(chapter 2. complex variables)

Hansung University Dept. of Information and Communication Eng.1 1Dept. Information and Communication Eng.

Signal and System (Chapter 2. Complex Variables)

Prof. Kwang-Chun [email protected]: 02-760-4253 Fax:02-760-4435


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


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


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



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



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


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


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


( Velocity and Position of Sine and Cosine)

Euler’s Identity


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


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



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


1 1 1z x jy 2 2 2z x jy


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:


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


Add the real parts and the imaginary parts:

Convert back to polar form, obtaining

Therefore, the final formula is


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



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


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


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;


x

y

01

23

45

6

0

0.2

0.4

0.6

0.8

1-2

-1

0

1

2


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



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



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 !



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


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 (Determinate Value)

( )( )( )

n xf xd x

( )( ) lim ( ) lim ( )( )x a x a

n af a f x f xd a

( )n x ( )d x



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

6(3)0

, F (Singularitiesor Pole)



[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

(chapter 2. complex variables)

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