signals and systems6.003: signals and systems collaboration policy • discussion of concepts in...

6.003: Signals and Systems

Signals and Systems

September 10, 2009


Today’s handouts: Single package containing

• Subject Information & Calendar

• Slides from Lecture 1

• Homework Assignment 1

Lecturer: Denny Freeman ([email protected])

Instructors: Elfar Adalsteinsson ([email protected])

Marc Baldo ([email protected])

TAs: Jennifer Roberts ([email protected])

Dennis Wei ([email protected])

TBA

Website: mit.edu/6.003

Text: Signals and Systems – Oppenheim and Willsky

mailto:[email protected]





http://mit.edu/6.003

6.003: Homework

Doing the homework is essential for understanding the content.

• where subject matter is/isn’t 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 (32-044)

• Mondays and Tuesdays, afternoons and early evenings


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

• Late homework will NOT be accepted unless excused by the

staff, a Dean, or Physician (except for single “extension”)

Homework Extension Policy

• Every student gets one extension

• Can be used for any weekly homework assignment and for any

reason

• Simply ask your TA for an extension before 11:59 pm on the day

preceding the due date (cannot be rescinded)

6.003 At-A-Glance

Sep 8 Registration DayR1: Continuous &

Discrete Systems

L1: Signals and

Systems

R2: Difference

Equations

Sep 15L2: Discrete-Time

Systems

HW1

due

R3: Discrete-Time

Systems

L3: Feedback,

Cycles, and Modes

R4: Feedback,

Cycles, and Modes

Sep 22L4: Feedback and

Position Control

HW2

due

R5: Feedback and

Position Control

L5: Feedback

Control Schemes

R6: Feedback

Control Schemes

Sep 29L6: CT Systems,

Differential Eqs.

HW3

due

R7: CT Systems,

Differential Eqs.

L7: Laplace and Z

Transforms

R8: Laplace and Z

Transforms

Oct 6L8: CT Operator

RepresentationsEX4

Exam 1

no recitation

L9: Second-Order

Systems

R9: CT Operator

Representations

Oct 13Columbus Day:

Monday Schedule

HW5

due

R10: Second-Order

Systems

L10: Convolution;

Impulse Response

R11: Convolution;

Impulse Response

Oct 20L11: Frequency

Response

HW6

due

R12: Frequency

Response

L12: Bode

Diagrams

R13: Bode

Diagrams

Oct 27L13: CT Feedback

and ControlEX7

Exam 2

no recitation

L14: CT Feedback

and Control

R14: CT Feedback

and Control

Nov 3L15: CT Fourier

Series

HW8

due

R15: CT Fourier

Series

L16: CT Fourier

Series

R16: CT Fourier

Series

Nov 10L17: CT Fourier

Transfm; Filtering

HW9

dueVeteran’s Day

L18: CT Fourier

Transforms

R17: CT Fourier

Transforms

Nov 17L19: DT Fourier

RepresentationsEX10

Exam 3

no recitation

L20: DT Fourier

Representations

R18: DT Fourier

Representations

Nov 24 L21: SamplingHW11

dueR19: Sampling Thanksgiving Vacation

Dec 1 L22: SamplingHW12

dueR20: Sampling L23: Modulation R21: Modulation

Dec 8L24: Applications

of 6.003EX13 R22: Review

Breakfast with

StaffStudy Period

Dec 15 finals finals 21/22 finals finals

Tuesday Wednesday Thursday Friday

Thanksgiving Vacation

Final Examination Period


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

Interested? ...send email to [email protected]


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

Example: Mass and Spring


x(t)

y(t)

mass &springsystem

x(t) y(t)


x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t)

tr0(t) r2(t)

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Cell Phone System

sound in

sound out

cellphonesystem

sound in sound out

Example: Cell Phone System

sound in

sound out

cellphonesystem

sound in sound out

t t

Signals and Systems: Widely Applicable

The Signals and Systems approach has broad application: electrical,

mechanical, optical, acoustic, biological, financial, ...

mass &springsystem

x(t) y(t)

t t

r0(t)

r1(t)

r2(t)

h1(t)

h2(t) tanksystem

r0(t) r2(t)

t t

cellphonesystem

sound in sound out

t t

Signals and Systems: Modular

The representation does not depend upon the physical substrate.

sound in

sound out

cellphone

tower towercell

phonesound

in

E/M optic

fiber

E/M soundout

focuses on the flow of information, abstracts away everything else

Signals and Systems: Hierarchical

Representations of component systems are easily combined.

Example: cascade of component systems

cellphone

tower towercell

phonesound

in

E/M optic

fiber

E/M soundout

Composite system

cell phone systemsound

insoundout

Component and composite systems have the same form, and are

analyzed with same methods.

Signals and Systems

Signals are mathematical functions.

• independent variable = time

• dependent variable = voltage, flow rate, sound pressure

mass &springsystem

x(t) y(t)

t t

tanksystem

r0(t) r2(t)

t t

cellphonesystem

sound in sound out

t t

Signals and Systems

continuous “time” (CT) and discrete “time” (DT)

t

x(t)

0 2 4 6 8 10n

x[n]

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.

