Digital Signal Processing Reference
In-Depth Information
APPENDIX
C
Complex Numbers
C.1 DEFINITION
A complex number has a real part and an imaginary part, and thus two quantities are contained in a single
complex number.
Real numbers are those used in everyday, non-scientific activities. Examples of real numbers are
0, 1.12, -3.37, etc. To graph real numbers, only a single axis extending from negative infinity to positive
infinity is needed, and any real number can be located and graphed on that axis. In this sense, real numbers
are one-dimensional.
Imaginar
y n
umbers are numbers that consist of a real number multiplied by the square root of
negative one (
√
−
1. In this topic,
j
will typically be used to represent the square root of negative one, although either
i
or
j
may be used in
m-code. Electrical engineers use
j
as the imaginary operator since the letter
i
is used to represent current.
Typical imaginary numbers might be 5
j
, -2.37
i
, etc.
Since complex numbers have two components, they naturally graph in a two-dimensional space,
or a plane, and thus two axes at right angles are used to locate and plot a complex number. In the case of
the complex plane, the x-axis is called the real axis, and it represents the real amplitude, and the y-axis is
called the imaginary axis and numbers are considered to be an amplitude multiplied by the square root
of negative one.
Typical complex numbers might be 1
1 ) which is usually called
i
or
j
, and has the property that
i
·
i
=−
+
i
, 2
.
2
−
0
.
3
j
, and so forth.
C.2 RECTANGULAR V. POLAR
A complex number can be located in the complex plane using either 1) rectangular coordinates (values
for horizontal and vertical axes, such as
x
and
y
) or 2) polar coordinates, in which a distance from the
origin (center of the plot where
x
and
y
are both zero) and an angle (measured counterclockwise starting
from the positive half of the real or x-axis) are specified.
Figure C.1 shows the complex number 0
.
5
0
.
6
j
plotted in the complex plane.
You can convert from rectangular coordinates to polar using these formulas:
+
(
Re
(W ))
2
(
Im
(W ))
2
Magnitude
=
+
arctan
(
Im
(W )
Angle
=
Re
(W )
)
(C.1)
Using 0
.
5
+
0
.
6
j
as the complex number, and plugging into the formula for magnitude, we get
(
0
.
5
)
2
(
0
.
6
)
2
Magnitude
=
+
=
0
.
781
and the angle would be
arctan
(
0
.
6
50
.
2
◦
=
=
=
Angle
0
.
5
)
0
.
876 06
radian
