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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
crossmultiply 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 realnumber 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 complexvalued 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 complexvalued 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
magnitudephase notation:
√
√
2 + 2i  = 2 2
−3 + 3i  = 3 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 √
3 2 ei 3π/4 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 )]. 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 i 2
ei π/4 ei π/4 3 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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This note was uploaded on 03/02/2012 for the course C S 450 taught by Professor Morse,b during the Winter '08 term at BYU.
 Winter '08
 Morse,B

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