 241 01
READING ASSIGNMENT 7
Tue 02/24 Lecture
Topic: introduction to matrix inversion; Gaussian elimination
Textbook References: sections 2.4, 2.5, 2.6
Key Points:
A n n matrix A is nonsingular, i.e., it possesses an inverse A1 , if and only if the

ENEE 241 02*
READING ASSIGNMENT 9
Thu 03/05 Lecture
Topics: inner products, norms and angles; projection
Textbook References: sections 2.10.12.10.4
Key Points:
The norm, or length, of a real-valued vector a is given by
a = a, a
1/2
,
where , denotes inne

ENEE 241 02
HOMEWORK ASSIGNMENT 10
Consider the complex-valued matrix
V=
4
= 1 3j
1 + 3j
v(1) v(2) v(3)
4
1 + 3j
1 3j
Due Tue 03/24
1
2
2
(i) (3 pts.) Evaluate the norms of the three columns v(1) , v(2) and v(3) .
(ii) (4 pts.) Show that the three columns

ENEE 241 02
HOMEWORK ASSIGNMENT 9
Due Thu 03/12
Solve by hand without using calculator matrix functions. Show all intermediate steps.
Let
v(1) = [ 3 1 5 3 ]T ,
v(2) = [ 1 1 6 4 ]T
and
v(3) = [ 13 7 1 1 ]T
Consider the tetrahedron with vertices A0 , A1 , A

S 15.1 (P 3.14)
_
x = [4 3 2 1 0 1 2 3].'
X = [A B C D E F G H].'
(i) x is real and circularly symmetric, thus X
is also real and circularly symmetric. It follows
that
B=H, C=G, and D=F
(ii) y = [0 1 2 3 4 3 2 1].' = (P^4)*x
so
Y[k] = exp(-j*(2*pi/8)*

S 17B.1
_
Using 2*cos(theta) = exp(j*theta) + exp(-j*theta), we see that
s(t) can be expressed as the (unweighted) sum of eight complex
sinusoids of the form
exp(2*pi*f*t) ,
where f (Hz) takes the four positive values (in Hz)
23+25+30
23+25-30
-23+25+30
2

Solved Example 18
(i) 0 = /3, hence T = 6.
s(t) is a linear combination of a constant and two rectangular pulse trains whose Fourier series
coecients are given by
sin(k/3)
sin(2k/3)
and
k
k
The corresponding duty factors are 1/3 and 2/3.
(ii) We have, for

Solved Example 19.1
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

S 22.1 (P 4.3)
_
Divide by 2*pi to express the three frequencies in cycles per
sample:
3/28, 9/35 and 17/48
Since these are rational (i.e., integer fractions), the
signal is periodic. The period is the smallest integer
L such that each frequency can be

ENEE 241 02
HOMEWORK ASSIGNMENT 1
Due Thu 02/05
Consider the complex numbers
z1 = 3 + 4j
and
z2 = 1 + 5j
(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).
(iii) (6 pts.) Express each

ENEE 241 02*
HOMEWORK ASSIGNMENT 2
Due Tue 02/10
Please turn in the two problems separately, with your name and section number on each sheet (or
set of sheets).
Problem 2A
Do not use a calculator for this problem. Express your answers using square roots a

ENEE 241 02*
HOMEWORK ASSIGNMENT 3
Due Thu 02/12
3
2
1
0
1
2
3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Two periods of the sinusoid x(t) = A cos(t + ) are plotted above. The value of x(0) equals
3 sin(/5).
(i) (6 pts.) Determine A, and . Express as an exact rati