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6.003: Signals andSystems

Discrete-TimeSystems

September

13,

2011

1

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Homework

Doing

the

homework

is

essential

to

understanding

the

content.

WeeklyHomeworkAssigments

tutor

(exam-type)

problems:

answers

are

automatically

checked

to

provide

quick

feedback

engineering

design

(real-world)

problems:

gradedbyahuman

Learning

doesnt

end

when

you

have

submitted

your

work!

solutions

will

be

posted

on

Wednesdays

at

5pm

read solutions tofind errorsand to seealternativeapproaches

mark the errors in yourpreviously submittedwork

submit

the

markup

by

Friday

at

5pm

identify

ALL

errors

and

get

back

half

of

the

points

you

lost!

2

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Discrete-TimeSystems

We

start

with

discrete-time

(DT)

systems

because

they

are

conceptually

simpler

than

continuous-time

systems

illustrate

same

important

modes

of

thinking

as

continuous-time

are increasingly important (digital electronicsandcomputation)

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MultipleRepresentations ofDiscrete-TimeSystems

Systems

can

be

represented

in

different

ways

to

more

easily

address

different typesof issues.

Verbal

description:

To

reduce

the

number

of

bits

needed

to

store

a

sequence

of

large

numbers

that

are

nearly

equal,

record

the

first

number,

and

then

record

successive

differences.

Difference equation:

y[n]=x[n]x[n1]Block

diagram:

1 Delay

+x[n] y[n]

We

will

exploit

particular

strengths

of

each

of

these

representations.

4

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DifferenceEquations

Difference

equations

are

mathematically

precise

and

compact.

Example:

y[n] =

x[n]

x[n

1]

Letx[n] equal theunit sample signal [n],

1,

if

n

= 0;[n] =

0, otherwise.

1 0 1 2 3 4n

x[n] =[n]

We

will

use

the

unit

sample

as

a

primitive

(building-block

signal)

to

construct

more

complex

signals.

5

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CheckYourself

Solvey[n]=x[n]

x[n

1]

given

x[n]=[n]

How

many

of

the

following

are

true?

1.

y[2]>y[1]

2.

y[3]>y[2]

3. y[2]=0

4. y[n]

y[n

1]=x[n]

2x[n

1]+x[n

2]

5. y[119]=0

6

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Step-By-StepSolutions

Difference

equations

are

convenient

for

step-by-step

analysis.

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Find y[n] given x[n] =[n]: y[n] =x[n] x[n 1]

y[1] =x[1] x[2] = 0 0 = 0

y[0] =x[0] x[1] = 1 0 = 1

y[1] =x[1] x[0] = 0 1 = 1

y[2] =x[2] x[1] = 0 0 = 0

y[3] =x[3] x[2] = 0 0 = 0

. . .

7

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CheckYourself

Solvey[n]=x[n]

x[n

1]

given

x[n]=[n]

How

many

of

the

following

are

true?

4

1.

y[2]>y[1]

2.

y[3]>y[2] X

3. y[2]=0

4. y[n]

y[n

1]=x[n]

2x[n

1]+x[n

2]

5. y[119]=0

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+x[n] y[n]

0

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+1 1

10

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

10

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+1 0

10 1

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+1 0 1

00 1

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+0 1

01

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+0 1

01 0

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

14

St B St S l ti

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+0 0

01 0

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

15

St B St S l ti

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+0 0

00

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

16

Ste B Ste Sol tio s

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Block

diagrams

are

also

useful

for

step-by-step

analysis.

Represent

y[n] =

x[n]

x[n

1]

with

a

block

diagram:

start

at

rest

1 Delay

+0 0

00

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

17

Check Yourself

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CheckYourself

DT systems can be described by difference equations and/or

blockdiagrams.

Difference

equation:

y[n]=x[n]x[n1]

Block

diagram:

1 Delay

+x[n] y[n]

In

what

ways

are

these

representations

different?

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Check Yourself

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CheckYourself

In

what

ways

are

difference

equations

different

from

block

diagrams?

Difference

equation:

y[n] =x[n]x[n1]Difference equationsaredeclarative.

They

tell

you

rules

that

the

system

obeys.

Block

diagram:

1 Delay

+x[n] y[n]

Block

diagrams

are

imperative.

They

tell

you

what

to

do.

Block

diagrams

contain

more

information

than

the

corresponding

difference

equation

(e.g.,

what

is

the

input?

what

is

the

output?)

