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:

(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
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
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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