Signals and Systems Note1

8/7/2019 Signals and Systems Note1

1/48

6.003:

Signals and

Systems

Signals and Systems

February 2,

2010


2/48

6.003:

Signals and

Systems

Todays handouts:

Lecturer: Instructors:

Website:

Text:

Single package

containing

Slides for Lecture 1 Subject Information & Calendar Denny

Freeman

Peter Hagelstein

Rahul

Sarpeshkar

mit.edu/6.003

Signals and

Systems

Oppenheim

and

Willsky


3/48

6.003:

Homework

Doing the

homework

is

essential

for

understanding

the

content.

where subject matter is/isnt learned equivalent to practice in sports or music

Weekly Homework

Assignments

Conventional Homework Problems plus Engineering Design Problems (Python/Matlab)

Open Office

Hours

!

Stata

Basement

Mondays and Tuesdays, afternoons and early evenings


4/48

6.003:

Signals and

Systems

Collaboration Policy

Discussion of concepts in homework is encouraged Sharing of homework or code is not permitted and will be re

ported to

the

COD

Firm Deadlines Homework must be submitted in recitation on due date Each student can submit one late homework assignment without

penalty.

Grades on other late assignments will be multiplied by 0.5 (unless excused

by

an

Instructor,

Dean,

or

Medical

Official).


5/48

spring spring spring spring spring

nals nals nals nals nals

6.003 At-A-Glance

Tuesday

Wednesday

Thursday

Friday

Feb

2

L1:

Signals and

Systems

R1:

Continuous &

Discrete

Systems

L2:

Discrete-Time

Systems

R2:

Difference

Equations

Feb 9

L3:

Feedback,

Cycles,

and

Modes

HW1

due

R3:

Feedback,

Cycles, and

Modes

L4:

CT Operator

Representations

R4:

CT

Systems

Feb 16

Presidents

Day:

Monday

Schedule

HW2

due

R5:

CT Operator

Representations

L5:

Second-Order

Systems

R6:

Second-Order

Systems

Feb 23

L6:

Laplace

and

Z

Transforms

HW3

due

R7:

Laplace and

Z

Transforms

L7:

Transform

Properties

R8:

Transform

Properties

Mar 2

L8: Convolution; Impulse

Response

EX4

Exam 1 no

recitation

L9: Frequency Response

R9: Convolution and

Freq.

Resp.

Mar 9

L10:

Bode

Diagrams

HW5

due

R10:

Bode

Diagrams

L11:

DT Feedback

and

Control

R11:

Feedback and

Control

Mar 16

L12:

CT

Feedback

and

Control

HW6

due

R12:

CT Feedback

and

Control

L13:

CT Feedback

and

Control

R13:

CT Feedback

and

Control

Mar 23

Spring

Week

Mar 30

L14:

CT

Fourier

Series

HW7

R14:

CT

Fourier

Series

L15:

CT Fourier

Series

R15:

CT Fourier

Series

Apr 6

L16:

CT

Fourier

Transform

EX8

due

Exam 2

no

recitation

L17:

CT Fourier

Transform

R16:

CT Fourier

Transform

Apr 13

L18:

DT

Fourier

Transform

HW9

due

R17:

DT Fourier

Transform

L19:

DT Fourier

Transform

R18:

DT Fourier

Transform

Apr 20

Patriots Day Vacation

HW10

R19: Fourier Transforms

L20: Fourier Relations

R20: Fourier Relations

Apr 27

L21:

Sampling

EX11

due

Exam 3

no

recitation

L22:

Sampling

R21:

Sampling

May 4

L23:

Modulation

HW12

due

R22:

Modulation

L24:

Modulation

R23:

Modulation

May 11

L25:

Applications

of 6.003 EX13 R24: Review

Breakfast

with

Staff Study Period

May 18

Final

Examination

Period


6/48

6.003:

Signals and

Systems

Weekly meetings

with

class

representatives

help staff understand student perspective learn about teaching One

representative

from

each

section

(4

total)

Tentatively

meet

on

Thursday

afternoon


7/48

The Signals

and

Systems

Abstraction

Describe

a

system

(physical,

mathematical,

or

computational)

by the

way

it

transforms

an

input

signal

into

an

output

signal .

systemsignal

in

signal out


8/48

Example:

Mass and

Spring


9/48

Example:

Mass and

Spring

x(t)

y(t)

mass & spring system

x(t)

y(t)


10/48

Example:

Mass and

Spring

x(t)

y(t)

x(t)

y(t)

mass & spring system

t

t


11/48

r 0 (t )

t

Example:

