(Chapter 2. Complex Variables) - HANSUNG · 2016. 3. 2. · Example 2.5: Find the sum of two...
Transcript of (Chapter 2. Complex Variables) - HANSUNG · 2016. 3. 2. · Example 2.5: Find the sum of two...
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Signal and System (Chapter 2. Complex Variables)
Prof. Kwang-Chun [email protected]: 02-760-4253 Fax:02-760-4435
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Origin of complex numbers :Who first thought up complex numbers?
Complex numbers were being used by mathematicians long before they were first properly defined, so it's difficult to trace the exact origin
The first reference (but there may be earlier ones) is by Cardan in 1545. Then, the notation was used in the sense of a convenient fiction to categorize the properties of some polynomials
Later Euler in 1777 first introduces the notation i and -i for the two different square roots of –1, and the notation a + bi for complex numbers
1
The Number System
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the numbers i and -i were called "imaginary", because their existence was still not clearly understood
Wessel in 1797 and Gauss in 1799 used the geometric interpretation of complex numbers as points in a plane, which made them somewhat more concrete and less mysterious
Rectangular/Polar Conversions
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Rectangular form of complex number:
Polar form : where and
Similarly,
z a j b (Real part of z)
(Imaginary part of z)z c
2 2c a b 1tan ba
Quadrant a b 1st 2nd 3rd 4th
cos , sina c b c
Rectangular/Polar Conversions
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Example 2.1:Find the polar form of complex number
Solution:As shown in figure, we have
MatLab code:
[Angle, Radius] = cart2pol(x,y)
1.5 0.5oz j
221.5 0.5 1.581c
1 0.5tan 18.431.5
18.43
Rectangular/Polar Conversions
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Definition:
A few laws of exponents:
Mathematical properties of Euler’s identity:
Example 2.2:
cos sinje j 2 2 1cos sin tan (tan ) 1je
(Rectangular form)
(Polar form)
, , 1/y xj x yjx jy jx jxy jy jx jxe e e e e e e e
2cos2 sin
j j
j j
e ee e j
cos , sin
2 2
j j j je e e ej
2 2( ) 2 4 4 2j x jx jx j xf x e e e e
2 2( ) 2 4
4cos 2 8cos
j x j x jx jxf x e e e e
x x
Express in terms of cosine function
Euler’s Identity
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Example 2.3: Plot a sinusoidal exponential signalcos sinjte t j t
fs = 500; %Sample rate (Hz)t = -10:1/fs:30; % Time index (s) y = exp(j*t);plot3(t,imag(y),real(y),'b');hold on;plot3(t,ones(size(t)),real(y),'r');plot3(t,imag(y),-ones(size(t)),'g');hold off;grid on;xlabel('Time (s)');ylabel('Imaginary Part');zlabel('Real Part');title('Complex:Blue,Real:Red,Imaginary:Green');
Euler’s Identity
-100
1020
30
-1-0.5
00.5
1-1
-0.5
0
0.5
1
Time (s)
Complex:Blue,Real:Red,Imaginary:Green
Imaginary Part
Rea
l Par
t
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tr
Re
Im
imy
rey
r
r
t
2
imy
rr
t
2
rey
( Math. Reference: Relation between Sinusoidal Signal and Complex Exponential Signal )
( ) cos sinj ty t re r t jr t
=2f is the angular frequency in rad/sec
f is the signal frequency in cycles per second or Hz
( )imy t( )rey t
(Phase vs. Angular Frequency) Phase, , is angle, usually
represented in radians
(circumference of unit circle) Angular Frequency, , is the
rate of change for phase
t
2 [radians] 360
Euler’s Identity
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( Velocity and Position of Sine and Cosine)
Euler’s Identity
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Addition/Subtraction: If and , then
Similarly,
If complex numbers are in polar form, add them after converting to rectangular forms
Multiplication: If and , then
If they are in polar form, it is particularly easyIf and , then
1 1 1z x jy 2 2 2z x jy 1 2 1 2 1 2z z x x j y y
1 2 1 2 1 2z z x x j y y
1 2 1 1 2 2 1 2 1 2 1 2 1 2z z x jy x jy x x y y j x y y x
1 1 1z c 2 2 2z c 1 21 21 2 1 2 1 2
jj jz z c e c e c c e
1 1 1z x jy 2 2 2z x jy
Complex-Number Arithmetic
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Repeated multiplication:
Complex conjugation: If , then ; or if , then
Conjugation Theorem: If z is an arithmetic expression of complex numbers, z* may be formed simply by replacing every j with –j
Mathematical properties:Addition:
Subtraction:
Multiplication:
nn j n jn nz ce c e c n
z a jb z a jb jz ce jz ce
2 2 Re( )z z a z
2 2 Im( )z z jb j z
2 2zz a jb a jb a b
Complex-Number Arithmetic
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Division: If and , then
If the numbers are in polar form, it is so easy!!
