ece3551 Microcomputer Systems : Complex Algebra Review.ppt

ece3551 Microcomputer Systems : Complex Algebra Review.ppt

1. Complex Algebra Review
Dr. V. Këpuska

2. September 22, 2025 Veton Këpuska 2
Complex Algebra Elements
Definitions:
 Note: Real numbers can be thought of as complex numbers with imaginary
part equal to zero.
CR
C
Ι
R
then If
NumbersComplex all ofSet :
NumbersImaginary all ofSet :
Numbers Real all ofSet :
1
number
complex a of
formCartezian


jyxzx,y
j


3. September 22, 2025 Veton Këpuska 3
Complex Algebra Elements


 z ofpart Imaginary
z ofpart Real
Im
Re
define then we If
0 If
0 If







zy
zx
jy xz
x zy
jy zx
R
I

4. September 22, 2025 Veton Këpuska 4
Euler’s Identity

j
ee
ee
je
je
je
jj
jj
j
j
j
2
sin
2
cos
sincos
sincos
sincos












 








5. September 22, 2025 Veton Këpuska 5
Polar Form of Complex Numbers
 Magnitude of a complex number z is a generalization of the absolute value
function/operator for real numbers. It is buy definition always non-negative.

z of argument)(or Angle z arg
z of Magnitude
radians ],-(
0r










z
rz
r
rez
j R

6. September 22, 2025 Veton Këpuska 6
Polar Form of Complex Numbers
Conversion between polar and
rectangular (Cartesian) forms.
For z=0+j0; called “complex zero” one can not define
arg(0+j0). Why?
  
 

































x
y
yxr
ry
rx
jy xjrr
jy xjr
jy xrez
j
1
22
tansin
cos
sincos
sincos






7. September 22, 2025 Veton Këpuska 7
Geometric Representation of
Complex Numbers.
Q1Q2
Q3 Q4
Im
Re
z
Re{z}
I
m
{
z
}
|
z
|

Complex or
Gaussian plane
Axis of
Reals
Axis of
Imaginaries

8. September 22, 2025 Veton Këpuska 8
Geometric Representation of
Complex Numbers.
Q1Q2
Q3 Q4
Im
Re
z
Re{z}
I
m
{
z
}
|
z
|

Complex or
Gaussian plane
Axis of
Reals
Axis of
Imaginarie
s
Complex
Number in
Quadrant
Condition 1 Condition 2
Q1 or Q2 Arg{z} ≥ 0 Im{z} ≥ 0
Q3 or Q4 Arg{z} ≤ 0 Im{z} ≤ 0
Q1 or Q4 Re{z} ≥ 0
Q2 or Q3 Re{z} ≤ 0

9. September 22, 2025 Veton Këpuska 9
Example

Im
Re
z
1 1
-1
-1-2
z
2
z
3
4
3
2
11
2
02
4
3
2
11
3
3
3
2
2
2
1
1
1












z
z
jz
z
z
jz
z
z
jz
{
{
{

10. September 22, 2025 Veton Këpuska 10
Conjugation of Complex Numbers
Definition: If z = x+jy ∈ C then z
*
= x-jy is called
the “Complex Conjugate” number of z.
Example: If z=re
j
(polar form) then what is z
*
also
in polar form?
 
   
  
 






j
j
rejrr
jrr
jrrz
jrrrez






sincos
coscos sincos
sinsin sincos
sincos


If z=re
j
then z*=re
-j

11. September 22, 2025 Veton Këpuska 11
Geometric Representation of
Conjugate Numbers
If z=re
j
then z*=re
-j
Im
Re
z
r

Complex or
Gaussian plane
-
r
x
y
-y
z
*

12. September 22, 2025 Veton Këpuska 12
Complex Number Operations
Extension of Operations for Real
Numbers
When adding/subtracting complex
numbers it is most convenient to use
Cartesian form.
When multiplying/dividing complex
numbers it is most convenient to use
Polar form.

