1. Complex Algebra Review
Digital Systems: Hardware Organization and Design
4/1/2017
Complex Algebra Review
Dr. V. Këpuska
Architecture of a Respresentative 32 Bit Processor

2. Complex Algebra Elements
Digital Systems: Hardware Organization and Design
4/1/2017
Complex Algebra Elements
Definitions:
Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero.
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

3. Complex Algebra Elements
Digital Systems: Hardware Organization and Design
4/1/2017
Complex Algebra Elements
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

4. Digital Systems: Hardware Organization and Design
4/1/2017
Euler’s Identity
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

5. Polar Form of Complex Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
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.
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

6. Polar Form of Complex Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
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?
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

7. Geometric Representation of Complex Numbers.
Digital Systems: Hardware Organization and Design
4/1/2017
Geometric Representation of Complex Numbers.
Axis of Imaginaries Im Q2 Q1 z
Axis of Reals
|z|
Im{z} 
Re{z} Re Q3 Q4
Complex or Gaussian plane
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

8. Geometric Representation of Complex Numbers.
Digital Systems: Hardware Organization and Design
4/1/2017
Geometric Representation of Complex Numbers.
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
Axis of Imaginaries Im Q2 Q1 z
Axis of Reals
|z|
Im{z} 
Re{z} Re Q3 Q4
Complex or Gaussian plane
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

9. Digital Systems: Hardware Organization and Design
4/1/2017
Example Im Re z1 1 -1 -2 z2 z3
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

10. Conjugation of Complex Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
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=rej (polar form) then what is z* also in polar form?
If z=rej then z*=re-j
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

11. Geometric Representation of Conjugate Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
Geometric Representation of Conjugate Numbers
If z=rej then z*=re-j Im z y r  x - Re r -y z*
Complex or Gaussian plane
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

12. Complex Number Operations
Digital Systems: Hardware Organization and Design
4/1/2017
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.
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

13. Addition/Subtraction of Complex Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
Addition/Subtraction of Complex Numbers
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

14. Multiplication/Division of Complex Numbers
Digital Systems: Hardware Organization and Design
4/1/2017
Multiplication/Division of Complex Numbers
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

15. Digital Systems: Hardware Organization and Design
4/1/2017
Useful Identities
z ∈ C,  ∈ R & n ∈ Z (integer set)
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

16. Digital Systems: Hardware Organization and Design
4/1/2017
Useful Identities
Example: z = +j0
=2 then arg(2)=0
=-2 then arg(-2)= Im j  z -2 -1 1 2 Re
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

17. Silly Examples and Tricks
Digital Systems: Hardware Organization and Design
4/1/2017
Silly Examples and Tricks Im j 
/2 -1
3/2 1 Re -j
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

18. Digital Systems: Hardware Organization and Design
4/1/2017
Division Example
Division of two complex numbers in rectangular form.
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

19. Digital Systems: Hardware Organization and Design
4/1/2017
Roots of Unity
Regard the equation:
zN-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; z3-1=0  z3=1 ⇒
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

20. Digital Systems: Hardware Organization and Design
4/1/2017
Roots of Unity
zN-1=0 has roots , k=0,1,..,N-1, where
The roots of are called Nth roots of unity.
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

21. Digital Systems: Hardware Organization and Design
4/1/2017
Roots of Unity
Verification:
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

22. Geometric Representation
Digital Systems: Hardware Organization and Design
4/1/2017
Geometric Representation Im j1 J1
2/3 J0
4/3
2/3 -1 1
2/3 Re J2
-j1
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor

23. Important Observations
Digital Systems: Hardware Organization and Design
4/1/2017
Important Observations
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).
The difference in angle between two consecutive roots is 2/N.
The roots, if complex, appear in complex-conjugate pairs. For example for N=3, (J1)*=J2. In general the following property holds: JN-k=(Jk)*
1 April 2017
Veton Këpuska
Architecture of a Respresentative 32 Bit Processor