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

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 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

ENEE 241 02
HOMEWORK ASSIGNMENT 7
Due Thu 03/05
Solve by hand without using calculator matrix functions. Show all intermediate steps.
Let
2
1
A=
1
3
4 2
2
1
4 7
3
1 3
8
1 6
(i) (8 pts.) Solve Ax = b, where b is the standard unit vector [ 1 0 0 0 ]T .
(

ENEE 241 02*
HOMEWORK ASSIGNMENT 6
Due Thu 02/26
Solve by hand without using your calculator. Show all intermediate steps.
Let
1
0
0
3
1
0
L=
3 3
1
5
3 3
0
0
0
1
(i) (4 pts.) Solve Lx = b for arbitrary x and b. Display L1 .
(ii) (5 pts.) Express the mat

ENEE 241 02*
Problem 5A
(i) (10 pts.) Let
HOMEWORK ASSIGNMENT 5
2
A 1 = u ,
0
1
A 2 = v
0
Due Tue 02/24
and
1
A 1 = w ,
1
where u, v and w are m 1 vectors.
Determine the dimensions of the matrix A and express each of its columns in terms of u, v and w.
Fo

ENEE 241 02*
HOMEWORK ASSIGNMENT 4
Due Thu 02/19
(i) (4 pts.) Consider the frequency f0 = 180 Hz and the sampling rate fs = 450 samples/sec. List
all the aliases of f0 with respect to fs in the frequency range 0.0 to 1.8 kHz.
(ii) (2 pts.) If we sample, a

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*
HOMEWORK ASSIGNMENT 3
Due Tue 02/17
Problem 3A
Consider the discrete-time sinusoids
x[n] = cos
7n
0.5
12
and
y [n] = cos (1.83n 0.5)
(i) (3 pts.) Which of the two sinusoids is periodic, and what is its fundamental period?
(ii) (3 pts.) Use M

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

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

S 14.1 (P 3.11)
_
x = [ a b c d e f ].'
X = [ A B C D E F ].'
For x_1:
x_1[n] = (-1)^n * x[n] = v^(3*n) * x[n]
By Property 6 (multiplication by Fourier sinusoid in
time domain), X is circularly shifted by +3 indices:
X_1 = P^(-3)*X = [ D E F A B C ].'

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

S 11.1 (P 3.4)
_
The four columns of V are orthogonal, with squared norm equal to 4.
Any s can be expressed as
s = c0*v0 + c1*v1 + c2*v2 + c3*v3
where
c# = (v#)'*s/4
Thus
4*c0 = [1 1 1 1]*[1 ; 4 ; -2 ; 5] = 8
4*c1 = [1 -j -1 j]*[1 ; 4 ; -2 ; 5] = 3 +

S 10.1
_
i) V = [ v1 v2 v3 ]
v1'*v2 = 3(a+bj) - 18j + 6j(a-jb)
v1'*v2 = 0 for orthogonality, so
Re( v1'*v2 ) = 3a + 6b = 0 -> a = -2b
Im( v1'*v2 ) = 3b - 18 + 6a = 0 -> b = -2, a = 4
The same terms are obtained in v2'*v3 and v3'*v1, so these inner
pro

S 8.1
_
A square matrix is tested for singularity in the
forward phase of Gaussian elimination. If a zero element
is encountered on the leading diagonal, then row interchanges
are used to replace that element by a nonzero value (from another
UNUSED row).