19

From Samples to Signals

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FromSamples toSignals

Lumping

all

of

the

(possibly

infinite)

samples

into

a

single

object

the

signal

simplifies

its

manipulation.

This lumping isanabstraction that isanalogous to

representing

coordinates

in

three-space

as

points

representing

lists

of

numbers

as

vectors

in

linear

algebra

creatinganobject inPython

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From Samples to Signals

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FromSamples toSignals

Operators

manipulate

signals

rather

than

individual

samples.

1 Delay

+x[n] y[n]

Nodes

represent

whole

signals

(e.g.,

X

and

Y

).

The

boxes

operate

on

those

signals:

Delay

=

shift

whole

signal

to

right

1

time

step

Add

=

sum

two

signals

1: multiplyby 1

Signals

are

the

primitives.

Operators

are

the

means

of

combination.

21

Operator Notation

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OperatorNotation

Symbols

can

now

compactly

represent

diagrams.

Let

R represent the right-shiftoperator:

Y

=

R{X} RX

where X represents the whole input signal (x[n] for all n) and Y

represents

the

whole

output

signal

(y[n] foralln)

Representing

the

difference

machine

1 Delay

+x[n] y[n]

with

R

leads

to

the

equivalent

representation

Y =X RX=(1 R)X

22

Operator Notation: Check Yourself

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OperatorNotation: CheckYourself

LetY =

RX. Whichof the following is/are true:

1. y[n]=x[n] foralln

2. y[n+1]=x[n] foralln

3. y[n]=x[n+1] foralln

4.

y[n

1]

=

x[n]

for

all

n

5.

none

of

the

above

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Check Yourself

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CheckYourself

Consider

a

simple

signal:

Then

Clearly

y[1]

=

x[0].

1 0 1 2 3 4n

X

1 0 1 2 3 4n

Y = RX

Equivalently,

if

n=0, then y[n+1]=x[n].

The same sortofargumentworks forallothern.

24

Operator Notation: Check Yourself

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OperatorNotation: CheckYourself

LetY =

RX. Whichof the following is/are true:

1. y[n]=x[n] foralln

2. y[n+1]=x[n] foralln

3. y[n]=x[n+1] foralln

4.

y[n

1]

=

x[n]

for

all

n

5.

none

of

the

above

25

OperatorRepresentation of aCascadedSystem

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Op p C Sy

System

operations

have

simple

operator

representations.

Cascade systems

multiplyoperator expressions.

1 Delay

+

1 Delay

+XY1

Y2

Using

operator

notation:

Y1 =(1 R)XY2 =(1 R)Y1

Substituting forY1:

Y2 =(1 R)(1 R)X

26

OperatorAlgebra

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p g

Operator

expressions

can

be

manipulated

as

polynomials.

1 Delay

+

1 Delay

+X

Y1Y2

Using

difference

equations:

y2[n]

=

y1[n]

y1[n

1]

= (x[n]x[n1])(x[n1]x[n2])=x[n]2x[n1]+x[n2]

Using

operator

notation:

Y2

=

(1

R)

Y1

=

(1

R)(1

R)

X

=(1 R)2X=(12R+R2)X

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OperatorApproach

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p pp

Applies

your

existing

expertise

with

polynomials

to

understand

block

diagrams,and therebyunderstand systems.

28

OperatorAlgebra

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Operator

notation

facilitates

seeing

relations

among

systems.

Equivalent

block

diagrams

(assuming

both

initially

at

rest):

1 Delay

+

1 Delay

+XY1

Y2

Delay

Delay

2

+X Y

Equivalent

operator

expressions:

(1 R)(1 R) = 12R+R2

The

operator

equivalence

is

much

easier

to

see.29

CheckYourself

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Operator

expressions

for

these

equivalent

systems

(if

started

at

rest)

obey

what

mathematical

property?

Delay1

+ DelayX Y

Delay

Delay Delay1

+X Y

1. commutate 2. associative

3.

distributive

4.

transitive

5.

none

of

the

above

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CheckYourself

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Delay1

+ DelayX Y

Y =R(1 R)X

Delay

Delay Delay1

+X Y

Y =(

R

R2)X

Multiplication

by

R

distributes

over

addition.

31

CheckYourself

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Operator

expressions

for

these

equivalent

systems

(if

started

at

rest)

obey

what

mathematical

property?

