Signals and Systems - MIT OpenCourseWare...sound in E/M optic ber E/M sound out Signals and Systems:...
Transcript of Signals and Systems - MIT OpenCourseWare...sound in E/M optic ber E/M sound out Signals and Systems:...
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6.01: Introduction to EECS I
Signals and Systems
February 15, 2011
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Module 1 Summary: Software Engineering
Focused on abstraction and modularity in software engineering.
Topics: procedures, data structures, objects, state machines
Lab Exercises: implementing robot controllers as state machines
BrainSensorInput Action
Abstraction and Modularity: Combinators
Cascade: make new SM by cascading two SM’s
Parallel: make new SM by running two SM’s in parallel
Select: combine two inputs to get one output
Themes: PCAP
Primitives – Combination – Abstraction – Patterns
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6.01: Introduction to EECS I
The intellectual themes in 6.01 are recurring themes in EECS:
• design of complex systems
• modeling and controlling physical systems
• augmenting physical systems with computation
• building systems that are robust to uncertainty
Intellectual themes are developed in context of a mobile robot.
Goal is to convey a distinct perspective about engineering.
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Module 2 Preview: Signals and Systems
Focus next on analysis of feedback and control systems.
Topics: difference equations, system functions, controllers.
Lab exercises: robotic steering
Themes: modeling complex systems, analyzing behaviors
steer left
steer left
straight ahead?
steer right
steer right
steer right
straight ahead?
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Analyzing (and Predicting) Behavior
Today we will start to develop tools to analyze and predict behavior.
Example (design Lab 2): use sonar sensors (i.e., currentDistance)
to move robot desiredDistance from wall.
desiredDistancecurrentDistance
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Analyzing (and Predicting) Behavior
Make the forward velocity proportional to the desired displacement.
desiredDistancecurrentDistance
>>> class wallFinder(sm.SM):
... startState = None
... def getNextValues(self, state, inp):
... desiredDistance = 0.5
... currentDistance = inp.sonars[3]
... return (state,io.Action(fvel=?,rvel=0))
Find an expression for fvel.
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Check Yourself
desiredDistancecurrentDistance
Which expression for fvel has the correct form?
1. currentDistance 2. currentDistance-desiredDistance
3. desiredDistance 4. currentDistance/desiredDistance
5. none of the above
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Check Yourself
desiredDistancecurrentDistance
Which expression for fvel has the correct form? 2
1. currentDistance 2. currentDistance-desiredDistance
3. desiredDistance 4. currentDistance/desiredDistance
5. none of the above
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aa
Analyzing (and Predicting) Behavior
Make the forward velocity proportional to the desired displacement.
desiredDistancecurrentDistance
>>> class wallFinder(sm.SM): ... startState = None ... def getNextValues(self, state, inp): ... desiredDistance = 0.5 ... currentDistance = inp.sonars[3] ... return (state,io.Action(
fvel=currentDistance-desiredDistance, rvel=0))
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Check Yourself
Which plot best represents currentDistance?
desiredDistancecurrentDistance
n
1.
n
2.
n
3.
n
4.
5. none of the above
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Check Yourself
Which plot best represents currentDistance? 2.
desiredDistancecurrentDistance
n
1.
n
2.
n
3.
n
4.
5. none of the above
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Check Yourself
Why does the distance undershoot?
n
2.
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Check Yourself
Why does the distance undershoot?
n
2.
The robot has inertia and there is delay in the sensors and actuators!
We will study delay in more detail over the next three weeks.
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Performance Analysis
Quantify performance by characterizing input and output signals.
n
desiredDistance
n
currentDistance
wallFinder
system
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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
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Example: Mass and Spring
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x(t)
y(t)
mass &springsystem
x(t) y(t)
Example: Mass and Spring
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x(t)
y(t)
mass &springsystem
x(t) y(t)
t t
Example: Mass and Spring
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Example: Tanks
r0(t)
r1(t)
r2(t)
h1(t)
h2(t)
tanksystem
r0(t)
tr0(t) r2(t)
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Example: Tanks
r0(t)
r1(t)
r2(t)
h1(t)
h2(t)
tanksystem
r0(t) r2(t)
t t
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sound in
sound out
cellphonesystem
sound in sound out
Example: Cell Phone System
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sound in
sound out
cellphonesystem
sound in sound out
t t
Example: Cell Phone System
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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
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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
© FreeFoto.com. CC by-nc-nd. Thiscontent is excluded from our CreativeCommons license. For more information,see http://ocw.mit.edu/fairuse.
