6.003: Signals and Systems

Lecture 1

Introduction to Signals and Systems

6.003: Signals and Systems

Today’s handouts: Single package containing• Subject Information• Lecture #1 slides (for today)• Recitation #2 handout (for tomorrow)

Lecturer Denny Freeman (freeman@mit.edu)

Instructors Qing Hu, Jeff Lang, Karen Livescu,Sanjoy Mahajan, Antonio Torralba

Head TA Demba Ba (demba@mit.edu)

TAs Paul Azunre, Dheera Venkatraman,Keng-Hoong Wee, Steve Zhou

Secretary Janice Balzer (balzer@mit.edu)

Text Signals and Systems by Oppenheim and Willsky

Web Site mit.edu/6.003

6.003: Signals and Systems

Homework: where subject matter is/isn’t learned.

equivalent to “practice” in sports or music.

• Weekly Homework Assignments− Conventional Homework Problems plus− Engineering Design Problems (often using Matlab, Oc-

tave, or Python)

• Homework Assignments are longer (by about 3 hours)than homework assignments in 12 unit subjects!− 15 units + 4 Engineering Design Points

• Open Office Hours!− Stata Basement (32-044)− Mondays and Tuesdays, afternoons and evenings

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

reported 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

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the day preceding the due date (cannot be rescinded)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 Calendar

• Basic Representations of Discrete-Time Systems (4weeks). difference equations, block diagrams, operatorexpressions, system functions, feedback and control, Ztransforms, convolution (O&W Chapters 1, 2, 10, and11).

• Basic Representations of Continuous-Time Systems

(3 weeks). differential equations, block diagrams, opera-tor expressions, system functions, feedback and control,Laplace transforms, convolution (O&W Chapters 1, 2, 9,and 11).

• Signal Processing (2 weeks). Fourier Series, FourierTransforms, Filtering (O&W Chapters 3, 4, 5, and 6).

• Sampling (2 weeks). Sampling, aliasing, DT processingof CT signals (O&W Chapter 7).

• Communications (2 weeks). modulation, AM, FM (O&WChapter 8).

6.003: Signals and Systems

Weekly meetings with class representatives

• help staff understand student perspective• learn about teaching

One representative from each section (6 total)

Tentatively meet on Thursday afternoon

Interested? ...send email to freeman@mit.edu

Lecture 1: The 6.003 Abstraction

systemsignal

in

signal

out

6.003 abstraction: describe a system (physical, mathemat-ical, or computational) by the way it transforms inputs intooutputs.

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

x(t)

y(t)

mass &springsystem

x(t) y(t)

t t

Example: Mass and Spring

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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(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)

tr0(t) r2(t)

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t t

Example: Tanks

r0(t)

r1(t)

r2(t)

h1(t)

h2(t)

tanksystem

r0(t) r2(t)

t 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: Uniform Representations

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

sound in

sound out

cellphonesystem

sound in sound out

t t

electrical, mechanical, chemical, optical, acoustic, biological,financial, ...

Signals and Systems: Uniformity → Modularity

sound in

sound out

cellphone

tower towercell

phonesound

in

E/M optic

fiber

E/M soundout

• focus on the flow of information

• abstract away everything else

Signals and Systems: Broad Applicability

mechanics circuits medical

vivoA

R1

R2

systemsignal

in

signal

out

Discrete-Time Systems

Example: 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 signal

2. as an output signal

3. none of the above

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$0.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$2.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$3.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$5.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$6.00

Example: Bank Account

Transactions (deposits/withdrawals) recorded daily (DT)

x[n] y[n]

n n

account

$ deposited today current balance

$7.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$0.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$2.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$3.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$2.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Withdrawals are negative deposits./

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.00

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.05

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.10

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.16

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.22

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.28

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$2.34

Example: Bank Account

Compound interest./

x[n] y[n]

n n

account

$ deposited today current balance

$1.41

Example: Bank Account

Early Retirement? How soon can you retire if• living expenses: $25,000 per year• rate of savings: $10,000 per year• 5% annual interest• live off your savings till age 80?

x[n]

y[n]

n

n

save retire

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

Population Growth

How does the number of pairs of rabbits grow?

1. logarithmic ( f [n] = O(log n))

2. polynomial ( f [n] = O(nk) for some k)

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