551
CHAPTER
30
h
’
&
gt
2
2
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vt
Complex Numbers
Complex numbers are an extension of the ordinary numbers used in everyday math.
They have
the unique property of representing and manipulating
two
variables as a
single
quantity.
This fits
very naturally with Fourier analysis, where the frequency domain is composed of two signals, the
real and the imaginary parts.
Complex numbers shorten the equations used in DSP, and enable
techniques that are difficult or impossible with real numbers alone.
For instance, the Fast Fourier
Transform is based on complex numbers.
Unfortunately, complex techniques are very
mathematical, and it requires a great deal of study and practice to use them effectively.
Many
scientists and engineers regard complex techniques as the dividing line between DSP as a
tool
,
and DSP as a
career
.
In this chapter, we look at the mathematics of complex numbers, and
elementary ways of using them in science and engineering.
The following three chapters discuss
important techniques based on complex numbers: the
complex Fourier transform
, the
Laplace
transform
, and the
ztransform
.
These complex transforms are the heart of theoretical DSP.
Get
ready, here comes the math!
The Complex Number System
To illustrate complex numbers, consider a child throwing a ball into the air.
For example, assume that the ball is thrown straight up, with an initial
velocity of 9.8 meters per second.
One second after it leaves the child's
hand, the ball has reached a height of 4.9 meters, and the acceleration of
gravity (9.8 meters per second
2
) has reduced its velocity to zero.
The ball
then accelerates toward the ground, being caught by the child two seconds
after it was thrown.
From basic physics equations, the height of the ball at
any instant of time is given by:
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552
t
’
1±
1
&
h
/4.9
where
h
is the height above the ground (in meters),
g
is the acceleration of
gravity (9.8 meters per second
2
),
v
is the initial velocity (9.8 meters per
second), and
t
is the time (in seconds).
Now, suppose we want to know
when
the ball passes a certain height.
Plugging in the known values and solving for
t
:
For instance, the ball is at a height of 3 meters
twice
:
(going up)
t
’
0.38
and
seconds (going down).
t
’
1.62
As long as we ask reasonable questions, these equations give reasonable
answers.
But what happens when we ask unreasonable questions?
For
example: At what time does the ball reach a height of 10 meters?
This
question has no answer in reality because the ball
never
reaches this height.
Nevertheless, plugging the value of
into the above equation gives two
h
’
10
answers:
and
.
Both these answers contain
t
’
1
%
&
1.041
t
’
1
&
&
1.041
the squareroot of a negative number, something that does not exist in the world
as we know it.
This unusual property of polynomial equations was first used
by the Italian mathematician Girolamo Cardano (15011576).
Two centuries
later, the great German mathematician Carl Friedrich Gauss (17771855)
coined the term
complex numbers
, and paved the way for the modern
understanding of the field.
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 Summer '09
 Complex Numbers, Complex number

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