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Speech Processing

Speech Processing. Complex Algebra Review. Complex Algebra Elements. Definitions: Note: Real numbers can be thought of as complex numbers with imaginary part equal to zero. Complex Algebra Elements. Euler’s Identity. Polar Form of Complex Numbers.

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Speech Processing

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  1. Speech Processing Complex Algebra Review

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

  3. Complex Algebra Elements Veton Këpuska

  4. Euler’s Identity 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. 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? Veton Këpuska

  7. 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 Veton Këpuska

  8. 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 Veton Këpuska

  9. Im z1 1 z2 -2 -1 Re z3 -1 Example 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=ej (polar form) then what is z* also in polar form? If z=rej then z*=re-j Veton Këpuska

  11. 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 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. Veton Këpuska

  13. Addition/Subtraction of Complex Numbers Veton Këpuska

  14. Multiplication/Division of Complex Numbers Veton Këpuska

  15. Useful Identities • z ∈ C,  ∈ R & n ∈ Z (integer set) Veton Këpuska

  16. Useful Identities • Example: z = +j0 • =2 then arg(2)=0 • =-2 then arg(-2)= Im j  z -2 -1 0 1 2 Re Veton Këpuska

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

  18. Division Example • Division of two complex numbers in rectangular form. Veton Këpuska

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