Tanks

r 0 (t)

h1 (t) r 1 (t)

h2 (t)

r 2 (t)

r 0 (t)

tank system

r 2 (t)


12/48

Example:

Tanks

r 0 (t)

h1 (t) r 1 (t)

h2 (t)

r 2 (t) r 0 (t)

r 2 (t)

tank systemt t


13/48

Example:

Cell Phone

System

sound out

sound in

sound in

cell

phone system

sound out


14/48

Example:

Cell Phone

System

sound out

sound in

sound in sound out

cell

phone system

t t


15/48

Signals and

Systems:

Widely

Applicable

The

Signals

and

Systems

approach

has

broad

application:

electrical,

mechanical, optical, acoustic, biological, nancial, ...

x(t)

y(t)

t

mass &

spring system

t

r 0 (t)

r 1 (t)

r 2 (t)

h1 (t)

h2 (t)

tank system

r 0 (t)

r 2 (t)

t

t

cell phone

system

sound in

sound

out

t

t


16/48

tower

E/M tower

sound E/M optic cell sound in phone

out

cellphone ber

focuses on

the

ow

of

information ,

abstracts

away

everything

else

Signals and

Systems:

Modular

The

representation

does

not

depend

upon

the

physical

substrate.

sound in

sound ou


17/48

Signals and

Systems:

Hierarchical

Representations

of

component

systems

are

easily

combined.

Example:

cascade of

component

systems

cell phone

tower tower cell phone sound in

E/M

optic ber

E/M sound out

Composite system

cell phone

system

sound

in

sound

out

Component and composite systems have the same form, and are analyzed

with

same

methods.


18/48

Signals and

Systems

Signals

are

mathematical

functions.

independent variable = time dependent variable = voltage, ow rate, sound pressure

mass &

spring system

x(t)

y(t)

t

t

r 0 (t) r2 (t)

tank systemt

t

sound in

sound

out

cell phone

system

t

t


19/48

Signals and

Systems

continuous

time

(CT)

and

discrete

time

(DT)

x(t) x[n]

nt 0 2 4 6 8 10 0 2 4 6 8 10

Many physical

systems

operate

in

continuous

time.

mass

and

spring

leaky tank Digital

computations

are

done

in

discrete

time.

state machines: given the current input and current state, what is the

next

output

and

next

state.


20/48

Signals and

Systems

Sampling:

converting

CT

signals

to

DT

x(t)

x[n] = x(nT

)

nt 0T 2T 4T 6T 8T 10T 0 2 4 6 8 10

T =

sampling

interval

Important for

computational

manipulation

of

physical

data.

digital representations of audio signals (e.g., MP3) digital representations of pictures (e.g., JPEG)


21/48

Signals and

Systems

Reconstruction:

converting

DT

signals

to

CT

zero-order hold

x[n]

x(t)

n

t 0 2 4 6 8 10

0 2T

4T

6T

8T

10T

T =

sampling

interval

commonly used

in

audio

output

devices

such

as

CD

players

Si l d S


22/48

Signals and

Systems

Reconstruction:

converting

DT

signals

to

CT

piecewise linear

x[n]

x(t)

n

t 0 2 4 6 8 10

0 2T

4T

6T

8T

10T

T =

sampling

interval

commonly used

in

rendering

images

Ch k Y lf


23/48

Check Yourself

Computer generated speech (by Robert Donovan)

t

f (t)

Listen to the following four manipulated signals: f 1 (t),

f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Ch k Y lf


24/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


25/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


26/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


27/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


28/48

Check Yourself

Computer generated speech (by Robert Donovaf (t)

n)

t


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


29/48

Check Yourself

Computer generated speech (by Robert Donovan) f (t)

t


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)


Check Yourself


30/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

f 1 (t) = f (2t) f 2 (t) = f (t) f 3 (t) = f (2t)

f 4 (t) = 2 f (t)

Check Yourself


31/48

Check Yourself


t

f (t)


f 2 (t),

f 3 (t),

f 4 (t).

How

many

of

the

following

relations

are

true?

2

f 1 (t) = f (2t) f 2 (t) = f (t) X f 3 (t) = f (2t) X

f 4 (t) = 2 f (t)

Check Yourself


32/48

y f (x,

y)

250 0 250

250

0

250

x

How many

images

match

the

expressions

beneath

them?

250

0

250

y

x

250

0

250

y

x

250

0

250

y

x

250

0

250

250

0

250

250

0

250 f 1 (x,

y)= f (2x,

y)

?

f 2 (x,

y)= f (2x250, y) ? f 3 (x, y)= f (x250, y) ?

