Week 2 (Chapter 2)

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• Week 2 (Chapter 2)
• Signals are Functions
• Systems

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A signal is a function

Each function has four things 0 the
name (f, g, sin, cos, sound, ) 1
the domain ( a set ) 2 the range (
a set ) 3 the graph or assignment (
for every domain element, a range
element )
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Function definition
Let f Domain ? Range such that ? x ? Domain,
f ( x ) expression
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Air pressure
Signal
Time
Voice Time ? Pressure Doman Time 0,1 Range
Pressure Reals graph(Voice) red points ?
Time X Pressure
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• Signals are modeled as functions
• But there are many choices in selecting the
  function domain and range
• The good engineer selects the choice appropriate
  to the task

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Examples from Topic notes/Signals/Sound
Voice 0,1 ? Pressure IntegerVoice 0,1 ?
Integers16 ComputerVoice DiscreteTime ?
Integers16 in which DiscreteTime 0, 1/8000,
2/8000,
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Examples from Topic notes/Signals/Images
Image VerticalSpace X HorizontalSpace ?
Intensity VerticalSpaceX HorizontalSpace
a,b X c,d Intensity black, white
Image(row,col) ? Intensity
DigitalImage VerticalSpaceX HorizontalSpace
1,2, ,300X1,2,,200 ? Intensity 0,1, ,
255 Integers8
ColorDigitalImage VerticalSpaceX HorizontalSpace
1,2, ,300X1,2,,200 ? Intensity3 0,1,
, 255 x 0, , 255x0, ,255 Integers83
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Two very important definitions 1. function
composition ?
If f X ? Y, g Y ? Z , g ? f X ? Z
is defined by ? x ? X, ( g ? f ) ( x )
g ( f ( x ) ) ? Z
x
g(f(x))
f(x)
X
Y
Z
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2. Function or Signal Space
For domain X, and range Y, X ? Y
f domain ( f ) X ? range ( f ) Y
set of all functions whose domain X, and
range Y
Sounds Time ? Pressure BitSequences Nats
? 0,1
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Images VerticalSpace x HorizontalSpace ?
Intensity3 is the set of all color
images. Suppose VerticalSpace 1, , 300,
HorizontalSpace 1,, 200
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Since Domain(Varaiya) 1, , 300 x 1, ,
200, is finite, graph (Varaiya) can be given as
this table
(1,1) Varaiya (1,1)
(1,2) Varaiya (1,2)

(1,200) Varaiya (1,200)
(2,1) Varaiya (2,1)


(200,300) Varaiya (200,300)
in which Varaiya (i,j) is an element of Intensity3
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or as a 300 x 200 matrix whose (i, j) th
element is Varaiya (i,j) ? Intensity3
Varaiya(1,1) Varaiya (1,200)



Varaiya (i, j)



Varaiya (300,1) Varaiya (300,200)
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So Images, the space of all images, can be
represented as the space of all 300 x 200
matrices, whose entries are elements of
Intensity3
Images 1, , 300 x 1, , 200 ? Intensity3
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Video (t) ? Images this is the tth frame
Video (t)(i, j) ? Intensity3 this is the
(i,j)th pixel of the tth frame
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An alternative signal is AltVideo DiscreteTime
x VerticalSpace x HorizontalSpace ?
Intensity3, in which AltVideo (t, i, j) is the
(i,j)th pixel value of frame t
The two definitions are related by ? t,
i, j AltVideo(t, i, j) Video (t)(i,j)
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Space of signals
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A space of signals is of the form X ? Y We will
be specially interested in the case where the
domain X represents Time Examples AnalogSounds
0,1 ? Pressure DigitalSounds 0,1/8000,
? Integers16
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PositionSpeed of a vehicle is a signal
in PositionSpeeds 0,5 ? Reals2
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(No Transcript)
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Image processing system
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Lane changing
Shock wave
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Lane changing
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Shock wave
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Image processing system (IPS)

• IPS FreewayVideos ?

Time ? Vehicle x Lane x Position or
Time x Vehicle s ? Lane x Position
IST
output signal
input signal
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StereoSounds Time ? Pressure2
Alternatively, SteroSounds LeftSounds x
RightSounds where LeftSounds RightSounds
Time ? Pressure
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A record or log of the successive buttons that
are pressed in the Cory Hall elevator may look
like 1, 4, Open, Close, 3, 2, B, 5
We could model this as a signal Record Indices ?
B,1,2,3,4,5, Open, Close where Indices
0,1,2, , N
The space of all such signals is Records
Indices ? Events Events B, 1, , Close
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In an event sequence, the domain Indices
represents succession, rather than quantitative
time.
Other event sequences DoorRecord Indices ?
enter, leave is a log of persons entering or
leaving 10 Evans eg. DoorRecord (e,l,l,e,e, )
NumberRecord Indices ? Integers is a log of
number of people in 10Evans after
each enter,leave event eg. NumberRecrod
(6,5,4,5,6, )
DoorRecord ? Indices ? enter, leave
NumberRecord ? Indices ? Integers
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Systems
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A system S is a function InputSignals ?
OutputSignals with InputSignals D ?
R, OutputSignals D ? R
S
y S(x) ? OutputSignals
x ? InputSignals
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10Evans Ints ? enter, leave ? Ints ?
Ints
OutputSignals
InputSignals
y (6,5,4,5,6, )
u (e,l,l,e,e, )
10Evans
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initial number in room
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The system 10Evans can be specified as follows
y 10Evans (u), and for all n ? Ints , y(n)
is given by
where
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How do you give S?
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How do you give Q?
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S
u?Accels
v?Speeds
y?Positions
Q
y(0)
v(0)
SºQ Accels ? Positions 0,5 ? Reals
? 0,5 ? Reals
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Other examples TopicNotes/Systems/DTMF
Sounds
KeypadSequences
DTMF