Signals and Systems

Sampling: converting CT signals to DT

t

x(t)

0T 2T 4T 6T 8T 10Tn

x[n] = x(nT )

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)

Signals and Systems

Reconstruction: converting DT signals to CT

zero-order hold

n

x[n]

0 2 4 6 8 10n

x(t)

0 2 4 6 8 10


commonly used in audio output devices such as CD players

Signals and Systems

Reconstruction: converting DT signals to CT

piecewise linear

n

x[n]

0 2 4 6 8 10n

x(t)

0 2 4 6 8 10


commonly used in rendering images

Check Yourself

Computer generated speech (by Robert Donovan)

t

f(t)

Listen to the following four manipulated signals:

f1(t), f2(t), f3(t), f4(t).How many of the following relations are true?

• f1(t) = f(2t)• f2(t) = −f(t)• f3(t) = f(2t)• f4(t) = 2f(t)

Check Yourself

Computer generated speech (by Robert Donovan)

t

f(t)

Listen to the following four manipulated signals:

f1(t), f2(t), f3(t), f4(t).How many of the following relations are true? 2

• f1(t) = f(2t)√

• f2(t) = −f(t) X• f3(t) = f(2t) X• f4(t) = 2f(t)

√

Check Yourself

−250 0 250

−25

00

250

y

x

f1(x, y)=f(2x, y) ?−250 0 250

−25

00

250

y

x

f2(x, y)=f(2x−250, y) ?−250 0 250

−25

00

250

y

x

f3(x, y)=f(−x−250, y) ?

−250 0 250−

250

025

0

y

x

f(x, y)

How many images match the expressions beneath them?

Check Yourself

−250 0 250

−25

00

250

y

x

f(x, y)−250 0 250

−25

00

250

y

x

f1(x, y) = f(2x, y) ?−250 0 250

−25

00

250

y

x

f2(x, y) = f(2x−250, y) ?−250 0 250

−25

00

250

y

x

f3(x, y) = f(−x−250, y) ?

x = 0 → f1(0, y) = f(0, y)√

x = 250 → f1(250, y) = f(500, y) X

x = 0 → f2(0, y) = f(−250, y)√

x = 250 → f2(250, y) = f(250, y)√

x = 0 → f3(0, y) = f(−250, y) X

x = 250 → f3(250, y) = f(−500, y) X

Check Yourself

−250 0 250

−25

00

250

y

x

f1(x, y)=f(2x, y) ?−250 0 250

−25

00

250

y

x

f2(x, y)=f(2x−250, y) ?−250 0 250

−25

00

250

y

x

f3(x, y)=f(−x−250, y) ?

−250 0 250−

250

025

0

y

x

f(x, y)

How many images match the expressions beneath them?

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

Example System: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

Deposits are an input (of money) into the system. How

are withdrawals represented in the framework of signals

and systems?

1. as an input signal2. as an output signal3. none of the above



x[n] y[n]

n n

account

$ deposited today current balance

$0.00



x[n] y[n]

n n

account


$1.00



x[n] y[n]

n n

account


$2.00



x[n] y[n]

n n

account


$3.00



x[n] y[n]

n n

account


$5.00



x[n] y[n]

n n

account


$6.00


Withdrawals are negative deposits (e.g., an input signal)./

x[n] y[n]

n n

account


$0.00



x[n] y[n]

n n

account


$1.00



x[n] y[n]

n n

account


$2.00



x[n] y[n]

n n

account


$1.00



x[n] y[n]

n n

account


$3.00



x[n] y[n]

n n

account


$2.00



x[n] y[n]

n n

account


$1.00


Compound interest./

x[n] y[n]

n n

account


$1.00


Compound interest./

x[n] y[n]

n n

account


$1.05


Compound interest./

x[n] y[n]

n n

account


$1.10


Compound interest./

x[n] y[n]

n n

account


$1.16


Compound interest./

x[n] y[n]

n n

account


$1.22


Compound interest./

x[n] y[n]

n n

account


$1.28


Compound interest./

x[n] y[n]

n n

account


$2.34


Compound interest./

x[n] y[n]

n n

account


$1.41


Early Retirement? How soon can you retire if

• rate of savings: $10,000 per year (prior to retirement)

• living expenses: $25,000 per year (before and after retirement)

• 5% annual interest (before and after retirement)

• live off your savings till age 80?

Check Yourself

Which curve shows the correct shape for the retirement

account?

y1[n]

y2[n]

y3[n]

n

n

n

Check Yourself

Which curve shows the correct shape for the retirement

account? y1[n]

y1[n]

y2[n]

y3[n]

n

n

n

Population Growth

Population Growth

Statement of the problem (1202 AD):

A certain man put a pair of rabbits in a place surrounded on all sides

by a wall. How many pairs of rabbits can be produced from that

pair in a year, if it is supposed that every month, each pair begets a

new pair, which, from the second month on, becomes productive?

Population Growth

Population Growth

How does the number of pairs of rabbits grow?

1. logarithmic (f [n] ∝ logn)2. polynomial (f [n] ∝ nk for some k)

3. exponential (f [n] ∝ zn for some z)

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