ECE 3250
HOMEWORK ASSIGNMENT I
Fall 2009
1. Find, read, and understand a proof of the Schrder-Bernstein Theorem, which states: o Let A and B be sets. If there exists an injective mapping f : A B and an injective mapping g : B A, then there exists a biject

ECE 3250
HOMEWORK ASSIGNMENT VI
Fall 2009
1. Show that every continuous-time system with input space X consisting of all decent signals and and system mapping of the form S (x) = Convh0 (x) +
N X k=1
dk Shifttk (x) ,
where h0 is decent and has nite durati

ECE 3250
HOMEWORK IV SOLUTIONS
Fall 2011
1. Note that h has nite duration.
(a) By Criterion 1, Dh = FZ because h has nite duration.
(b) S (x) = Convh (x), so for every n Z,
`
10
X
X
3 1 711
14 `
k
S (x)(n) =
h(k)x(n k) =
3(7 ) =
1 711 ; .
=
1 1/7
3
k=
k=0

ECE 3250
HOMEWORK VI SOLUTIONS
Fall 2009
1. By denition, a LTI system has a frequency response if and only if for all R the signal t ej t is an admissible input for the system. A signal x is an admissible input for a system such as the one in the problem

ECE 3250
HOMEWORK ASSIGNMENT IX
Fall 2009
1. Let x1 be a 1024-point signal and x2 be a 1025-point signal. Note that x1 x2 is a 2048-point signal. (a) Show that it takes a worst-case 512.5 211 multiplications to compute x1 x2 directly. (Suggestion: it take

ECE 3250
HOMEWORK I SOLUTIONS
Fall 2009
1. Most of the proofs Ive seen involve drawing pictures and dening bijective mappings between various subsets of A and various subsets of B (where A and B are the two sets youre trying to dene a bijection between).

ECE 3250
HOMEWORK ASSIGNMENT VIII
Fall 2009
1. Note that N = ej N is already in polar form.
k (a) Show that cfw_N : 0 k < N is the set of so-called N th roots of unity. I.e., this is the complete set of solutions to the polynomial equation z N 1 = 0. (b)

ECE 3250
HOMEWORK I SOLUTIONS
Fall 2011
1. Most of the proofs Ive seen involve drawing pictures and dening bijective mappings
between various subsets of A and various subsets of B (where A and B are the two sets
youre trying to dene a bijection between).

ECE 3250
HOMEWORK ASSIGNMENT V
Fall 2009
1. In each case, h is the impulse response of a LTI system and x is an input signal. Your task is to nd S (x). (a) h(t) = e3t u(t) for all t R and x(t) = e5t u(t) for all t R. (b) h(t) = e3t u(t) for all t R and x(

ECE 3250
HOMEWORK VIII SOLUTIONS
Fall 2013
1.
b be the frequency response of the composite system. You can see in at least
(a) Let H
b =H
b1 H
b2 .
two ways that H
jt
b
If you put t 7 ejt into the composite system, then t 7 H()e
comes
jt
b 1 ()ejt comes

ECE 3250
PRELIM I
Fall 2015
Problems 1 through 11 are worth 8 points each. Problem 12 is worth 12 points. Throughout, denotes convolution.
1. A is a set, B is a proper subset of A (i.e. B A and B 6= A), and f : A B is a
mapping. You can be certain that
(i

ECE 3250
HOMEWORK IV SOLUTIONS
Fall 2016
1. Note that h has finite duration.
(a) By Criterion 1, Dh = FZ because h has finite duration.
(b) S(x) = h x, so for every n Z,
`
10
X
X
7 1 311
21 `
k
S(x)(n) =
=
h(k)x(n k) =
1 311 ; .
7(3 ) =
1 1/3
2
k=
k=0
Not

ECE 3250
HOMEWORK III SOLUTIONS
Fall 2016
Note: Ive expressed the solutions to the problems involving calculating convolutions in
particular ways that match up well with some stuff well be doing later in the course. Your
answers might not look exactly lik

ECE 3250
HOMEWORK ASSIGNMENT VIII
Fall 2016
1.
b of the LTI system that has output
(a) Find the frequency response H
`
t 7 et 2e3t + e5t u(t)
when its input is t 7 8e5t u(t).
(b) Find the impulse response h of the ideal low-pass filter with cutoff frequen

ECE 3250
HOMEWORK ASSIGNMENT VII
Fall 2016
1. In Chapter 9 of the monograph I asserted in passing that l2 is an inner-product space.
Heres how to prove that
X
hx, yi =
x(k)y(k)
k=
2
defines an inner product on l . First show that show that for any complex

