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Copyright Monash University 2009
Signal Processing
First
Lecture2Phase&
TimeShiftComplex
Exponentials
1
Copyright Monash University 2009
READING ASSIGNMENTS
ThisLecture: Chapter2,Sects.23to25
AppendixA:ComplexNumbers AppendixB:MATLAB NextLecture:finishChap.2,
Section26toend
2
Copyright Monash University 2009
LECTURE OBJECTIVES
DefineSinusoidFormulafromaplot RelateTIMESHIFTtoPHASE
3
tjXetz )(
Introduce an ABSTRACTION:Complex Numbers represent SinusoidsComplex Exponential Signal
Copyright Monash University 2009
SINUSOIDAL SIGNAL
FREQUENCY Radians/sec or,Hertz(cycles/sec)
PERIOD (insec)
AMPLITUDE Magnitude
PHASE
4
)cos( tA
f)2(
21
f
T
A
Copyright Monash University 2009
PLOTTING COSINE SIGNAL from the FORMULA
Determineperiod:
Determineapeak locationbysolving
Peakatt=4
5
)2.13.0cos(5 t
0)( t
3/203.0/2/2 T
Copyright Monash University 2009
ANSWER for the PLOT
UseT=20/3andthepeaklocationatt=4
6
)2.13.0cos(5 t
320
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TIME-SHIFT
Inamathematicalformulawecanreplacetwithttm
Thenthet=0pointmovestot=tm
Peakvalueofcos((ttm))isnowatt=tm
7
))(cos()( mm ttAttx
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TIME-SHIFTED SINUSOID
8
))4((3.0cos(5))4(3.0cos(5)4( tttx
Copyright Monash University 2009
PHASE TIME-SHIFT
Equatetheformulas:
andweobtain:
or,
9
)cos())(cos( tAttA m mt
mt
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SINUSOID from a PLOT
Measure theperiod,T Betweenpeaksorzerocrossings
Compute frequency: =2/T
Measure timeofapeak:tm Compute phase: =tm
Measure heightofpositivepeak:A
10
3 steps
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(A, , ) from a PLOT
11
25.0))(200( mm tt
20001.022 T1001period1 sec01.0 T
sec00125.0mt
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SINE DRILL (MATLAB GUI)
12
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PHASE is AMBIGUOUS
Thecosinesignalisperiodic Periodis2
Thusaddinganymultipleof2 leavesx(t)unchanged
13
Ttt
t
mm
m
2)2(2
then, if
)cos()2cos( tAtA
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COMPLEX NUMBERS
Tosolve:z2 =1 z=j MathandPhysicsusez=i
Complexnumber:z=x+jy
14
x
y zCartesiancoordinatesystem
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PLOT COMPLEX NUMBERS
15
Copyright Monash University 2009
COMPLEX ADDITION = VECTOR Addition
1626
)53()24()52()34(
213
jj
jjzzz
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*** POLAR FORM ***
VectorForm Length =1 Angle =
CommonValues j hasangleof0.5 1hasangleof j hasangleof1.5 also,angleofjcould be0.51.52 becausethePHASEisAMBIGUOUS
17
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POLAR RECTANGULAR
Relate(x,y)to(r,)
18
r
x
y
Need a notation for POLAR FORM
xyyxr
1
222
Tan
sincos
ryrx
Most calculators doPolar-Rectangular
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Eulers FORMULA
ComplexExponential Realpartiscosine Imaginarypartissine Magnitudeisone
19
)sin()cos( jrrre j
)sin()cos( je j
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COMPLEX EXPONENTIAL
Interpretthisasa RotatingVector t Anglechangesvs.time ex:rad/s Rotates in0.01secs
20
)sin()cos( tjte tj
)sin()cos( je j
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cos = REAL PART
21
Real Part of Eulers}{)cos( tjeet
General Sinusoid )cos()( tAtx
So,
}{}{)cos( )(
tjj
tj
eAeeAeetA
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REAL PART EXAMPLE
22
Answer:
Evaluate:
tjj eAeetA )cos( tjjeetx 3)(
)5.0cos(33
)3()(5.0
teee
ejetxtjj
tj
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COMPLEX AMPLITUDE
23
Then, any Sinusoid = REAL PART of Xejt
tjjtj eAeeXeetx )(
General Sinusoid
tjj eAeetAtx )cos()(Complex AMPLITUDE = X
jtj AeXXetz )(