Digital Signal Processing Reference
In-Depth Information
Complex numbers in Cartisian representation
•
Multiplication with a real absolute term k
(
)
ˆ
(
)
k
U
=+=
k
a
ib
kU
cos
ϕ
+
i
sin
ϕ
bw.
z
(
)
(
)
ˆ
ˆ
ˆ
ˆ
(
)
(
)
ª
º
k
ª
a
+±+
ib
a
i
b
º
=
k
U
cos
ϕ
±
U
cos
ϕ
+
i
U
sin
ϕ
±
U
sin
ϕ
¬
¼
¬
¼
1
1
2
2
1
1
2
2
1
1
2
2
Result
: the operation equates to a dilation of the vector length bei the
factor k
•
Multiplication of two complex numbers
ˆ
ˆ
(
)(
)
(
)
(
)
aiba ib U s
+
+
=
ϕ
+
i
sin
ϕ
U s
ϕ
+
i
sin
ϕ
=
1
1
2
2
1
1
1
2
2
2
ˆˆ
(
)
(
)
(
)
U
U
ª
cos
ϕϕ
c
o
s
−
sin
ϕϕ
sin
+
i
sin
ϕϕ
cos
+
si
n
ϕ
cos
ϕ
º
¬
¼
12
1
2
1
2
1
2
2
1
Result
: products of angle functions yield . Geometrical interpretation is
unsure!
•
Division of two complex numbers
ˆ
(
)
Ucos
ϕ
+
i
sin
ϕ
aib
aib
Ucos
+
1
1
=
1
1
1
=
ˆ
(
)
+
ϕ
+
i
sin
ϕ
2
2
2
2
2
Tricky expansion by using (a+b)(a-b) = a
2
- b
2
(
)(
)
(
)
(
)
aiba ib
+
−
aa bb i ba ab
+
+
−
aa
bb
ab
+
ba
−
+
ab
1
1
2
2
12
12
12
12
=
=
12
12
+
i
12
12
=
(
)(
)
2
2
2
2
2
2
ai
+
b
aib
−
a
+
b
+
ab
2
2
2
2
2
2
2
2
2
2
ˆ
ˆ
(
)
(
)
Ucoscos
Uco
ϕϕ
+
−
sin
ϕϕ
sin
U
si
n
ϕϕ
c
os
−
cos
ϕϕ
sin
1
1
2
1
2
1
1
2
1
2
+
i
(
)
(
)
ˆ
ˆ
s
2
ϕ
sin
2
ϕ
U
cos
2
ϕ
−
sin
2
ϕ
2
2
2
2
2
2
Result
: products of angle functions yield . Calculation is extensive .
Geometrical interpretation is unsure !
Solution
: multiplication and division of two complex numbers in
exponential notation with the help of the EULERian relation .
Illustration 302:
Multiplication and division of two complex numbers in Cartesian representation
The
multiplication
and division of complex numbers in Cartesian representation tend to be rather compli-
cated. The solution of this problem is indicated below and specified in Illustration 303.
Multiplication, division, sampling, absolute value, windowing and quantization were
explained as non- linear processes. For complex calculation the latter with the exception
of multiplication and division, are of less importance.
The multiplication and division of complex numbers in Cartesian representation proves
very complicated (see Illustration 301). Above all, the appropriate geometrical interpre-
tation is very difficult.
