Digital Signal Processing
Prof. Nizamettin AYDIN
[email protected]
http://www.yildiz.edu.tr/~naydin
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Digital Signal Processing
Lecture 3
Phasor Addition Theorem
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READING ASSIGNMENTS
• This Lecture:
– Chapter 2, Section 2-6
• Other Reading:
– Appendix A: Complex Numbers
– Appendix B: MATLAB
– Next Lecture: start Chapter 3
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LECTURE OBJECTIVES
• Phasors = Complex Amplitude
– Complex Numbers represent Sinusoids
z (t )  Xe
jt
j
 ( Ae )e
jt
• Develop the ABSTRACTION:
– Adding Sinusoids = Complex Addition
– PHASOR ADDITION THEOREM
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Z DRILL (Complex Arith)
MATLAB Command Name:
spfirst
zdrill
ZDrill is a program that
tests the users ability to
calculate the result of
simple operations on
complex numbers. The
program emphasizes the
vectorial view of a complex
number. The following six
operations are supported:
•Add
•Subtract
•Multiply
•Divide
•Inverse
•Conjugate
Complex drill
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AVOID Trigonometry
• Algebra, even complex, is EASIER !!!
• Can you recall cos(q1+q2) ?
(q
+q
)
j
1
2
• Use: real part of e
= cos(q +q )
1
e
j (q1 +q 2 )
jq1
e e
2
jq 2
 (cosq1 + j sin q1)(cosq 2 + j sin q 2 )
 (cosq1 cosq 2  sin q1 sin q 2 ) + j(...)
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Euler’s FORMULA
• Complex Exponential
– Real part is cosine
– Imaginary part is sine
– Magnitude is one
jq
e  cos(q ) + j sin( q )
e
jt
 cos( t ) + j sin(  t )
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Real & Imaginary Part Plots
PHASE DIFFERENCE
= p/2
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COMPLEX EXPONENTIAL
e
j t
 cos( t ) + j sin(  t )
• Interpret this as a Rotating Vector
q  t
Angle changes vs. time
ex: 20p rad/s
Rotates 0.2p in 0.01 secs
jq
e  cos(q ) + j sin(q )
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Rotating Phasor
See Demo on CD-ROM
Chapter 2
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Cos = REAL PART
Real Part of Euler’s
cos( t)  ee
j t

General Sinusoid
x(t)  Acos(t +  )
So,
A cos( t +  )  eAe

 
 eAe e 
j ( t + )
j
j t
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COMPLEX AMPLITUDE
General Sinusoid
x(t)  Acos( t +  )  eAe e
j j t

Sinusoid = REAL PART of (Aejf)ejt
x(t)  e Xe
j t
 ez(t)
Complex AMPLITUDE = X
z(t)  Xe
j t
X  Ae
j
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POP QUIZ: Complex Amp
• Find the COMPLEX AMPLITUDE for:
x(t )  3 cos(77p t + 0.5p )
• Use EULER’s FORMULA:

 e 3e
x(t )  e 3e
X  3e
j ( 77p t +0.5p )
j 0.5p
e
j 77p t


j 0.5p
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WANT to ADD SINUSOIDS
• ALL SINUSOIDS have SAME FREQUENCY
• HOW to GET {Amp,Phase} of RESULT ?
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ADD SINUSOIDS
• Sum Sinusoid has SAME Frequency
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PHASOR ADDITION RULE
Get the new complex amplitude by complex addition
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Phasor Addition Proof
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POP QUIZ: Add Sinusoids
• ADD THESE 2 SINUSOIDS:
x1 (t )  cos(77p t )
x2 (t )  3 cos(77p t + 0.5p )
• COMPLEX ADDITION:
1e + 3e
j0
j 0.5p
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POP QUIZ (answer)
• COMPLEX ADDITION:
1 + j 3  2e
j 3  3e
jp / 3
j 0.5p
1
• CONVERT back to cosine form:
x3 (t )  2 cos( 77p t +
p)
3
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ADD SINUSOIDS EXAMPLE
x1 (t )
tm1
x2 (t )
tm2
x3 (t )  x1 (t ) + x2 (t )
tm3
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Convert Time-Shift to Phase
• Measure peak times:
tm1=-0.0194, tm2=-0.0556, tm3=-0.0394
• Convert to phase (T=0.1)
f1=-tm1 = -2p(tm1 /T) = 70p/180,
f2= 200p/180
• Amplitudes
A1=1.7, A2=1.9, A3=1.532
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Phasor Add: Numerical
• Convert Polar to Cartesian
X1 = 0.5814 + j1.597
X2 = -1.785 - j0.6498
sum =
X3 = -1.204 + j0.9476
• Convert back to Polar
X3 = 1.532 at angle 141.79p/180
This is the sum
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ADD SINUSOIDS
X1
VECTOR
(PHASOR)
ADD
X3
X2
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