Review Complex Numbers Complex Numbers Complex Arithmetic The

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					Review: Complex Numbers                                                  Review: Complex Numbers
                                                                           Basics




                                                                         Complex Numbers


                          Review: Complex Numbers                              A complex number is one of the form
           CS 450: Introduction to Digital Signal and Image Processing                                               a + bi

                                                                               where                                         √
                                   Bryan Morse                                                                      i=           −1
                               BYU Computer Science
                                                                                                   a: real part
                                                                                                   b: imaginary part




Review: Complex Numbers                                                  Review: Complex Numbers
  Basics                                                                   Basics




Complex Arithmetic                                                       The Complex Plane
                                                                               Complex numbers can be thought of as points in the
             When you add two complex numbers, the real parts and
                                                                               complex plane:
             imaginary parts add independently:

                          (a + bi) + (c + di) = (a + c) + (b + d)i                                                Imaginary


             When you multiply two complex numbers, you
                                                                                                                         i
             cross-multiply them like you would polynomials:

                  (a + bi) ∗ (c + di) = ac + a(di) + (bi)c + (bi)(di)                                     -1                                  Real
                                        = ac + (ad + bc)i + (bd)(i 2 )                                                                1

                                        = ac + (ad + bc)i − bd
                                        = (ac − bd) + (ad + bc)i                                                    -i




Review: Complex Numbers                                                  Review: Complex Numbers
  Magnitude and Phase                                                      Magnitude and Phase




Magnitude and Phase                                                      Magnitude and Phase in the Complex Plane
             The length is called the magnitude:
                                                                                                               Imaginary
                                     |a + bi| =   a2   +   b2                                                                             z
                                                                                                                             |z|
             The angle from the real-number axis is called the phase:                                                    i

                                                           b
                                   φ(a + bi) = tan−1
                                                           a                                             -1                  ϕ(z)
                                                                                                                                               Real
                                                                                                                                      1
             When you multiply two complex numbers, their magnitudes
             multiply:
                                     |xy| = |x||y|                                                                 -i
             and their phases add:
                                    φ (xy) = φ (x) + φ (y )
Review: Complex Numbers                                                      Review: Complex Numbers
  Complex Conjugates                                                           Complex Conjugates




Complex Conjugates                                                           Complex Conjugates in the Complex Plane

                                                                                                               Imaginary
              Complex number z:                                                                                                     z
                                           z = a + bi
                                                                                                                       i
              Its complex conjugate:
                                          z ∗ = a − bi
                                                                                                         -1                                  Real
                                                                                                                                    1
              The complex conjugate       z∗   has
                    the same real part but opposite imaginary part, and
                    the same magnitude but opposite phase.                                                        -i

                                                                                                                                    z*




Review: Complex Numbers                                                      Review: Complex Numbers
  Complex Conjugates                                                           Complex Conjugates




Complex Conjugates                                                           Linear Algebra with Complex Numbers

                                                                                           The inner product of two complex-valued vectors involves
              Adding z + z ∗ , cancels the imaginary parts to leave a real                 multiplying each component of one of the vector not by the
              number:                                                                      other but by the complex conjugate of the other:
                                 (a + bi) + (a − bi) = 2a
                                                                                                              u·v =            u[k] v [k]∗
              Multiplying z ∗ z ∗ gives the real number equal to |z|2 :
                                                                                                                           k
                              (a + bi)(a − bi) = a2 − (bi)2
                                                                                           The length of a complex-valued vector is thus a real
                                                     = a2 + b 2
                                                                                           number:
                                                                                                           u 2 =u·u =      u[k ] u[k]∗
                                                                                                                                k




Review: Complex Numbers                                                      Review: Complex Numbers
  Euler Notation                                                               Euler Notation




Magnitudes and Phases - revisited                                            Euler’s Formula