13. September 22, 2025 Veton Këpuska 13
Addition/Subtraction of Complex
Numbers
  
 
 
2121
2121
212121
222111
III
ReReRe
:Thus

then
& ,
Let
zmzmzzm
zzzz
yyjxxzz
jyxzjyxz





14. September 22, 2025 Veton Këpuska 14
Multiplication/Division of Complex
Numbers


 

2121
2121
2121
212121
2211


:Therefore


then
&
Let
21
2121
21
zzzz
zzzz
errzz
eerrererzz
erzerz
j
jjjj
jj








 
21
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1


:Therefore


Olso
21
21
2
1
zz
z
z
z
z
z
z
e
r
r
z
z
ee
r
r
er
er
z
z
j
jj
j
j




































15. September 22, 2025 Veton Këpuska 15
Useful Identities
z ∈ C,  ∈ R & n ∈ Z (integer set)
  

 

   








































n
n
n
n
zz
znnzzz
zzzzz
zzzz
zzzzz
zzzz
z
z
z
z
zzzz
zzzz
zzzz
)16
)15)14
0 if
0 if 0
)13)12
ImIm)11ReRe)10
)9)8
)7
)6)5
ImIm)4ReRe)3
)2)1
2
2121
2
1
2
1
2121





16. September 22, 2025 Veton Këpuska 16
Useful Identities
Example: z = +j0
=2 then arg(2)=0
=-2 then arg(-2)=
Im
Re
j
-1-2
z
210


17. September 22, 2025 Veton Këpuska 17
Silly Examples and Tricks

 
 

  1012sin2cos
10
2
3
sin
2
3
cos
101sincos
10
2
sin
2
cos
1010sin0cos
2
2
3
2
0





























jje
jjje
jje
jjje
jje
j
j
j
j
j








Im
Re
j
-1 10

-j
/2
3/2
jjjjjjjj
jjjj
jjjjjjjj
jjjj




151173
141062
13951
12840
1111
1111

1
0
222
2





j
jj
j
eeejjj
ejjj

18. September 22, 2025 Veton Këpuska 18
Division Example
Division of two complex numbers in
rectangular form.
 
 
 
 
  
 
 
 
 
 

































2
1
22
2
1
22
22
2
2
Im
22
2112
Re
22
2121
2
1
22
21122121
22
22
22
11
22
11
2
1
222111 ,
z
z
z
z
zz
yx
yxyx
j
yx
yyxx
z
z
yx
yxyxjyyxx
jyx
jyx
jyx
jyx
jyx
jyx
z
z
jyxzjyxz

19. September 22, 2025 Veton Këpuska 19
Roots of Unity
Regard the equation:
z
N
-1=0, where z ∈ C & N ∈ Z
+
(i.e. N>0)
The fundamental theorem of algebra
(Gauss) states that an Nth degree algebraic
equation has N roots (not necessarily
distinct).
Example:
N=3; z
3
-1=0  z
3
=1 ⇒













)root 3(?
)root 2(?
)root 1(11
rd
3
nd
2
st
1
3
z
z
zz

20. September 22, 2025 Veton Këpuska 20
Roots of Unity
z
N
-1=0 has roots , k=0,1,..,N-1, where
The roots of
are called N
th
roots of unity.
N
j
e
2

1,...,1,0,
2
 Nke
N
k
j
k

k


21. September 22, 2025 Veton Këpuska 21
Roots of Unity
Verification:
 
 
 


1,...,1,0for trueis wich
02sin
12cos
012sin2cos
012sin2cos
2sin2cos
Identity Eulers Applying
0101
2
2
2





















Nk
k
k
jkjk
kjk
kjke
ee
kj
kj
N
N
k
j









22. September 22, 2025 Veton Këpuska 22
J
1
Geometric Representation
2
1
01
3
3
4
3
22
2
3
2
3
12
1
3
02
0




kee
kee
ke
N
jj
jj
j



Im
Re
1
-j1
-1
J
2
0
j1
2

/
3
4

/
3 J
0
2

/
3
2/3

23. September 22, 2025 Veton Këpuska 23
Important Observations
1.Magnitude of each root are equal to 1. Thus, the Nth roots of unity are
located on the unit circle. (Unit circle is a circle on the complex plane with
radius of 1).
2.The difference in angle between two consecutive roots is 2/N.
3.The roots, if complex, appear in complex-conjugate pairs. For example for
N=3, (J
1
)
*
=J
2
. In general the following property holds: J
N-k
=(J
k
)
*

ke
N
k
j
 ,1||
2
DEQ
N
e
N
j
k
k
kk
..
2
2
1
1
1





 





*2
*
1213
*
*
*
222
2
222
1&3For
1

















kN
eeeeeee
kkN
kN
k
j
N
k
j
N
k
j
jN
k
j
N
N
j
N
kN
j
kN


