Digital Signal Processing Reference
In-Depth Information
C.5 DIVISION AND COMPLEX CONJUGATE
C.5.1 USING RECTANGULAR COORDINATES
The quotient of two complex numbers can be computed by using complex conjugates. The complex
conjugate of any complex number
a
bj
.
Suppose we wished to simplify an expression of the form
+
bj
is simply
a
−
+
a
bj
dj
which is one complex number divided by another. This expression can be simplified by multiplying
both numerator and denominator by the complex conjugate of the denominator, which yields for the
denominator a single real number which does not affect the ratio of the real and imaginary parts (now
isolated in the numerator) to each other.
c
+
bdj
2
a
+
bj
c
−
dj
ac
−
adj
+
bcj
−
(ac
+
bd)
+
j(bc
−
ad)
dj
·
dj
=
=
+
−
c
2
−
+
−
d
2
j
2
c
2
+
d
2
c
c
cdj
cdj
The ratio of the real part to the imaginary part is
ac
+
bd
ad
If a complex number
W
is multiplied by its conjugate
W
∗
, the product is
bc
−
2
, or the magnitude
|
W
|
squared of the number
W
.
WW
∗
= |
W
|
2
As a concrete example, let
W
=
1
+
j
Then
W
∗
=
1
−
j
and the product is
j
2
1
−
=
2
which is the same as the square of the magnitude of
W
:
(
1
2
1
2
)
2
+
=
2
Example C.6.
Simplify the ratio 1/j
We multiply numerator and denominator by the complex conjugate of the denominator, which is
−
j
:
(
1
j
)(
−
j
−
j
j
)
=
1
)
=−
j
−
−
(
−