ECE 3250
HOMEWORK V SOLUTIONS
Fall 2016
1.
(a) By linearity, S(x) = S(x + d) = y + S(d). Now, h is an l1 signal and d is an l
signal, so, by Criterion 3,
kS(x) yk = kS(d)k = kh dk khk1 kdk .
Since kdk = and
khk1 =
X
|h(n)| =
n=0
1/5
= 3/20 ,
1 + 1/3
it fo

ECE 3250
HOMEWORK V SOLUTIONS
Fall 2013
1.
(a) By linearity, S(x) = S(x + d) = y + S(d). Now, h is an l1 signal and d is an l
signal, so, by Criterion 3,
kS(x) yk = kS(d)k = kh dk khk1 kdk .
Since kdk = and
khk1 =
X
|h(n)| =
n=0
1/7
= 5/42 ,
1 + 1/5
ar st

ECE 3250
HOMEWORK ASSIGNMENT IV
Fall 2013
1. A certain LTI system has impulse response h whose specification is
n
7
if 0 n < 11
h(n) =
0
otherwise.
(a) What is Dh , the set of all x FZ for which h x exists?
(b) Find the output S(x) of the system when the

ECE 3250
HOMEWORK VI SOLUTIONS
Fall 2016
1. In each case, S(x) = h x .
(a) S(x) = h x, so
Z
S(x)(t)
h(t )x( )d
=
Z
=
e3(t ) u(t )e5 u( )d
Z
=
e3(t ) u(t )e5 d
3t R t 8
e
e d if t 0
0
0
if t < 0
(1/8)(e5t e3t ) if t 0
0
if t < 0 .
0
=
=
The equality on

Syllabus Math 2930 Fall 2015
Notes:
Under Reading and Problems, 1.1 means section 1.1 in the textbook.
IDE = Interactive Differential Equations, a collection of applets for solving and visualizing differential
equations, with labs and homework problems

ECE 3250
HOMEWORK ASSIGNMENT VIII
Fall 2013
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
b 1 and H
b 2 be the frequency responses of ideal low-pass and band-pass filters.
1. Let H
Specifically, let
1 if | 3700
b 1 () =
H
0 otherwise
and
1 if

ECE 3250
HOMEWORK ASSIGNMENT IX
Fall 2013
1. Suppose xc (t) is a continuous-time signal and we find that
(
n1
n odd
3(1) 2
xc (n ) =
7
0
n even.
I.e., we sample xc (t) every T =
7
seconds and get the indicated results.
ar stu
ed d
vi y re
aC s
o
ou urc
rs

ECE 3250
HOMEWORK ASSIGNMENT II
Fall 2016
1. If cfw_cn is a convergent sequence of real numbers, does there necessarily exist R > 0
such that |cn | R for every n N? Equivalently, is cfw_cn : n N necessarily a bounded
set of real numbers? Explain why or w

ECE 3250
HOMEWORK ASSIGNMENT II
Fall 2016
1. If cfw_cn is a convergent sequence of real numbers, does there necessarily exist R > 0
such that |cn | R for every n 2 N? Equivalently, is cfw_cn : n 2 N necessarily a bounded
set of real numbers? Explain why

Contents
Chapter 1. Numbers
Sets, mappings, cardinality, and the natural numbers
The integers and rational numbers
Convergent sequences, Cauchy sequences, and the real numbers
The complex numbers
Decimal expansions of real numbers
Cantors diagonal argumen

ECE 3250
HOMEWORK ASSIGNMENT IV
Fall 2015
1. A certain LTI system has impulse response h whose specification is
n
3
if 0 n < 11
h(n) =
0
otherwise.
(a) What is Dh , the set of all x FZ for which Convh (x) exists?
(b) Find the output S(x) of the system wh

ECE 3250
HOMEWORK ASSIGNMENT III
Fall 2016
For many of the problems, you might find the following geometric-series identities useful.
In the equations, is a real or complex number.
(
M
X
M + 1 if = 1
m
=
1 M +1
if 6= 1 .
1
m=0
Furthermore,
X
m =
m=0
1
1

ECE 3250
HOMEWORK I SOLUTIONS
Fall 2016
1. Most of the proofs Ive seen involve drawing pictures and defining bijective mappings
between various subsets of A and various subsets of B (where A and B are the two sets
youre trying to define a bijection betwee

ECE 3250
HOMEWORK ASSIGNMENT V
Fall 2016
1. A certain discrete-time LTI system with system mapping S has impulse response h
with specification
(3)n
h(n) =
u(n) for all n Z .
5
(a) Suppose that x is a bounded signal, and that S(x) = y. Suppose d is another