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**Unformatted text preview: **Review: Complex Numbers Review: Complex Numbers
CS 450: Introduction to Digital Signal and Image Processing Bryan Morse
BYU Computer Science Review: Complex Numbers
Basics Complex Numbers
A complex number is one of the form
a + bi
where
i=
a: real part
b: imaginary part √ −1 Review: Complex Numbers
Basics Complex Arithmetic
When you add two complex numbers, the real parts and
imaginary parts add independently:
(a + bi ) + (c + di ) = (a + c ) + (b + d )i
When you multiply two complex numbers, you
cross-multiply them like you would polynomials:
(a + bi ) ∗ (c + di ) = ac + a(di ) + (bi )c + (bi )(di )
= ac + (ad + bc )i + (bd )(i 2 )
= ac + (ad + bc )i − bd
= (ac − bd ) + (ad + bc )i Review: Complex Numbers
Basics The Complex Plane
Complex numbers can be thought of as points in the
complex plane:
Imaginary i -1 Real
1 -i Review: Complex Numbers
Magnitude and Phase Magnitude and Phase
The length is called the magnitude:
|a + bi | = a2 + b 2 The angle from the real-number axis is called the phase:
φ(a + bi ) = tan−1 b
a When you multiply two complex numbers, their magnitudes
multiply:
|xy | = |x ||y |
and their phases add:
φ (xy ) = φ (x ) + φ (y ) Review: Complex Numbers
Magnitude and Phase Magnitude and Phase in the Complex Plane
Imaginary
z
i |z| ϕ(z) -1 Real
1 -i Review: Complex Numbers
Complex Conjugates Complex Conjugates
Complex number z :
z = a + bi
Its complex conjugate:
z ∗ = a − bi
The complex conjugate z ∗ has
the same real part but opposite imaginary part, and
the same magnitude but opposite phase. Review: Complex Numbers
Complex Conjugates Complex Conjugates in the Complex Plane
Imaginary
z
i -1 Real
1 -i
z* Review: Complex Numbers
Complex Conjugates Complex Conjugates
Adding z + z ∗ , cancels the imaginary parts to leave a real
number:
(a + bi ) + (a − bi ) = 2a
Multiplying z ∗ z ∗ gives the real number equal to |z |2 :
(a + bi )(a − bi ) = a2 − (bi )2
= a2 + b 2 Review: Complex Numbers
Complex Conjugates Linear Algebra with Complex Numbers
The inner product of two complex-valued vectors involves
multiplying each component of one of the vector not by the
other but by the complex conjugate of the other:
u [k ] v [k ]∗ u·v =
k The length of a complex-valued vector is thus a real
number:
u 2 =u·u =
u [k ] u [k ]∗
k Review: Complex Numbers
Euler Notation Magnitudes and Phases - revisited
Remember that under complex multiplication
magnitudes multiply
phases add We can do the same thing using exponents:
(a1 eb1 )(a2 eb2 ) = a1 a2 e(b1 +b2 )
Let’s encode complex numbers using exponential notation
to make it easier to work with magnitude and phase Review: Complex Numbers
Euler Notation Euler’s Formula
Euler’s formula uses exponential notation to encode
complex numbers—uses i in the exponent to differentiate
from real numbers
Euler’s formula (deﬁnition):
ei θ = cos θ + i sin θ
ei θ is the vector with magnitude 1.0 and phase θ
Any complex number z can be written as
z = |z | ei φ(z ) Review: Complex Numbers
Euler Notation Euler’s Formula: Graphical Interpretation
ei θ
Imaginary i eiθ θ -1 Real
1 -i Review: Complex Numbers
Euler Notation Euler’s Formula: Graphical Interpretation
z = |z |ei φ(z )
Imaginary
z
i |z| ϕ(z) -1 Real
1 -i Review: Complex Numbers
Euler Notation Euler’s Formula: Application
What is (2 + 2i )(−3 + 3i )?
Suppose that we already have these numbers in
magnitude-phase notation:
√
√
|−3 + 3i | = 3 2
|2 + 2i | = 2 2
π
φ (2 + 2√ = π
i) 4
φ (−3 + 3√ = 34
i)
2 + 2i = 2 2 ei π/4 −3 + 3i = 3 2 ei 3π/4 (2 + 2i )(−3 + 3i ) √
2 2 ei π/4 = = 12 ei π
= −12
Review: Complex Numbers
Euler Notation Powers of Complex Numbers
Suppose that we take a complex number
z = |z | ei φ(z )
and raise it to to some power n:
zn = |z | ei φ(z ) n = |z |n einφ(z )
z n has magnitude |z |n and phase n [φ (z )]. √
3 2 ei 3π/4 Review: Complex Numbers
Powers of Complex Numbers Powers of Complex Numbers: Example
What is i n for various n?
Imaginary i = ei π/2
i 0 = ei 0 = 1
i 1 = ei π/2 = i
i 2 = ei 2π/2 = −1
i 3 = ei 3π/2 = −i
i 4 = ei 4π/2 = 1
.
.
. i -1 Real
1 -i Review: Complex Numbers
Powers of Complex Numbers Powers of Complex Numbers: Example
What is ei π/4 n for various n?
Imaginary ei π/4 0 ei π/4 1 ei π/4 2 ei π/4 3 i -1 Real 4
ei π/4 1 -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)
ei θ = cos θ + i sin θ
z = |z | ei φ(z )
Complex conjugate: z ∗ = a − bi = |z | e−i φ(z )
Raising a complex number to a power:
n
z n has magnitude |z | and phase n [φ (z )] ...

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