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8/7/2019 Signals and Systems note3
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6.003: Signals and SystemsFeedback, Poles, and Fundamental Modes
February9,2010
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Last Time: Multiple Representations of DT SystemsVerbal descriptions: preserve the rationale.To reduce thenumberofbitsneeded to storea sequenceoflargenumbersthatarenearlyequal, recordthefirstnumber,and then record successivedifferences.
Difference equations: mathematicallycompact.y[n] = x[n] x[n 1]
Block diagrams: illustrate signalflowpaths.1 Delay
+x[n] y[n]
Operator representations: analyze systemsaspolynomials.Y = (1R
) X
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Last Time: Feedback, Cyclic Signal Paths, and ModesSystems with signals that depend on previous values of the samesignalare said tohave feedback.Example: Theaccumulator systemhas feedback.
+X YDelay
Bycontrast, thedifferencemachinedoesnothave feedback.1 Delay
+X Y
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
Delay
+p0
X Y
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
Delay
+p0
X Y
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
Delay
+p0
X Y
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
X + YDelayp0
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
X + YDelayp0
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.
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Last Time: Feedback, Cyclic Signal Paths, and ModesThe effect of feedback can be visualized by tracing each cyclethrough thecyclic signalpaths.
X + YDelayp0
x[n] = [n] y[n]
n n1 0 1 2 3 4 1 0 1 2 3 4Eachcyclecreatesanother sample in theoutput.The responsewillpersisteven though the input is transient.
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Geometric Growth: PolesTheseunit-sampleresponsescanbecharacterizedbyasinglenumber the polewhich is thebaseof thegeometric sequence.
Delay
+p0
X Y
ny[n] =
p0, if n >= 0;0, otherwise.
y[n] y[n] y[n]
n n n
1 0 1 2 3 4 1 0 1 2 3 4 1 0 1 2 3 4p0 = 0.5 p0 = 1 p0 = 1.2
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Check YourselfHowmanyof the following unit-sample responses can berepresentedbya singlepole?
n n
n n
n
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Check YourselfHowmanyof the following unit-sample responses can berepresentedbya singlepole? 3
n n
n n
n
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Geometric GrowthThevalueof p0 determines the rateofgrowth.
y[n] y[n] y[n] y[n]
1 0 1z
p0 < 1:1< p0 < 0:
0< p0 < 1:p0 > 1:
magnitudediverges,alternating signmagnitudeconverges,alternating signmagnitudeconvergesmonotonicallymagnitudedivergesmonotonically
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Second-Order SystemsThe unit-sample responses ofmore complicated cyclic systems aremorecomplicated.
y[n]
+R
R1.6
0.63
X Y
n
1 0 1 2 3 4 5 6 7 8Notgeometric. This responsegrows thendecays.
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Factoring Second-Order SystemsFactortheoperatorexpressiontobreakthesystem intotwosimplersystems (divideandconquer).
+R
R1.6
0.63
X Y
Y =X+ 1.6RY0.63R2Y(11.6R+ 0.63R2)Y =X(1
0.7
R)(1
0.9
R)Y =X
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Factoring Second-Order SystemsThe
factored
form
corresponds
to
a
cascade
of
simpler
systems.
(10.7R)(10.9R)Y =X
X +0.7 R
+0.9 R
YY2
(10.7R)Y2 =X (10.9R)Y =Y2
+0.9
R+
0.7 RY
Y1X
(10.9R)Y1 =X (10.7R)Y =Y1
Theorderdoesntmatter (if systemsare initiallyat rest).
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Factoring Second-Order SystemsThe
unit-sample
response
of
the
cascaded
system
can
be
found
by
multiplying thepolynomial representationsof the subsystems.Y 1 1 1X
= (1
0.7R
)(1
0.9R
)= (1
0.7R
)
(1
0.9R
)
= (1 + 0.7R+ 0.72
R2+ 0.73
R3+
)
(1 + 0.9
R+ 0.92
R2+ 0.93
R3+
)
Multiply, thencollect termsofequalorder:Y = 1 + (0.7 + 0.9)R+ (0.72+ 0.7 0.9 + 0.92)R2X
+ (0.73+ 0.72 0.9 + 0.7 0.92+ 0.93)R3+
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Multiplying PolynomialGraphical
representation
of
polynomial
multiplication.
Y = (1 + aR+ a2R2+ a3R3+ ) (1 + bR+ b2R2+ b3R3+ )X1
a Ra2 R2
a3 R3
+
1b Rb2 R2
b3 R3
+
... ... ... ...