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cellphone
tower towercell
phonesound
in
E/M optic
fiber
E/M soundout
Signals and Systems: Hierarchical
Representations of component systems are easily combined.
Example: cascade of component systems
Composite system
cell phone systemsound
insoundout
Component and composite systems have the same form, and are
analyzed with same methods.
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The Signals and Systems Abstraction
Our goal is to develop representations for systems that facilitate
analysis.
systemsignal
in
signal
out
Examples:
• Does the output signal overshoot? If so, how much?
• How long does it take for the output signal to reach its final value?
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Continuous and Discrete Time
or discrete time.
Inputs and outputs of systems can be functions of continuous time
We will focus on discrete-time systems.
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Difference Equations
Difference equations are an excellent way to represent discrete-time
systems.
Example:
y[n] = x[n]− x[n − 1]
Difference equations can be applied to any discrete-time system;
they are mathematically precise and compact.
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We will use the unit sample as a “primitive” (building-block signal)
to construct more complex signals.
{
Difference Equations
Difference equations are mathematically precise and compact.
Example:
y[n] = x[n]− x[n − 1]
Let x[n] equal the “unit sample” signal δ[n],
1, if n = 0; δ[n] =
0, otherwise.
−1 0 1 2 3 4n
x[n] = δ[n]
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{
Difference Equations
Difference equations are mathematically precise and compact.
Example:
y[n] = x[n]− x[n − 1]
Let x[n] equal the “unit sample” signal δ[n],
1, if n = 0; δ[n] =
0, otherwise.
We will use the unit sample as a “primitive” (building-block signal)
−1 0 1 2 3 4n
x[n] = δ[n]
to construct more complex signals.
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Difference
Step-By-Step Solutions
equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Step-By-Step Solutions
Difference equations are convenient for step-by-step analysis.
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n]− x[n− 1]
y[−1] = x[−1]− x[−2] = 0− 0 = 0y[0] = x[0]− x[−1] = 1− 0 = 1y[1] = x[1]− x[0] = 0− 1 = −1y[2] = x[2]− x[1] = 0− 0 = 0y[3] = x[3]− x[2] = 0− 0 = 0. . .
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Multiple Representations of Discrete-Time Systems
Block diagrams are useful alternative representations that highlight
visual/graphical patterns.
Difference equation:
y[n] = x[n]− x[n − 1]
• difference equations are mathematically compact
• block diagrams illustrate signal flow paths
Block diagram:
Delay−1
+x[n] y[n]
Same input-output behavior, different strengths/weaknesses:
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Block
Step-By-Step Solutions
diagrams are also useful for step-by-step analysis.
Represent y[n] = x[n]− x[n − 1] with a block diagram:
−1 Delay
+x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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
0−1
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
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Block
Step-By-Step Solutions
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
0−1→ 0
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step Solutions
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
0−1→ 0
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
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Block
Step-By-Step Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step
Block diagrams are also useful for step-by-step an
Solutions
alysis.
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Step-By-Step
Block diagrams are also useful for step-by-step
Solutions
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] = δ[n]
−1 0 1 2 3 4n
y[n]
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Check Yourself
DT systems can be described by difference equations and/or
block diagrams.
Difference equation:
y[n] = x[n]− x[n − 1]
Block diagram:
−1 Delay
+x[n] y[n]
In what ways are these representations different?
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Check Yourself
In what ways are difference equations different from block diagrams?
Difference equation:
y[n] = x[n]− x[n − 1]
Difference equations are “declarative.”
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?)
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Multiple Representations of Discrete-Time Systems
Block diagrams are useful alternative representations that highlight
visual/graphical patterns.
Difference equation:
y[n] = x[n]− x[n − 1]
• difference equations are mathematically compact
• block diagrams illustrate signal flow paths
Block diagram:
Delay−1
+x[n] y[n]
Same input-output behavior, different strengths/weaknesses:
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From Samples to Signals
Lumping all of the (possibly infinite) samples into a single object
– the signal – simplifies its manipulation.
This lumping is analogous to
• representing coordinates in three-space as points
• representing lists of numbers as vectors in linear algebra
• creating an object in Python
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From Samples to Signals
Operators manipulate signals rather than individual samples.
Delay−1
+X Y
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: multiply by −1•
Signals are the primitives.
Operators are the means of combination.
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Symbols
Operator Notation
can now compactly represent diagrams.