3

Delay1

+ DelayX Y

Delay

Delay Delay1

+X Y

1. commutate 2. associative

3.

distributive

4.

transitive

5.

none

of

the

above

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CheckYourself

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How

many

of

the

following

systems

are

equivalent

to

Y

=

(4R2+

4R

+

1)

X

?

Delay 2 + Delay 2 +X Y

Delay + Delay 4 +X Y

Delay 4 +

Delay

+X Y

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CheckYourself

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Y

=

(2R

+

1)(2R

+

1)

X


Y

=

(4R2

+

4R

+

1)

X

Delay 4 +

Delay

+X Y

Y

=

(4R2

+

4R

+

1)

X

All

implement

Y

=

(4R2

+ 4R+1)X34

CheckYourself

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How

many

of

the

following

systems

are

equivalent

to

Y

=

(4R2+

4R

+

1)

X

?

3



Delay 4 +

Delay

+X Y

35

Operator

Algebra:

Explicit

and

Implicit

Rules

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Recipes

versus

constraints.

Recipe:

subtract

a

right-shifted

version

of

the

input

signal

from

a

copy

of

the

input

signal.

1 Delay

+X Y

Y =(1

R)X

Constraint:

the

difference

between

Y

and

RY

is

X.

Delay

+X YY =RY +X(1R)Y =X

But

how

does

one

solve

such

a

constraint?

36

Example:

Accumulator

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Try

step-by-step

analysis:

it

always

works.

Start

at

rest.

+

Delay

x[n] y[n]

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Find y[n] given x[n] =[n]: y[n] =x[n] +y[n 1]

y[0] =x[0] +y[1] = 1 + 0 = 1

y[1] =x[1] +y[0] = 0 + 1 = 1y[2] =x[2] +y[1] = 0 + 1 = 1

. . .

37

Example:

Accumulator

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Try

step-by-step

analysis:

it

always

works.

Start

at

rest.

+

Delay

x[n] y[n]

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Find y[n] given x[n] =[n]: y[n] =x[n] +y[n 1]

y[0] =x[0] +y[1] = 1 + 0 = 1

y[1] =x[1] +y[0] = 0 + 1 = 1y[2] =x[2] +y[1] = 0 + 1 = 1

. . .

Persistent

response

to

a

transient

input!

38

Example:

Accumulator

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The

response

of

the

accumulator

system

could

also

be

generated

by

asystemwith infinitelymanypaths from inputtooutput,eachwith

oneunitofdelaymore than theprevious.

Y =(1 +R+R2+R3+ )X

Delay

Delay Delay

Delay Delay Delay

+

... ...

X Y

39

Example:

Accumulator

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These

systems

are

equivalent

in

the

sense

that

if

each

is

initially

at

rest,

they

will

produce

identical

outputs

from

the

same

input.

(1 R)Y1 =X1 ? Y2 =(1 +R+R2+R3+ )X2

Proof:

AssumeX2 =X1:

Y2

=

(1 +

R

+

R2

+

R3

+

)

X2

=(1 +R+R2+R3+ )X1=(1 +R+R2+R3+ )(1 R)Y1=((1+R+R2+R3+ )(R+R2+R3+ ))Y1=

Y1

It follows thatY2 =Y1.

It

also

follows

that

(1

R)

and

(1

+

R

+

R2

+

R3

+

)

are

reciprocals.

40

Example:

Accumulator

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The

reciprocal

of1Rcanalsobeevaluatedusingsyntheticdivision.

Therefore

1 +R +R2

+R3

+ 1 R 11 R

R

R R2

R2

R2 R3

R3

R3 R

4

1= 1 +R+R2+R3+R4+

1 R

41

Feedback

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Systems

with

signals

that

depend

on

previous

values

of

the

same

signalare said tohave feedback.

Example:

The

accumulator

system

has

feedback.

Delay

+X Y

By contrast, thedifferencemachinedoesnothave feedback.

1 Delay

+x[n] y[n]

42

Cyclic

Signal

Paths,

Feedback,

and

Modes

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Block

diagrams

help

visualize

feedback.

Feedbackoccurswhen there isa cyclic signalflowpath.

R

R

2

+X Y

Delay

+X Y

acyclic

cyclic

Acyclic:

all

paths

through

system

go

from

input

to

output

with

no

cycles.

Cyclic:

at

least

one

cycle.

43

Feedback,

Cyclic

Signal

Paths,

and

Modes

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The

effect

of

feedback

can

be

visualized

by

tracing

each

cycle

through thecyclic signalpaths.

