Digital Signal Processing Reference
In-Depth Information
Display of complex numbers in Cartesian coordinates
Imaginary axis
Complex numbers as vector
GAUSSian plane
a
+
ib
=
i b = i
Û
sin(
ϕ
)
ˆ
(
)
Û
(cos
ϕ
+
i sin
ϕ
)
U
=+ =
a
i
b
U
co
s
ϕ
+
i
si
n
ϕ
ϕ
a =
Û
cos(
Real axis
ϕ
)
Imaginary axis
ib
1
a
1
+ ib
1
Addition of two complex numbers
GAUSSian plane
Addition
of two complex numbers
(
)
(
)
a
+++ =
+
ib
a
ib
(a
1
+a
2
)+ i(b
1
+ b
2
)
1
1
2
2
ˆ
(
)
ˆ
(
)
U
co
s
ϕ
i
sin
ϕ
+
U
cos
ϕ
+
i
sin
ϕ
1
1
1
2
2
2
a
2
a
1
Real axis
(
)
aa
+
+
ib
(
+
b
)
=
1
2
1
2
imaginar
y
re
a
l
(
)
(
)
Attention: here b
2
< 0
e.g. negativ
ˆ
ˆ
ˆ
ˆ
U
c
o
s
ϕ
+
U
c s
o
ϕ
+
iU
sin
ϕ
+
U
s
i
n
ϕ
ib
2
1
1
2
2
1
1
2
2
a
2
+ ib
2
rl
ea
im
aginar
y
Imaginary axis
(a
1
+ib
1
) - i(a
2
+ b
2
)
Subtraction of two complex numbers
ib
1
a
1
+ ib
1
GAUSSian plane
Subtraction
of two complex numbers
(
)
(
)
a
+−+ =
+
ib
a
ib
-
(a
2
+ ib
2
)
1
1
2
2
ˆ
(
)
ˆ
(
)
U
co
s
ϕ
i
sin
ϕ
−
U
cos
ϕ
+
i
sin
ϕ
1
1
1
2
2
2
a
2
Real axis
(
)
a
1
aa
−
+
ib
(
−
b
)
=
1
2
1
2
imaginar
y
re
a
l
(
)
(
)
ˆ
ˆ
ˆ
ˆ
U
c
o
s
ϕ
−
U
c s
o
ϕ
+
iU
sin
ϕ
−
U
s
i
n
ϕ
1
1
2
2
1
1
2
2
rl
ea
im
aginar
y
ib
2
a
2
+ ib
2
Illustration 301:
Display of a complex number, adding and subtraction in Cartesian coordinates
The vector is illustrated by the connecting line between the points (0, 0) and (a, b) and (0,i0) and (a, ib).
The projection on the real and imaginary axis is defined by the angle functions. If the vector length (as
used in electro- technology) is defined as Ы, a = Ы sin (
). Addition and subtraction are
very easy and easy to interpret geometrically. Addition and subtraction can be carried out graphically
simply by using ruler, set square and pencil.
ϕ
) and b= Ы sin (
ϕ
There is no standardized identification for complex numbers. In electro- technology, usually
u
(t),
i(
t) and
Z
for voltage, current and AC resistance (impedance) are used. Mathematics and physics often do without the
underline (in the following c
k
, X(f), Y(f)and H(f) etc.).
So far, complex numbers have been defined by (a, ib), the projection of the vectors on the
real and imaginary axis. This is the
Cartesian form
of a complex number. As Illustration
300 and Illustration 301 show, adding and subtraction as well as multiplication with a con-
stant can be carried out very easily and graphically in this way.
