Complex Algebra Review Digital Systems: Hardware Organization and Design 2/25/2019 Complex Algebra Review Dr. V. Këpuska Architecture of a Respresentative 32 Bit Processor

Complex Algebra Elements Digital Systems: Hardware Organization and Design 2/25/2019 Complex Algebra Elements Definitions: Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Complex Algebra Elements Digital Systems: Hardware Organization and Design 2/25/2019 Complex Algebra Elements 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Euler’s Identity 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Polar Form of Complex Numbers Digital Systems: Hardware Organization and Design 2/25/2019 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. 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Polar Form of Complex Numbers Digital Systems: Hardware Organization and Design 2/25/2019 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? 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Geometric Representation of Complex Numbers. Digital Systems: Hardware Organization and Design 2/25/2019 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 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Geometric Representation of Complex Numbers. Digital Systems: Hardware Organization and Design 2/25/2019 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 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Example Im Re z1 1 -1 -2 z2 z3 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Conjugation of Complex Numbers Digital Systems: Hardware Organization and Design 2/25/2019 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=ej (polar form) then what is z* also in polar form? If z=rej then z*=re-j 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Geometric Representation of Conjugate Numbers Digital Systems: Hardware Organization and Design 2/25/2019 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 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Complex Number Operations Digital Systems: Hardware Organization and Design 2/25/2019 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. 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Addition/Subtraction of Complex Numbers Digital Systems: Hardware Organization and Design 2/25/2019 Addition/Subtraction of Complex Numbers 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Multiplication/Division of Complex Numbers Digital Systems: Hardware Organization and Design 2/25/2019 Multiplication/Division of Complex Numbers 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Useful Identities z ∈ C,  ∈ R & n ∈ Z (integer set) 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Useful Identities Example: z = +j0 =2 then arg(2)=0 =-2 then arg(-2)= Im j  z -2 -1 1 2 Re 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Silly Examples and Tricks Digital Systems: Hardware Organization and Design 2/25/2019 Silly Examples and Tricks Im j  /2 -1 3/2 1 Re -j 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Division Example Division of two complex numbers in rectangular form. 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 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 ⇒ 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Roots of Unity zN-1=0 has roots , k=0,1,..,N-1, where The roots of are called Nth roots of unity. 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Digital Systems: Hardware Organization and Design 2/25/2019 Roots of Unity Verification: 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Geometric Representation Digital Systems: Hardware Organization and Design 2/25/2019 Geometric Representation Im j1 J1 2/3 J0 4/3 2/3 -1 1 2/3 Re J2 -j1 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor

Important Observations Digital Systems: Hardware Organization and Design 2/25/2019 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)* 25 February 2019 Veton Këpuska Architecture of a Respresentative 32 Bit Processor