lecture # ch2-3 complex exponentials & complex numbers€¦ · lecture objectives introduce...
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DSP-First, 2/e
LECTURE # CH2-3
Complex Exponentials
& Complex Numbers
Aug 2016 1© 2003-2016, JH McClellan & RW Schafer
TLH MODIFIED
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 2
READING ASSIGNMENTS
This Lecture:
Chapter 2, Sects. 2-3 to 2-5
Appendix A: Complex Numbers
Complex Exponentials
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LECTURE OBJECTIVES
Introduce more tools for manipulating
complex numbers Euler Eq.
Conjugate
Multiplication & Division
Powers
N-th Roots of unity
Aug 2016 3© 2003-2016, JH McClellan & RW Schafer
re j rcos() jrsin()
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 4
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LECTURE OBJECTIVES
Phasors = Complex Amplitude
Complex Numbers represent Sinusoids
Take Real or Complex part
}){()cos( tjj eAetA
Aug 2016 5© 2003-2016, JH McClellan & RW Schafer
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Phasors Not Phasers
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 6
Captain Kirk &
Sally Kellerman STAR TREK
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WHY? What do we gain?
Sinusoids are the basis of DSP,
but trig identities are very tedious
Abstraction of complex numbers
Represent cosine functions
Can replace most trigonometry with algebra
Avoid (Most) all Trigonometric
manipulations
Aug 2016 7© 2003-2016, JH McClellan & RW Schafer
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COMPLEX NUMBERS
To solve: z2 = -1
z = j
Math and Physics use z = i
Complex number: z = x + j y
x
y z
Cartesian
coordinate
system
Aug 2016 8© 2003-2016, JH McClellan & RW Schafer
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PLOT COMPLEX NUMBERS
}{zx
}{zy
Real part:
Imaginary part:}05{5 j
5}52{ j
2}52{ j
Aug 2016 9© 2003-2016, JH McClellan & RW Schafer
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COMPLEX ADDITION = VECTOR Addition
26
)53()24(
)52()34(213
j
j
jj
zzz
Aug 2016 10© 2003-2016, JH McClellan & RW Schafer
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*** POLAR FORM ***
Vector Form
Length =1
Angle =
Common Values j has angle of 0.5p
1 has angle of p
j has angle of 1.5p
also, angle of j could be 0.5p1.5p2p
because the PHASE is AMBIGUOUS
Aug 2016 11© 2003-2016, JH McClellan & RW Schafer
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POLAR <--> RECTANGULAR
Relate (x,y) to (r,) r
x
y
Need a notation for POLAR FORM
x
y
yxr
1
222
Tan
sin
cos
ry
rx
Most calculators do
Polar-Rectangular
Aug 2016 12© 2003-2016, JH McClellan & RW Schafer
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Euler’s FORMULA
Complex Exponential
Real part is cosine
Imaginary part is sine
Magnitude is one
re j rcos() jrsin()
)sin()cos( je j
Aug 2016 13© 2003-2016, JH McClellan & RW Schafer
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Cosine = Real Part
Complex Exponential
Real part is cosine
Imaginary part is sine
)sin()cos( jrrre j
)cos(}{ rre j
Aug 2016 14© 2003-2016, JH McClellan & RW Schafer
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Common Values of exp(j)
Changing the angle
p njj eej 200110
4/21 pjej
ppp )2/12(2/2/ njj eej
ppp )12(011 njj eej
pppp )2/12(2/2/32/3 njjj eeej
Aug 2016 15© 2003-2016, JH McClellan & RW Schafer
𝑎2 + 𝑏2 = 𝑐2 θ = arctan(𝑏
𝑎)
IF 𝑎 ≥ 0
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COMPLEX EXPONENTIAL
Interpret this as a Rotating Vector
t
Angle changes vs. time
ex: 20prad/s
Rotates 0.2p in 0.01 secs
)sin()cos( tjte tj
)sin()cos( je j
Aug 2016 16© 2003-2016, JH McClellan & RW Schafer
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Cos = REAL PART
}{
}{)cos( )(
tjj
tj
eAe
AetA
So,
Real Part of Euler’s
}{)cos( tjet
General Sinusoid
)cos()( tAtx
Aug 2016 17© 2003-2016, JH McClellan & RW Schafer
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COMPLEX AMPLITUDE
}{)cos()( tjj eAetAtx
General Sinusoid
)}({}{)( tzXetx tj
Sinusoid = REAL PART of complex exp: z(t)=(Aejf)ejt
Complex AMPLITUDE = X, which is a constant
tjj XetzAeX )(when
Aug 2016 18© 2003-2016, JH McClellan & RW Schafer
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POP QUIZ: Complex Amp
Find the COMPLEX AMPLITUDE for:
Use EULER’s FORMULA:
p5.03 jeX
)5.077cos(3)( pp ttx
}3{
}3{)(
775.0
)5.077(
tjj
tj
ee
etx
pp
pp
Aug 2016 19© 2003-2016, JH McClellan & RW Schafer
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POP QUIZ-2: Complex Amp
Determine the 60-Hz sinusoid whose
COMPLEX AMPLITUDE is:
Convert X to POLAR:
33 jX
)3/120cos(12)( pp ttx
}12{
})33{()(
1203/
)120(
tjj
tj
ee
ejtx
pp
p
Aug 2016 20© 2003-2016, JH McClellan & RW Schafer
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Aug 2016 © 2003-2016, JH McClellan & RW Schafer 21
Note atan (3/√3) –is atan(√3) = π/3
Remember 60 degree angle
Cos = 1/2 Sin = √3/2
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COMPLEX CONJUGATE (z*)
Useful concept: change the sign of all j’s
RECTANGULAR: If z = x + j y, then the
complex conjugate is z* = x – j y
POLAR: Magnitude is the same but angle
has sign change
z re j z* re j
Aug 2016 22© 2003-2016, JH McClellan & RW Schafer
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COMPLEX CONJUGATION
Flips vector
about the real
axis!