Example 2.4: If and , then find
Solution:
1 1 2 2 1 2 1 2 1 2 1 21 1 12 2
2 2 2 2 2 2 2 2 2
x jy x jy x x y y j y x x yz x jyz x jy x jy x jy x y
1
1 2
2
1 1 1 1 1
2 2 2 2 2
jj
j
z c c e c ez c c e c
1 21
2
z z zz
1 1 3z j 2 2 1z j
21 3 2 1 1 3 2 1
1 3 1 32 1 5
4 25 5
j j j jj j
j
j
rationalizing
Complex-Number Arithmetic
1 1 1z x jy 2 2 2z x jy
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Phasor Addition Rule:The phasor representation of cosine signals can be
used to show the following result
Example 2.5: Find the sum of two signals
Solution:Represent and by the phasors:
Convert both phasors to rectangular form:
Complex-Number Arithmetic
1
( ) cos cosN
k kk
x t A t A t
1 2( ) 1.7cos 120 70 /180 , ( ) 1.9cos 120 200 /180x t t x t t
1( )x t 2 ( )x t1 270 /180 200 /180
1 1 2 21.7 , 1.9j jj jX A e e X A e e
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Add the real parts and the imaginary parts:
Convert back to polar form, obtaining
Therefore, the final formula is
Complex-Number Arithmetic
1 20.5814 1.597, 1.785 0.6498X j X j
3 1 2
0.5814 1.597 1.785 0.6498 1.204 0.9476X X X
j j j
141.79 /1803 1.532 jX e
3 1 2( ) ( ) ( ) 1.532cos 120 141.79 /180x t x t x t t
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Complex-Number Arithmetic
141.79 /1803 1.532 jX e
3 1 2( ) ( ) ( ) 1.532cos 120 141.79 /180x t x t x t t
Adding sinusoids by doing a phasor addition, which is actually a graphical vector sum.
The time of the signal maximum is marked on each plot( )ix t
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A function of a complex variable is complex function, which has a real and an imaginary parts like complex number
Example 2.6: If , where , under what conditions,
if any, does ?
Solution: dividing function into real and imaginary parts and solving give
( ) 6 /F z z z x jy
Re( ) 3F
2 2 2 2
6 6 6( ) x yF z jx jy x y x y
=3
2 21 1x y
Circle centered at z=1+j0
Function of a Complex Variable
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Also, we can visualize the function, and plot each part separately over z-plane
Example 2.7:Plot the real part of complex function ( ) sin( )F z z
clear all;
x = 0:0.05:6;y = 0:0.05:1;
[X,Y] = meshgrid(x,y);Z = real(sin(X+i*Y));
mesh(X,Y,Z); grid on;
Function of a Complex Variable
x
y
01
23
45
6
0
0.2
0.4
0.6
0.8
1-2
-1
0
1
2
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However, when we hope to visualize a function over entire complex number plane, a pole-zero diagramprovides the informationPole: Locate where function
is infiniteIndicate with an “X”
Zero: Locate where functionis zero
Indicate with an “O”
Multiplicity of root likeIndicate by placing the value of n
(Double Zero)
( )nz a
Function of a Complex Variable
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Example 2.8: Provide a pole-zero plot for the following function
Solution:Placing F in factored form gives
MatLab code: zplane(Z,P)
Example 2.9:Find the magnitude of
along the path
Solution:
2
2
2 4( )
2
zF z
z z z
2 2 2( )
1 2z j z j
F zz z z
Poles: z=0, +1, or –2Zeros: z=+j2 or –j2
2 4( )
1z
F zz
2z x j
Zeros Z and Poles P are in column vectors
Function of a Complex Variable
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It seems that it will dip as passing a zero, and peak as passing a pole
Let’s evaluate F along the path:
2
2
2 2 4 2 6 20( 2)2 1 2 5
x j x x jF x jx j x x
Note that the maximum and minimum points do not correspond to the points of pole and zero !
Function of a Complex Variable
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What is indeterminate values ? If a function takes the form at a
point, we say that the value of the function is indeterminate at that pointWill interpret this as meaning that the function is hiding
its true value from us at this point
So we should investigate to find the appropriate value
As example, what is the value of at m=0 ?The value is indeterminate at m=0 because F(0)=0/0
If so, how can get the determinate value at m=0 ?
0 / 0, 0 , , /
sin( )
amF m
m
Indeterminate Value
Indeterminate Values
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Answer: After integration
Before integration
A method to have a determinate value from indeterminate value is L’Hopital’s Rule If becomes indeterminate at x=a, then
where and are the derivatives of n(x) and d(x)evaluated at x=a
0
0
sin 0( ) cos( )0
a
m
amF m mx dx
m
0 0( 0) cos(0 ) 1
a aF m x dx dx a= = ⋅ = =ò ò(Determinate Value)
( )( )( )
n xf xd x
( )( ) lim ( ) lim ( )( )x a x a
n af a f x f xd a
( )n a ( )d a
Indeterminate Values
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Example 2.10:Evaluate F(z) at z=1, 2, and 3, where
Solution: Substituting directly, we find
First method to solve these problems: Use factorization
Second method (preferred one) : Use L’Hopital’s rule (Can determine the indeterminate
value !)
2
2
3 2( ) 35 6
z zF zz z
0(1) 0, 0(2) ?02
FF (Indeterminate Values),
2
2
1 23 2 1( ) 3 3 35 6 2 3 3
z zz z zF zz z z z z
(2) 3F
2
2 3lim ( 3) 32 5x a
z
zF zz
(Singularitiesor Pole)
Indeterminate Values
6(3)0
F
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[Problem 1]
Use MatLab to plot the phase of along the path
[Problem 2]
Sketch a fully labeled pole-zero diagram for the following complex function:
Obtain expression for the real and imaginary parts of along the path , and sketch each for
Identify actual values at
[Problem 3]
Evaluate at z=0, 2, and 10:
2 4( )
1z
F zz
2z x j
2( )2
F zz
( )F z1z jy
1y
( )F z3 2
2
6 40( )12 20
z z zF zz z
Homework Assignment #2