Defining Signals and Systems
Last time we
• Introduced mathematical notation to help define sets
• Learned the names of commonly used sets (Reals, etc.)
• Created multi-dimensional sets using set product
• Used notation to define domain & range for common signals
• Saw multiple ways to define a particular set
Today we will
• List common ways to define functions (signals or systems)
• Touch on declarative vs. imperative definitions
• Talk about domain and range of systems
• Consider several common ways of defining systems
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Defining Functions
A set is a collection of elements.
You can define a set by listing the elements in curly brackets:
EE20Profs = {Ross, Lee, Varaiya, ElGhaoui, Henzinger}
Sometimes there are too many elements to list individually.
Example: Set of all real numbers between 0 and 1
You can define an interval using square & round parentheses.
A square bracket includes the endpoint, round excludes.
(0,1)
all real numbers strictly between 0 and 1
[5, ∞)
all real numbers greater than or equal to 5
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Famous Sets
Some sets that we will use routinely include:
Reals
Reals+
Integers
Integers+
Naturals
Naturals0
Bools
Char
Complex
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Set of all real numbers 
Set of all nonnegative real numbers +
Set of all integers 
Set of all nonnegative integers +
Set of all positive integers {1, 2, …}
Set of all nonnegative integers (Integers+) 
{true, false}
set of all alphanumeric characters
set of all complex numbers 
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Famous Symbols
Set inclusion:
belong to”

“belongs to”
1 [0,1]
Quantifiers:
exists”
"
Logical operators:
“not”
 “and”
Logical relations:
 “implies”
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
“does not
1 [0,1)
$ “there
“for all”

“or”

 “is equivalent to”
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Defining Sets Using Predicates
We can define a new set NS, a subset of some set S, in the
following way:
We say that some general element x S is also a member of
NS if and only if the statement Pred(x) is true for that x.
Here, x is called a variable and Pred(x) is called a predicate.
Using mathematical notation, we define NS:
NS = {x S | Pred(x) }
Example: Write a definition for the set of even numbers, Evens.
Evens = {x Integers | x/2 Integers }
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Set Operations
Subset 
A  B means “all elements in A are also in B”
Example:
Naturals 
Integers
Union 
A B is the set of x for which x  A or x  B
Example:
(0,5)  [4,8] = (0,8]
Intersection 
A B is the set of x for which x  A and x B
Example:
(0,5)  [4,8] = [4,5)
Subtraction \
A \ B is the set of x for which x  A and x  B
Example:
(0,5) \ [4,8] = (0,4)
Empty Set 
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Practice Using Mathematical Notation
Define the following sets using mathematical notation:
A B =
{x  A | x  A  x  B}
Rationals =
(rational numbers,
)
{x  Reals | $ p  Integers, q  Naturals with x = p/q}
Integers16 =
(integers representable by 16 bits)
{x  Integers | x ≥ -32768  x ≤ 32767}
or
Integers  [-32768, 32767]
or
{-32768, -32767, … , 32767}
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Product Sets
Sets we commonly refer to as “multi-dimensional” can often be
expressed as a Cartesian product of two or more sets.
A X B = {(a,b) | a  A  b  B}
The product set A X B consists of all ordered pairs (a,b) with a
being an element in set A and b being an element in set B.
Example: Define the 1st quadrant of the real plane, Reals+2
Reals+2 = Reals+ X Reals+
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About Product Sets
A X B = {(a,b) | a  A  b  B}
B X A = {(b,a) | a  A  b  B}
A X A = {(a1, a2) | a1  A  a2  A}
A X A = A2
A X A X A X … X A = An
n
A1 X A2 X A3 X … X An =  Ai
i=1
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Product Set Examples
Set of pixels on an old VGA monitor (640x480), VGAScreen
VGAScreen = {1, 2, … , 640} X {1, 2, …, 480}
or
[1, 640] X [1, 480] Integers2
Set of seats in this classroom, 145DwinelleSeats
145DwinelleSeats =
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Signals Using Product Sets
Prof. Ross’s computer desktop image is a 1024 x 768 pixel, 24-bit
color image of Brett Favre. The image may be viewed as a signal.
This means that each picture element (pixel) in the grid has its
color described by a 24-bit (3 byte) number.
What is the domain of this signal? {1, 2, … 1024} X {1, 2, … 768}
What is the range? {1, 2, 3, …, 224} = {1, 2, 3, …, 16777216}
The 24 bits can represent color as follows: the first 8 bits
represent the red intensity, the next 8 bits represent the green
intensity, and the last 8 bits represent blue intensity.
What is another way to represent the range?
{1, 2, … , 28} X {1, 2, … , 28} X {1, 2, … , 28} = {1, 2, … , 28}3
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Systems Using Products of Signal Sets
Recall that systems map signals to new signals.
The domain and range of a system are sets of signals.
Example: Consider the system FreezeFrame. It takes a video
composed of a sequence of 1024 x 768 pixel, 24-bit color
images as input. It takes another input, DesiredFrame, which
is an integer between 1 and MaxFrames. DesiredFrame
changes over time.
The system “freezes” the video on the frame DesiredFrame, and
this image is the system output. The output changes over time
as DesiredFrame changes.
What is the domain of this system?
What is the range of this system?
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