Digital Signal Processing Reference
In-Depth Information
M
θ
where
M
is the magnitude and
θ
is the angle, which may be expressed in either degrees or radians.
C.3 ADDITION AND SUBTRACTION
The rule for adding complex numbers is as follows:
d)
In other words, for addition or subtraction, just add or subtract the real parts and then the imaginary
parts, keeping them separate.
(a
+
bj )
+
(c
+
dj )
=
(a
+
c)
+
j(b
+
C.4 MULTIPLICATION
C.4.1 RECTANGULAR COORDINATES
A real number times a real number is a real number, i.e.,
(
6
)(
−
2
)
=−
12
A real number times an imaginary number is an imaginary number, i.e.,
(
6
)(
−
2
j)
=−
12
j
An imaginary number times an imaginary number is negative one times the product of the two
remaining real numbers. For example
(
6
j)(
−
2
j)
=
(
6
)(j)(
−
2
)(j)
=
(
6
)(
−
2
)(j)(j)
=
(
−
12
)(
−
1
)
=
12
The general rule is
(a
+
bj)(c
+
dj )
=
(ac
−
bd)
+
j(ad
+
bc)
C.4.2 POLAR COORDINATES
The product of two complex numbers expressed in polar coordinates is a complex number having a
magnitude equal to the product of the two magnitudes, and an angle equal to the sum of the two angles:
(M
1
θ
1
)(M
2
θ
2
)
=
M
1
M
2
(θ
1
+
θ
2
)
Example C.4.
Multiply the complex number 0.9
90 by itself.
Multiply the magnitudes and add the angles to get 0.81
180 which is -0.81.
Example C.5.
Multiply
(
0
.
707
+
0
.
707
j)
and
(
0
.
707
−
0
.
707
j)
.
(0.707 + 0.707j) = 1
45 and (0.707 - 0.707j) = 1
−
45 (angles in degrees); hence the product is the
product of the magnitudes and th
e
sum of the
an
gles = 1
0 = 1. Doing the multiplication in rectangular
coordinates gives a product of (
√
2
/
2)(1 + j)(
√
2
/
2)(1 - j) = (2/4)(2) = 1.
