To prove the above, we may use the fact that the space of polynomials with nite order is dense in the space of
continuous functions on a compact set under the sup norm. One example is the Bernstein po
Substitution leads to:
A
(f )ei2f t df
lim
A
f =A
A
( )ei2f d ei2f t df
= lim
A
f =A
A
= lim
ei2f ei2f t d df
( )
A
f =A
A
= lim
ei2f ( t) df d
( )
A
(4.3)
f =A
A
1
By Lemma 4.5.1, the fact that f =A
Chapter 1
Banach and Hilbert Signal Spaces and
Representation of Signals
In this chapter, some of the discussions we had in class are presented in a compactied form, with some additional
topics includ
Let X be a normed space of signals and let us dene the dual space of X as the set of linear and continuous
functions on X to R or C, and let us denote this space by X . This space is a linear space, u
Theorem 2.2.1 For any singular distribution, there exists a sequence of regular distributions which converges
to the singular distribution.
The proof of this argument follows from the following. Let f
and call this value the frequency response of the system for frequency f , whenever it exists.
A similar discussion applies for a discrete-time system.
Let u l (Z; R). If u(n) = ei2f n , then
y(n) =
h
Queens University
Mathematics and Engineering and Mathematics and Statistics
MTHE/MATH 335
Mathematics of Engineering Systems
Supplemental Course Notes - Winter 2013
Serdar Y ksel
u
March 18, 2013
Thi
MTHE/MATH 335
Winter 2013
Sample Midterm II
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [40 Points]
Let x, y f (Z; R); that is, x, y map Z to
MTHE/MATH 335
Winter 2013
Sample Midterm I
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [40 Points]
Let X be a Hilbert space and x, y X. Prove
MTHE/MATH 335
Winter 2013
Sample Midterm III
There are four questions.
Be as neat as possible, clearly state your results
Student Number:
Problem 1 [30 Points]
Consider a discrete-time system describe