Delay

+

p0

X Y

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Each

cycle

creates

another

sample

in

the

output.

44

Feedback,

Cyclic

Signal

Paths,

and

Modes

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The

effect

of

feedback

can

be

visualized

by

tracing

each

cycle


Delay

+

p0

X Y

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Each

cycle

creates

another

sample

in

the

output.

45

Feedback,

Cyclic

Signal

Paths,

and

Modes

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The

effect

of

feedback

can

be

visualized

by

tracing

each

cycle


Delay

+

p0

X Y

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Each

cycle

creates

another

sample

in

the

output.

46

Feedback,

Cyclic

Signal

Paths,

and

Modes

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The

effect

of

feedback

can

be

visualized

by

tracing

each

cycle


Delay

+

p0

X Y

1 0 1 2 3 4n

x[n] =[n]

1 0 1 2 3 4n

y[n]

Each

cycle

creates

another

sample

in

the

output.

The

response

will

persist

even

though

the

input

is

transient.

47

Check

Yourself

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How

many

of

the

following

systems

have

cyclic

signal

paths?

R

R+ +X Y + +

R R

X Y

R

+ +X Y + +

R R

X Y

48

Check

Yourself

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How

many

of

the

following

systems

have

cyclic

signal

paths?

3

R

R+ +X Y + +

R R

X Y

R

+ +X Y + +

R R

X Y

49

Finite

and

Infinite

Impulse

Responses

The imp lse response of an ac clic s stem has finite d ration hile

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The

impulse

response

of

an

acyclic

system

has

finite

duration,

while

thatofa cyclic system canhave infiniteduration.

1 Delay

+X Y

Delay

+X Y

1 0 1 2 3 4

n

1 0 1 2 3 4

n

50

Analysis

of

Cyclic

Systems:

Geometric

Growth

If traversing the cycle decreases or increases the magnitude of the

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If

traversing

the

cycle

decreases

or

increases

the

magnitude

of

the

signal, then the fundamentalmodewilldecayorgrow, respectively.

If

the

response

decays

toward

zero,

then

we

say

that

it

converges.

Otherwise,we itdiverges.

51

Check

Yourself

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How

many

of

these

systems

have

divergent

unit-sample

responses?

Delay0.5

+X Y

Delay1.2

+X Y

Delay0.5

+ 1.2DelayX Y

52

Check

Yourself

y[n]

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Delay0.5

+X Y

1 0 1 2 3 4n

y[ ]

Delay1.2

+X Y

1 0 1 2 3 4n

y[n]

Delay0.5

+ 1.2DelayX Y

1 0 1 2 3 4n

y[n]

53

Check

Yourself

y[n]

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X

X

Delay0.5

+X Y

1 0 1 2 3 4n

[ ]

Delay1.2

+X Y

1 0 1 2 3 4n

y[n]

Delay0.5

+ 1.2DelayX Y

1 0 1 2 3 4n

y[n]

54

Check

Yourself

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How

many

of

these

systems

have

divergent

unit-sample

responses?

1

Delay0.5

+X YX

Delay1.2

+X Y

Delay0.5

+ 1.2DelayX Y

X

55

Cyclic

Systems:

Geometric

Growth

If traversing the cycle decreases or increases the magnitude of the

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If

traversing

the

cycle

decreases

or

increases

the

magnitude

of

the

signal, then the fundamentalmodewilldecayorgrow, respectively.

Delay0.5

+X Y

Delay1.2

+X Y

1 0 1 2 3 4n

y[n]

1 0 1 2 3 4n

y[n]

These

are

geometric

sequences:

y[n]=(0.5)n and (1.2)n forn0.Thesegeometric sequencesare called fundamentalmodes.

56

Multiple

Representations

of

Discrete-Time

Systems

Now you know four representations of discrete-time systems.

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Now

you

know

four

representations

of

discrete time

systems.

Verbal

descriptions: preserve the rationale.

To

reduce

the

number

of

bits

needed

to

store

a

sequence

of

large

numbers

that

are

nearly

equal,

record

the

first

number,

and

then

record

successive

differences.

Difference

equations: mathematicallycompact.

y[n] =

x[n]

x[n

1]

Block

diagrams: illustrate signalflowpaths.

1 Delay

+x[n] y[n]

Operator

representations:

analyze

systems

as

polynomials.

Y =(1 R)X

57

MIT OpenCourseWarehttp://ocw.mit.edu
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6.003 Signals and SystemsFall 2011

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