Aug 2016 23© 2003-2016, JH McClellan & RW Schafer
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USES OF CONJUGATION
Conjugates useful for many calculations
Real part:
Imaginary part:
}{2
)()(
2
*zx
jyxjyxzz
}{2
2
2
*zy
j
yj
j
zz
Aug 2016 24© 2003-2016, JH McClellan & RW Schafer
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Inverse Euler Relations
Cosine is real part of exp, sine is imaginary part
Real part:
Imaginary part:
)cos(2
}{,
}{2
*
jjjj ee
eez
zzz
)sin(2
}{,
}{2
*
j
eeeez
zyj
zz
jjjj
Aug 2016 25© 2003-2016, JH McClellan & RW Schafer
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Mag & Magnitude Squared
Magnitude Squared (polar form):
Magnitude Squared (Cartesian form):
Magnitude of complex exponential is one:
22 ||))((* zrrerezz jj
|e j |2 cos2 sin2 1
zz* (x jy) (x jy) x2 j2y2 x2 y2
Aug 2016 26© 2003-2016, JH McClellan & RW Schafer
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COMPLEX MULTIPLY =
VECTOR ROTATION
Multiplication/division scales and rotates
vectors
Aug 2016 27© 2003-2016, JH McClellan & RW Schafer
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POWERS
Raising to a power N rotates vector by Nθ
and scales vector length by rN
zN re j N
rNe jN
Aug 2016 28© 2003-2016, JH McClellan & RW Schafer
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MORE POWERS
Aug 2016 29© 2003-2016, JH McClellan & RW Schafer
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ROOTS OF UNITY
We often have to solve zN=1
How many solutions?
kjNjNN eerz p 21
N
kkNr
pp
22,1
1,2,1,0,2 Nkez Nkj p
Aug 2016 30© 2003-2016, JH McClellan & RW Schafer
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ROOTS OF UNITY for N=6
Solutions to zN=1
are N equally
spaced vectors on
the unit circle!
What happens if
we take the sum
of all of them?
Aug 2016 31© 2003-2016, JH McClellan & RW Schafer
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Sum the Roots of Unity
Looks like the answer is zero (for N=6)
Write as geometric sum
e j2p k /N
k 0
N1
0?
rk
k0
N1
1 rN
1 rthen let r e j2p /N
01)(11Numerator 2/2 pp jNNjN eer
Aug 2016 32© 2003-2016, JH McClellan & RW Schafer
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Needed later to describe periodic signals
in terms of sinusoids (Fourier Series) Especially over one period
j
ee
j
ede
jajbb
a
b
a
jj
0110/2
0
/2
jj
eedte
jTTjT
Ttjp
p
Integrate Complex Exp
Aug 2016 33© 2003-2016, JH McClellan & RW Schafer
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BOTTOM LINE
CARTESIAN: Addition/subtraction is most
efficient in Cartesian form
POLAR: good for multiplication/division
STEPS:
Identify arithmetic operation
Convert to easy form
Calculate
Convert back to original form
Aug 2016 34© 2003-2016, JH McClellan & RW Schafer
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Review Appendix A and
TLH Ch2 on Course Website
Aug 2016 © 2003-2016, JH McClellan & RW Schafer 43
Harman Chapter 2 Pages 55-60 Complex Numbers
and
MATLAB Complex Numbers