Check Yourself


33/48

y y y y

250

0

250

250

0

250

250

0

250

250

0

250

250 0 250 x

250 0 250 x

250 0 250 x

250 0 250 f (x, y )

f 1 (x, y ) = f (2x, y)

?

f 2 (x, y ) =

f (2x250, y) ? f 3 (x, y ) = f (x250, y) ?

x = 0

f 1 (0, y) =

f (0, y)

x =

250

f 1 (250, y) = f (500, y) X x

= 0

f 2 (0, y) = f (250, y) x = 250 f 2 (250, y) = f (250, y) x

= 0

f 3 (0, y) = f (250, y) X x

=

250

f 3 (250, y) = f (500, y) X

x

Check Yourself


34/48

y f (x,

y)

250 0 250

250

0

250

x

How many

images

match

the

expressions

beneath

them?

250

0

250

250

0

250

y

x

f 1 (x, y)= f (2x,

y)

?

250

0

250

250

0

250

y

x

f 2 (x, y)= f (2x250, y) ?

250

0

250

250

0

250

y

x

f 3 (x, y)= f (x250, y) ?

The Signals and Systems Abstraction


35/48

Describe

a

system

(physical,

mathematical,

or

computational)

by

the way it transforms an input signal into an output signal .

systemsignal in

signal out

Example System: Leaky Tank


36/48

Formulate

a

mathematical

description

of

this

system.

r 0 (t)

h1 (t) r 1 (t)

What determines

the

leak

rate?

Check Yourself


37/48

The holes

in

each

of

the

following

tanks

have

equal

size.

Which

tank

has

the

largest

leak

rate

r 1 (t)?

3.

4.

1. 2.

Check Yourself


38/48

The holes

in

each

of

the

following

tanks

have

equal

size.

Which

tank

has

the

largest

leak

rate

r 1 (t)?

2

3.

4.

1. 2.



39/48

Formulate

a

mathematical

description

of

this

system.

r 0 (t)

r 1 (t)

h1 (t)

Assume linear

leaking:

r 1 (t)

h1 (t)

What determines the height h1 (t)?



40/48

Formulate

a

mathematical

description

of

this

system.

r 0 (t)

r 1 (t)

h1 (t)

Assume linear

leaking:

r 1 (t)

h1 (t)

Assume water is conserved: dh 1 (t)

r0 (t) r 1 (t)dt

Solve:

dr 1 (t)

r 0 (t) r 1 (t)dt

Check Yourself


41/48

What are

the

dimensions

of

constant

of

proportionality

C ?

dr1

(t) dt

= C r 0 (t) r 1 (t)

Check Yourself


42/48

What are

the

dimensions

of

constant

of

proportionality

C ?

inverse

time

(to

match

dimensions

of

dt )

dr 1 (t) dt

=

C

r 0 (t)

r 1 (t)

Analysis of the Leaky Tank


43/48

Call

the

constant

of

proportionality

1/

.

Then

is called

the

time

constant

of

the

system.

dr 1 (t)

r 0 (t)

r 1 (t)=

dt

Check Yourself


44/48

Which of

the

following

tanks

has

the

largest

time

constant

?

3.

4.

1. 2.

Check Yourself


45/48

Which of

the

following

tanks

has

the

largest

time

constant

?

4

3.

4.

1. 2.

Analysis of

the

Leaky

Tank


46/48

Call the

constant

of

proportionality

1/

.

Then is called the time constant of the system.

dr 1 (t)

r 0 (t)

r 1 (t)=

dt Assume

that

the

tank

is

initially

empty,

and

then

water

enters

at

a

constant

rate

r 0 (t) = 1 .

Determine

the

output

rate

r 1 (t).

r 1 (t)

time (seconds)

1 2 3

Explain the

shape

of

this

curve

mathematically.

Explain

the

shape

of

this

curve

physically.

Leaky Tanks

and

Capacitors


47/48

Although derived

for

a

leaky

tank,

this

sort

of

model

can

be

used

to

represent a variety of physical systems. Water

accumulates

in

a

leaky

tank.

r 0 (t)

h1 (t)

r1

(t) Charge accumulates

in

a

capacitor.

i i

io

+ vC

dv i i io dh = i i io analogous to r 0 r 1dt C dt

MIT OpenCourseWarehttp://ocw mit edu
http://ocw.mit.edu/http://ocw.mit.edu/


48/48

http://ocw.mit.edu

6.003 Signals and SystemsSpring 20 10

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
http://ocw.mit.edu/http://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/termshttp://ocw.mit.edu/

Signals and Systems Note1

Documents

Transcript of Signals and Systems Note1