Let R represent the right-shift operator:
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] for all n)
Representing the difference machine
Delay−1
+X Y
with R leads to the equivalent representation
Y = X −RX = (1 −R)X
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Operator Notation: Check Yourself
Let Y = RX. Which of the following is/are true:
1. y[n] = x[n] for all n
2. y[n + 1] = x[n] for all n
3. y[n] = x[n + 1] for all n
4. y[n − 1] = x[n] for all n
5. none of the above
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Operator Notation: Check Yourself
Let Y = RX. Which of the following is/are true: 2.
1. y[n] = x[n] for all n
2. y[n + 1] = x[n] for all n
3. y[n] = x[n + 1] for all n
4. y[n − 1] = x[n] for all n
5. none of the above
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Operator Representation of a Cascaded System
System operations have simple operator representations.
Cascade systems multiply operator expressions. →
Using operator notation:
Delay−1
+
Delay−1
+XY1
Y2
Y1 = (1 −R)X
Y2 = (1 −R)Y1
Substituting for Y1:
Y2 = (1 −R)(1 −R)X
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Operator Algebra
Operator expressions expand and reduce like polynomials.
Using difference equations:
Delay−1
+
Delay−1
+XY1
Y2
y2[n] = y1[n]− y1[n − 1]
= (x[n]− x[n − 1]) − (x[n − 1]− x[n − 2])
= x[n]− 2x[n − 1] + x[n − 2]
Using operator notation:
Y2= (1 −R)Y1 = (1 −R)(1 −R)X
= (1 −R)2X
= (1 − 2R +R2)X
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Operator Approach
Applies your existing expertise with polynomials to understand block
diagrams, and thereby understand systems.
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Operator Algebra
Operator notation facilitates seeing relations among systems.
“Equivalent” block diagrams (assuming both initially at rest):
Delay−1
+
Delay−1
+XY1
Y2
Delay
Delay
−2
+X Y
Equivalent operator expression:
(1−R)(1 −R) = 1 − 2R +R2
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Operator Algebra
Operator notation prescribes operations on signals, not samples:
X :
−2RX :
+R2X :
y = X − 2RX +R2X :
e.g., start with X, subtract 2 times a right-shifted version of X, and
add a double-right-shifted version of X!
−1 0 1 2 3 4 5 6n
−1 0 1 2 3 4 5 6n
−1 0 1 2 3 4 5 6n
−1 0 1 2 3 4 5 6n
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Operator Algebra
Expressions involving R obey many familiar laws of algebra, e.g.,
commutativity.
R(1−R)X = (1 −R)RX
This is easily proved by the definition and it implies thatof R,
cascaded systems commute (assuming initial rest)
Delay−1
+ DelayX Y
is equivalent to
Delay−1
+DelayX Y
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Operator Algebra
Multiplication distributes over addition.
Equivalent systems
Delay−1
+ DelayX Y
−1
+
Delay Delay
DelayX Y
Equivalent operator expression:
R(1−R) = R−R2
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( ) ( )
The
Operator Algebra
associative property similarly holds for operator expressions.
Equivalent systems
Delay
Delay Delay Delay−1
2
−1
+ +X Y
Delay
Delay
Delay Delay−1
2
−1
+ +X Y
Equivalent operator expression:
(1−R)R (2−R) = (1 −R) R(2−R)
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Check Yourself
How many of the following systems are equivalent?
Delay 2 + Delay 2 +X Y
Delay + Delay 4 +X Y
Delay 4 +
Delay
+X Y
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Check Yourself
How many of the following systems are equivalent? 3
Delay 2 + Delay 2 +X Y
Delay + Delay 4 +X Y
Delay 4 +
Delay
+X Y
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Explicit and Implicit Rules
Recipes versus constraints.
Y = (1 −R)X
Recipe: between input signal and
right-shifted input signal.
Delay−1
+X Y
output signal equals difference
Y = RY +X
= X(1−R)Y
Constraints: find the signal Y such that the difference between Y
and RY is X. But how?
Delay
+X Y
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Example: Accumulator
Try step-by-step analysis: it always works. Start “at rest.”
+
Delay
x[n] y[n]
−1 0 1 2 3 4n
x[n] = δ[n]
−1 0 1 2 3 4n
y[n]
Find y[n] given x[n] = δ[n]: y[n] = x[n] + y[n− 1]
y[0] = x[0] + y[−1] = 1 + 0 = 1y[1] = x[1] + y[0] = 0 + 1 = 1y[2] = x[2] + y[1] = 0 + 1 = 1. . .
Persistent response to a transient input!
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Delay
Delay Delay
Delay Delay Delay
+
... ...
X Y
Example: Accumulator
The response of the accumulator system could also be generated by
a system with infinitely many paths from input to output, each with
one unit of delay more than the previous.