X Y
Collect termsofequalorder:Y = 1 + (a+ b)R+ (a
2
+ ab+ b2)R2
+ (a3+ a2b+ ab2+ b3)R3
+ X
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Multiplying PolynomialsTabular representationofpolynomialmultiplication.
(1 + aR+ a2R2+ a3R3+ ) (1 + bR+ b2R2+ b3R3+ )1 bR b2R2 b3R3
1 1 bR b2R2 b3R3 aR aR abR2 ab2R3 ab3R4
a2R2 a2R2 a2bR3 a2b2R4 a2b3R5 a3
R3 a3
R3 a3b
R4 a3b2
R5 a3b3
R6
Group samepowersofRby following reversediagonals:
Y = 1 + (a+ b)R+ (a2
+ ab+ b2)R2
+ (a3+ a2b+ ab2+ b3)R3
+ X y[n]
n1 0 1 2 3 4 5 6 7 8
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Partial FractionsUse
partial
fractions
to
rewrite
as
a
sum
of
simpler
parts.
+
R
R1.6
0.63
X Y
Y 1 1 4.5 3.5X = 11.6R+ 0.63R2 = (10.9R)(10.7R) = 10.9R 10.7R
S d O d S t E i l t F
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Second-Order Systems: Equivalent FormsThe
sum
of
simpler
parts
suggests
a
parallel
implementation.
Y 4.5 3.5X = 10.9R 10.7R
+0.9 R
4.5 +
3.5R0.7
+
X YY1
Y2
n nIf x[n] = [n] then y1[n] = 0.9 and y2[n] = 0.7 for n0.n
Thus, y[n] = 4.5(0.9)n3.5(0.7) for n0.
P ti l F ti
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Partial FractionsGraphical
representation
of
the
sum
of
geometric
sequences.
ny1[n] = 0.9 for n0n1 0 1 2 3 4 5 6 7 8
ny2[n] = 0.7 for n0n1 0 1 2 3 4 5 6 7 8
ny[n] = 4.5(0.9)n3.5(0.7)for n0
n1 0 1 2 3 4 5 6 7 8
Partial Fractions
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Partial FractionsPartial fractionsprovidesa remarkableequivalence.
R
R1.6
0.63
+X Y
+0.9
R4.5 +
3.5R0.7
+
X YY1
Y2
follows from thinkingabout systemaspolynomial (factoring).
Poles
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PolesThe
key
to
simplifying
a
higher-order
system
is
identifying
its
poles.
Poles are the roots of the denominator of the system functionalwhenR 1.zStartwith system functional:
Y 1 1 1= = =X 1 1.6R+0.63R2 (1p0R)(1p1R) (10.7R) (10.9R) p0=0.7 p1=0.91SubstituteR andfind rootsofdenominator:zY 1 z2 z2= = =X 1.6 0.63 z21.6z +0.63 (z0.7) (z0.9)1 + 2
z z z0=0.7 z1=0.9
Thepolesareat0.7and0.9.
Check Yourself
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Check YourselfConsider the systemdescribedby
y[n] =14 y[n1]+18
y[n2]+x[n1]12 x[n2]Howmanyof the followingare true?
1. Theunit sample responseconverges to zero.2. Therearepolesat z= 12 and z= 14 .3. There isapoleat z= 12 .4. Thereare twopoles.5. Noneof theabove
Check Yourself
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Check Yourself1 1 1y[n] =4
y[n 1] +8y[n 2]+x[n 1]2
x[n 2](1 +4
1R 81R2)Y = (R 2
1R2)XR R2R
12
1 121 1
2z 14zH(
R) =
YX
2zz
1 += = =11 +4 R218 14 1 18 1 18z2+ 2zz
z12z =
12
14z+
1. Theunit sample responseconverges to zero.12 and z= 14. X2. Therearepolesat z=12. X 3. There isapoleat z=
4. Thereare twopoles.5. Noneof theabove X
Check Yourself
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Check YourselfConsider the systemdescribedby
y[n] =14 y[n1]+18
y[n2]+x[n1]12 x[n2]Howmanyof the followingare true? 2
1. Theunit sample responseconverges to zero.2. Therearepolesat z= 12 and z= 14 .3. There isapoleat z= 12 .4. Thereare twopoles.5. Noneof theabove
Population Growth
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Population Growth
Population Growth
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Population Growth
Population Growth
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Population Growth
Population Growth
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opu at o G owt
Population Growth
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p
Population Growth
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p
Population Growth
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Population Growth
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Population Growth
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Population Growth
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Check Yourself
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Whatare thepole(s)of theFibonacci system?1.
