Digital Signal Processing
An Overview of Complex Numbers
Moslem Amiri, V´aclav Pˇrenosil
Embedded Systems Laboratory
Faculty of Informatics, Masaryk University
Brno, Czech Republic
amiri@mail.muni.cz
prenosil@fi.muni.cz
February 24, 2012
The Complex Number System
Example
Consider throwing a ball straight up into air with an initial velocity (vi )
of 9.8 meters/sec
Height of ball (h) at any instant of time (t)
h =
−gt2
2
+ vi t −→ t = 1 ± 1 − h/4.9
E.g., ball reaches h = 3 meters twice: t = 0.38 (going up), t = 1.62
(going down)
h = 10 −→ t =?
Never in reality
But h = 10 −→ t = 1 +
√
−1.041 and t = 1 −
√
−1.041
Contain square-root of a negative number
Called complex numbers
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The Complex Number System
Every complex number is sum of two components
A real part: an ordinary number
An imaginary part: square-root of a negative number
Imaginary part is usually reduced to an ordinary number times
√
−1
Example
Consider
t = 1 +
√
−1.041
= 1 +
√
1.041
√
−1
= 1 + 1.02
√
−1
Real part: 1
Imaginary part: 1.02
√
−1
Abstract term
√
−1 is given a special symbol: j (sometimes i)
Therefore, t = 1 + 1.02j
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The Complex Number System
Complex numbers are represented by locations in a two-dimensional
display called complex plane
Horizontal axis = real part of complex number
Vertical axis = imaginary part
Chapter 30- Complex Numbers 553
Real axis
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
2 + 6 j
-4 - 1.5 j
3 - 7 j
8j
7j
6j
5j
4j
3j
2j
1j
0j
-1j
-2j
-3j
-4j
-5j
-6j
-7j
-8j
ne. Every complex number
ation in the complex plane,
the three examples shown
ntal axis represents the real
vertical axis represents the
Imaginaryaxis
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The Complex Number System
In equations, a complex number is represented by a single variable
Example
A = 2 + 6j
Re A = 2, Im A = 6
Complex numbers follow the same algebra as ordinary numbers,
treating j as a constant
Addition, subtraction, multiplication, and division of complex numbers
(a + bj) + (c + dj) = (a + c) + j(b + d)
(a + bj) − (c + dj) = (a − c) + j(b − d)
(a + bj)(c + dj) = (ac − bd) + j(bc + ad)
(a + bj)
(c + dj)
=
ac + bd
c2 + d2
+ j
bc − ad
c2 + d2
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The Complex Number System
Complex conjugate
A complex number with sign of imaginary part switched
Z = a + bj
Z∗
= a − bj
Some properties of complex numbers
1 Commutative property
AB = BA
2 Associative property
(A + B) + C = A + (B + C)
3 Distributive property
A(B + C) = AB + AC
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Polar Notation
Complex numbers can be expressed in two notations
Rectangular notation (which was just described)
Polar notation
In polar notation
Magnitude
Length of vector starting at origin and ending at complex point
Phase angle
measured between this vector and positive x-axis
Rectangular-to-polar conversion
M = (Re A)2 + (Im A)2
θ = arctan
Im A
Re A
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Polar Notation
Real axis
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
2 + 6 j or
M = %40
2= arctan (6/2)
3 - 7 j or
M = %58
2= arctan (-7/3)
-4 - 1.5 j or
M = %18.25
2= arctan (-1.5/-4)
8j
7j
6j
5j
4j
3j
2j
1j
0j
-1j
-2j
-3j
-4j
-5j
-6j
-7j
-8j
bers in polar form. Three
s in the complex plane are
r coordinates. Figure 30-1
ame points in rectangular
Imaginaryaxis
This brings up a giant leap in the mathematics. (Yes, this means you should
pay extra attention). A complex number written in rectangular notation
Polar-to-rectangular conversion
Re A = M cos(θ)
Im A = M sin(θ)
Using above equations
a + jb = M(cos θ + j sin θ)
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Polar Notation
Euler’s relation
ejx = cos x + j sin x
Rewriting equation
a + jb = M(cos θ + j sin θ)
using Euler’s relation results in a complex exponential
a + jb = Mejθ
Using exponential polar form makes multiplication and division simple
M1ejθ1
M2ejθ2
= M1M2ej(θ1+θ2)
M1ejθ1
M2ejθ2
=
M1
M2
ej(θ1−θ2)
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Polar Notation
In Euler’s relation
ejx = cos(x) + j sin(x)
or e−jx = cos(x) − j sin(x)
one complex expression is equal to another complex expression
This is not useful
Rearranging relations
cos(x) =
ejx
+ e−jx
2
sin(x) =
ejx
− e−jx
2j
=
j(e−jx
− ejx
)
2
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Using Complex Numbers
Question
How to use a mathematics that has no connection with everyday
experience?
Answer
Change physical problem into a complex number form
Manipulate complex numbers
Then change back into a physical answer
Two ways that physical problems can be represented using complex
numbers
Substitution
Mathematical equivalence
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Using Complex Numbers by Substitution
Substitution
Takes two real physical parameters
Places one in real part of complex number and one in imaginary part
After mathematical operations, complex number is separated into its
real and imaginary parts corresponding to physical parameters
Substitution allows two values to be manipulated as a single complex
number
Example
A boat is pushed in one direction by wind, and in another direction by ocean
current
Resulting force is vector sum of two force vectors
Use complex numbers
Place east/west coordinate into real part of a complex number
North/south coordinate into imaginary part
Substitution allows us to treat each vector as a single complex number
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Using Complex Numbers by Substitution
Example (Continued)
Wind (2 parts to east, 6 parts to north) −→ A: 2 + 6j
Ocean current (4 parts to east, 3 parts to south) −→ B : 4 − 3j
Sum, C : 6 + 3j −→ 6 parts to east, 3 parts to northChapter 30- Complex Numbers 559
Real axis
-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
B
A+B=C
North
South
8j
7j
6j
5j
4j
3j
2j
1j
0j
-1j
-2j
-3j
-4j
-5j
-6j
-7j
-8j
C
AE 30-3
vectors with complex numbers. The
A & B represent forces measured
pect to north/south and east/west.
/west dimension is replaced by the
of the complex number, while the
uth dimension is replaced by the
ry part. This substitution allows
x mathematics to be used with an
real problem.
Imaginaryaxis
West
East
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Using Complex Numbers by Substitution
Substitution method is mathematically awkward
There is no equation, there is representation
Example
When A equals B, we know countless consequences: 5A = 5B,
1 + A = 1 + B, A/x = B/x, etc.
When A represents B, without additional information, we know nothing
E.g., when sinusoids are represented by complex numbers, we allow addition
and subtraction, but prohibit multiplication and division
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Using Complex Numbers by Mathematical Equivalence
Mathematical equivalence is a way of making complex numbers
mathematically equivalent to physical problem
Example
In DSP, sine and cosine waves can be described as having a positive
frequency or a negative frequency
Substitution method ignores negative frequencies
Since there are applications where negative frequencies are important,
mathematical equivalence is of help here
This will be discussed throughout this course
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References
Steven W. Smith, The Scientist & Engineer’s Guide to Digital
Signal Processing, California Technical Pub, 1997.
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