Week 2 (Chapter 2) 1.Signals are Functions 2.Systems

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 ) A signal is a function

Function definition Let f : Domain  Range such that  x  Domain, f ( x ) = expression

Voice: Time  Pressure Doman = Time =[0,1] Range = Pressure = Reals graph(Voice) = { red points}  Time X Pressure Time Air pressure Signal

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

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, …}

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, …,300}X{1,2,…,200}  Intensity = {0,1, …, 255} = Integers8 ColorDigitalImage: VerticalSpaceX HorizontalSpace = {1,2, …,300}X{1,2,…,200}  Intensity 3 = {0,1, …, 255} x {0, …, 255}x{0, …,255} = Integers8 3

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 X YZ f(x) g(f(x))

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}]

Images = [VerticalSpace x HorizontalSpace  Intensity 3 ] is the set of all color images. Suppose VerticalSpace = {1, …, 300}, HorizontalSpace = {1,…, 200}  1  i  300, 1  j  200 Varaiya(i, j) = (n R, n G, n B ), where (n R, n G, n B ) are the RGB values of the pixel (i, j) Consider a particular image Varaiya  Images

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 Intensity 3

or as a 300 x 200 matrix whose (i, j) th element is Varaiya (i,j)  Intensity 3 Varaiya(1,1)Varaiya (1,200) Varaiya (i, j) Varaiya (300,1)Varaiya (300,200)

So Images, the space of all images, can be represented as the space of all 300 x 200 matrices, whose entries are elements of Intensity 3 Images = [{1, …, 300} x {1, …, 200}  Intensity 3 ]

A video movie is a signal (30 frames/sec) VideoVideo: DiscreteTime  Images DiscreteTime = {0, 1/30, 2/30, …} Domain (Video) = DiscreteTime, Range(Video) = Images Video (t)  Images [this is the t th frame] Video (t)(i, j)  Intensity 3 [this is the (i,j) th pixel of the t th frame]

An alternative signal is AltVideo: DiscreteTime x VerticalSpace x HorizontalSpace  Intensity 3, 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)

Space of signals

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] [0,5] Position The position of a vehicle is a signal in Positions = [[0, 5]  Reals]

Position&Speed of a vehicle is a signal in Position&Speeds = [[0,5]  Reals 2 ] [0,5] Position&Speed

[0,5] Position [0,5] Speed Alternatively, Position&Speed = (Position, Speed) with Position  Positions = [[0,5]  Reals] Speed  Speeds = [[0,5]  Reals]

Image processing system

Lane changingShock wave

Lane changing

Shock wave

Image processing system (IPS) IPS: [FreewayVideos]  [Time  Vehicle # x Lane x Position] or [Time x Vehicle #s  Lane x Position] IST input signal output signal

StereoSounds = [Time  Pressure 2 ] Alternatively, SteroSounds = LeftSounds x RightSounds where LeftSounds = RightSounds = [Time  Pressure] t LeftSound(t) t RightSound(t)

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}

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 + ]

Systems

A system S is a function : InputSignals  OutputSignals with InputSignals = [D  R], OutputSignals = [D’  R’] S x  InputSignals y = S(x)  OutputSignals

10Evans: [Ints +  {enter, leave}]  [Ints +  Ints + ] 10Evans u = (e,l,l,e,e,… ) y = (6,5,4,5,6,… ) initial number in room 5 InputSignals OutputSignals

The system 10Evans can be specified as follows: y = 10Evans (u), and for all n  Ints +, y(n) is given by where

How do you ‘give’ S? S u  Speeds y=S(u)  Positions Speeds = [[0,5]  Reals], Positions = [[0,5]  Reals] y(0) initial position

How do you ‘give’ Q? Q u  Accels v = Q(u)  Speeds Accels = [[0,5]  Reals], Speeds = [[0,5]  Reals] v(0) initial speed

u  Accelsv  Speedsy  Positions Q v(0) S y(0) SºQ: Accels  Positions : [[0,5]  Reals]  [[0,5]  Reals]

Other examples: TopicNotes/Systems/DTMFTopicNotes/Systems/DTMF KeypadSequences Sounds DTMF