1
2. 1and13. 1and24. 1.618. . . and0.618. . . 5. noneof theabove
Check Yourself
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Whatare thepole(s)of theFibonacci system?Differenceequation forFibonacci system:
y[n] =
x[n] +
y[n
1]
+
y[n
2]
System functional:
Y 1H= =X 1RR2Denominator is secondorder2poles.
Check Yourself
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Find thepolesby substitutingR 1/z in system functional.Y 1 1 z2H= = =X 1RR2 11z 12 z2 z 1z
Polesareat1 5 1z= =,2
where represents thegolden ratio1 + 5= 1.6182
The twopolesareat1z0 = 1.618 and z1 = 0.618
Check Yourself
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Whatare thepole(s)of theFibonacci system? 41. 12. 1and13. 1and24. 1.618. . . and0.618. . . 5. noneof theabove
Example: Fibonaccis Bunnies
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Eachpolecorresponds toa fundamentalmode.1 1.618 and 0.618
1 n
n
n
n
1 0 1 2 3 4 1 0 1 2 3 4Onemodediverges,onemodeoscillates!
Example: Fibonaccis Bunnies
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The unit-sample response of the Fibonacci system can be writtenasaweighted sumof fundamentalmodes.
1Y 1 5 5H= =
= +
X 1RR2 1 R 1 +1Rh[n] = n+ 1 ()n; n 05
5
Butwealreadyknow that h[n] is theFibonacci sequence f:
f : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, . . . Thereforewecancalculate f[n]withoutknowing f[n 1]or f[n 2]!
Complex Poles
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What ifapolehasanon-zero imaginarypart?Example:
Y 1=X 1R+R21 z2= =11z +z12 z2 z+ 1 Polesare z=
1
2 23j= ej/3.Whatare the implicationsofcomplexpoles?
Complex Poles
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Partial fractionsworkevenwhen thepolesarecomplex.
Y 1 1 1 ej/3 ej/3X = 1ej/3R 1ej/3R = j3 1ej/3R 1ej/3R
Thereare two fundamentalmodes (bothgeometric sequences):ejn/3 =cos(n/3) +jsin(n/3) and ejn/3 =cos(n/3)jsin(n/3)
n n
Complex Poles
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Complexmodesareeasier tovisualize in thecomplexplane.ejn/3 =cos(n/3)+jsin(n/3)
j2/3 Im ej1/3ej3/3 j0/3e e Re n
j4/3 j5/3e eejn/3 =cos(n/3)jsin(n/3)
j4/3 Im ej5/3ej3/3 j0/3e e Re n
j2/3 j1/3e e
Complex Poles
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Theoutputofareal systemhas realvalues.y[n] =x[n] +y[n 1] y[n 2]
Y 1H= =X 1R+R21 1= 1 ej/3R 1 ej/3R
1 ej/3 ej/3= j 3 1 ej/3
R 1 ej/3
Rh[n] = 1 ej(n+1)/3 ej(n+1)/3 =2 sin(n+ 1)j 3 3 3
h[n]1
1
n
Check Yourself
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Unit-sample responseofa systemwithpolesat z =rej.
n
Which
of
the
following
is/are
true?
1. r < 0.5and 0.52. 0.5< r < 1and 0.53. r < 0.5and 0.084. 0.5< r < 1and 0.085. noneof theabove
Check Yourself
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Unit-sample responseofa systemwithpolesat z =rej.
n
Whichof the following is/are true? 21. r < 0.5and 0.52. 0.5< r < 1and 0.53. r < 0.5and 0.084. 0.5< r < 1and 0.085. noneof theabove
Check Yourself
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R R R+X Y
Howmanyof the following statementsare true?1.This systemhas3 fundamentalmodes.2.Allof the fundamentalmodescanbewrittenasgeometrics.3.Unit-sample response is y[n] : 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 . . . 4.Unit-sample response is y[n] : 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 . . . 5.One of the fundamental modes of this system is the unitstep.
Check Yourself
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R R R+X Y
Howmanyof the following statementsare true? 41.This systemhas3 fundamentalmodes.2.Allof the fundamentalmodescanbewrittenasgeometrics.3.Unit-sample response is y[n] : 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 . . . 4.Unit-sample response is y[n] : 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1 . . . 5.One of the fundamental modes of this system is the unitstep.
SummaryS d f dd i d d l b h i d
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Systemscomposedofadders,gains,anddelayscanbecharacterizedby theirpoles.Thepolesofa systemdetermine its fundamentalmodes.Theunit-sampleresponseofasystemcanbeexpressedasaweightedsumof fundamentalmodes.Theseproperties follow fromapolynomial interpretationofthesystem functional.
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6.003 Signals and Systems
Spring 2010
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