Download presentation

Presentation is loading. Please wait.

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

Similar presentations

Presentation is loading. Please wait....

OK

6.6 The Fundamental Theorem of Algebra

6.6 The Fundamental Theorem of Algebra

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on media research center Ppt on review of related literature powerpoint Ppt on national education day in the us Ppt on paintings and photographs related to colonial period dates Ppt on main bodies of uno Ppt on c language fundamentals Ppt on reported speech in english grammar Ppt on water and air pollution Ppt on council of ministers eu Ppt on life history of william shakespeare