Speech Processing Complex Algebra Review Veton Kpuska Complex

Speech Processing Complex Algebra Review Veton Këpuska

Complex Algebra Elements u Definitions: u Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. 10/3/2020 Veton Këpuska 2

Complex Algebra Elements 10/3/2020 Veton Këpuska 3

Euler’s Identity 10/3/2020 Veton Këpuska 4

Polar Form of Complex Numbers u 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. 10/3/2020 Veton Këpuska 5

Polar Form of Complex Numbers u u Conversion between polar and rectangular (Cartesian) forms. For z=0+j 0; called “complex zero” one can not define arg(0+j 0). Why? 10/3/2020 Veton Këpuska 6

Geometric Representation of Complex Numbers. Im z Q 1 Axis of Reals |z | Im{z} Q 2 Axis of Imaginaries Re{z} Q 3 Re Q 4 Complex or Gaussian plane 10/3/2020 Veton Këpuska 7

Geometric Representation of Complex Numbers. Arg{z} ≥ 0 Q 3 or Q 4 Arg{z} ≤ 0 Q 1 or Q 4 Re{z} ≥ 0 Q 2 or Q 3 Re{z} ≤ 0 10/3/2020 Im{z} ≥ 0 Im Q 2 Im{z} ≤ 0 Axis of Reals | Q 1 or Q 2 Axis of Imaginarie s z Q 1 Condition 2 |z Condition 1 Im{z} Complex Number in Quadrant Re{z} Q 3 Re Q 4 Complex or Gaussian plane Veton Këpuska 8

Example Im z 1 1 z 2 -2 -1 z 3 10/3/2020 Re -1 Veton Këpuska 9

Conjugation of Complex Numbers u u 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 10/3/2020 Veton Këpuska 10

Geometric Representation of Conjugate Numbers u If z=rej then z*=re-j Im z r y x Re r - -y Complex or Gaussian plane 10/3/2020 Veton Këpuska z* 11

Complex Number Operations u u u 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. 10/3/2020 Veton Këpuska 12

Addition/Subtraction of Complex Numbers 10/3/2020 Veton Këpuska 13

Multiplication/Division of Complex Numbers 10/3/2020 Veton Këpuska 14

Useful Identities u z ∈ C, ∈ R & n ∈ Z (integer set) 10/3/2020 Veton Këpuska 15

Useful Identities u Example: z = +j 0 n n =2 then arg(2)=0 =-2 then arg(-2)= Im j z -2 10/3/2020 -1 0 1 Veton Këpuska 2 Re 16

Silly Examples and Tricks Im j /2 -1 3 /2 0 1 Re -j 10/3/2020 Veton Këpuska 17

Division Example u Division of two complex numbers in rectangular form. 10/3/2020 Veton Këpuska 18