                                                                                           Euler’s formula uses exponential notation to encode
              Remember that under complex multiplication                                   complex numbers—uses i in the exponent to differentiate
                    magnitudes multiply                                                    from real numbers
                    phases add
                                                                                           Euler’s formula (definition):
              We can do the same thing using exponents:
                                                                                                               eiθ = cos θ + i sin θ
                               (a1 eb1 )(a2 eb2 ) = a1 a2 e(b1 +b2 )
                                                                                           eiθ is the vector with magnitude 1.0 and phase θ
              Let’s encode complex numbers using exponential notation                      Any complex number z can be written as
              to make it easier to work with magnitude and phase
                                                                                                                   z = |z| eiφ(z)
Review: Complex Numbers                                                                 Review: Complex Numbers
  Euler Notation                                                                          Euler Notation




Euler’s Formula: Graphical Interpretation                                               Euler’s Formula: Graphical Interpretation
      eiθ                                                                                     z = |z|eiφ(z)
                                       Imaginary                                                                                  Imaginary
                                                                                                                                                     z
                                                                                                                                             |z|
                                              i                                                                                          i
                                                    eiθ


                               -1             θ                     Real                                                    -1               ϕ(z)
                                                                                                                                                                 Real
                                                            1                                                                                       1


                                         -i                                                                                         -i




Review: Complex Numbers                                                                 Review: Complex Numbers
  Euler Notation                                                                          Euler Notation




Euler’s Formula: Application                                                            Powers of Complex Numbers
      What is (2 + 2i)(−3 + 3i)?
                                                                                              Suppose that we take a complex number
      Suppose that we already have these numbers in
      magnitude-phase notation:                                                                                                    z = |z| eiφ(z)
                             √                     √
                 |2 + 2i| = 2 2       |−3 + 3i| = 3 2                                         and raise it to to some power n:
                 φ (2 + 2i) = π
                          √ 4         φ (−3 + 3i) = 3π
                                                √    4                                                                                                       n
                              iπ/4 −3 + 3i = 3 2 ei3π/4
               2 + 2i = 2 2 e                                                                                                    zn =      |z| eiφ(z)
                                                                                                                                   = |z|n einφ(z)
                                                   √                 √
                   (2 + 2i)(−3 + 3i)    =         2 2 eiπ/4         3 2 ei3π/4
                                                                                              z n has magnitude |z|n and phase n [φ (z)].
                                                       iπ
                                         = 12 e
                                         = −12

Review: Complex Numbers                                                                 Review: Complex Numbers
  Powers of Complex Numbers                                                               Powers of Complex Numbers




Powers of Complex Numbers: Example                                                      Powers of Complex Numbers: Example

                                                                                                                      n
      What is i n for various n?                                                              What is eiπ/4               for various n?
                                                   Imaginary
                                                                                                                                              Imaginary
      i=    eiπ/2                                                                                          0
                                                                                                eiπ/4
      i 0 = ei0 = 1                                             i                                          1
      i 1 = eiπ/2 = i                                                                           eiπ/4                                                    i
                                                                                                      2
      i 2 = ei2π/2 = −1                                                                         eiπ/4
      i 3 = ei3π/2 = −i                     -1                                   Real           eiπ/4
                                                                                                           3
                                                                                                                                        -1
                                                                      1                                                                                                 Real
      i 4 = ei4π/2 = 1                                                                                4                                                           1
      .
      .                                                                                         eiπ/4
      .
                                                        -i
                                                                                                                                                    -i
Review: Complex Numbers
  Summary




Summary: Complex Numbers

             Can represent in (real,imaginary) Cartesian form
             Can represent in (magnitude,phase) polar form
             Magnitude = distance from 0 (same idea as absolute value)
             Phase = angle with the real axis
             Euler’s theorem: exponential notation for (magnitude,phase)
                                   eiθ = cos θ + i sin θ
                                       z = |z| eiφ(z)

             Complex conjugate: z ∗ = a − bi = |z| e−iφ(z)
             Raising a complex number to a power:
                                  n
             z n has magnitude |z| and phase n [φ (z)]

				
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