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8/11/2019 MIT6_003F11_lec02
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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)
3
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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
8
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Step-By-StepSolutions
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]
9
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Step-By-StepSolutions
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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Step-By-StepSolutions
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]
11
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Step-By-StepSolutions
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]
12
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Step-By-StepSolutions
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]
13
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Step-By-StepSolutions
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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Step-By-StepSolutions
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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Step-By-StepSolutions
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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Step-By-StepSolutions
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?
18
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
20
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
23
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
27
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
30
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
32
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
33
CheckYourself
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Delay 2 + Delay 2 +X Y
Y
=
(2R
+
1)(2R
+
1)
X
Delay + Delay 4 +X Y
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 2 + Delay 2 +X Y
Delay + Delay 4 +X Y
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
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.
45
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.
46
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.
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
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6.003 Signals and SystemsFall 2011
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