Y = (1 + R +R2 +R3 + )X· · ·
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Example: Accumulator
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: Assume X2 = 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 that Y2 = Y1.
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Example: Accumulator
The system functional
Delay
+X Y
for the accumulator is the reciprocal of a
polynomial in R.
= X(1−R)Y
The product (1−R)× (1 +R +R2 +R3 + ) equals 1.· · ·
Therefore the terms (1−R) and (1 +R +R2 +R3 + ) are reciprocals. · · ·
Thus we can write
Y = 1 = 1 + R +R2 +R3 +R4 +X
· · · 1−R
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Example: Accumulator
The reciprocal of 1−R can also be evaluated using synthetic division.
Therefore
1 +R +R2 +R3 + · · ·1−R 1
1 −RRR −R2
R2
R2 −R3
R3
R3 −R4· · ·
1 4 +1−R
= 1 + R +R2 +R3 +R · · ·
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Check Yourself
A system is described by the following operator expression:
Y X
= 1 1 + 2R
.
Determine the output of the system when the input is a
unit sample.
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Check Yourself
Evaluate the system function using synthetic division.
Therefore the system function can be written as
Y = X
1 −2R +4R2 −8R3 + · · ·1 + 2R 1
1 +2R−2R−2R −4R2
4R2
4R2 +8R3
−8R3
−8R3 −16R4· · ·
1 1 + 2R
= 1 − 2R + 4R2 − 8R3 + 16R4 + · · ·
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a
Check Yourself
Now find Y given that X is delta function.
x[n] = δ[n]
Think about the “sample” representation of the system function:
Y 2= 1 − 2R + 4R − 8R3 + 16R4 +X
· · ·
y[n] = (1− 2R + 4R2 − 8R3 + 16R4 +
) δ[n]· · ·
y[n] = δ[n]− 2δ[n − 1] + 4δ[n − 2]− 8δ[n − 3] + 16δ[n − 4] +· · ·
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0
y[n]
Check Yourself
A system is described by the following operator expression:
Y X
= 1 1 + 2R
.
Determine the output of the system when the input is a
unit sample.
y[n] = δ[n]− 2δ[n − 1] + 4δ[n − 2]− 8δ[n − 3] + 16δ[n − 4] + · · ·
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Linear Difference Equations with Constant Coefficients
Any system composed of adders, gains, and delays can be repre
sented by a difference equation.
y[n] + a1y[n − 1] + a2y[n − 2] + a3y[n − 3] + · · ·
= b0x[n] + b1x[n − 1] + b2x[n − 2] + b3x[n − 3] + · · ·
Such a system can also be represented by an operator expression.
(1 + a1R + a2R2 + a3R3 + )Y = (b0 + b1R + b2R2 + b3R3 + )X· · · · · ·
We will see that this correspondence provides insight into behavior.
This correspondence also reduces algebraic tedium.
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Check Yourself
Determine the difference equation that relates x[·] and y[·].
Delay
Delay
+x[n] y[n]
1. y[n] = x[n − 1] + y[n − 1] 2. y[n] = x[n − 1] + y[n − 2] 3. y[n] = x[n − 1] + y[n − 1] + y[n − 2] 4. y[n] = x[n − 1] + y[n − 1]− y[n − 2] 5. none of the above
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Check Yourself
Determine a difference equation that relates x[·] and y[·] below.
Delay
Delay
+x[n] y[n]
Assign names to all signals. Replace Delay with R.
R
R
+X YE
W
Express relations among signals algebraically.
E = X +W ; Y = RE ; W = RY
Solve: Y = RE = R(X +W ) = R(X +RY ) → RX = Y − R2Y
Corresponding difference equation: y[n] = x[n − 1] + y[n − 2]
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Check Yourself
Determine the difference equation that relates x[·] and y[·]. 2.
Delay
Delay
+x[n] y[n]
1. y[n] = x[n − 1] + y[n − 1] 2. y[n] = x[n − 1] + y[n − 2] 3. y[n] = x[n − 1] + y[n − 1] + y[n − 2] 4. y[n] = x[n − 1] + y[n − 1]− y[n − 2] 5. none of the above
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Signals and Systems
Block diagrams: illustrate signal flow paths.
Delay−1
+x[n] y[n]
Operator representations: analyze systems as polynomials.
Multiple representations of discrete-time systems.
Difference equations: mathematically compact.
y[n] = x[n]− x[n − 1]
Y = (1 −R)X
Labs: representing signals in python
controlling robots and analyzing their behaviors.
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6.01SC Introduction to Electrical Engineering and Computer Science Spring 2011
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