Digital Signal Processing Reference
In-Depth Information
and
√
−
j0
.
3524
π
.
y
=
2
−
j4
=
20e
Repeat Example B.1 but by selecting one of the two formats (rectangular or
polar) for which the arithmetic operation is computationally simpler.
Solution
(1) The real and imaginary components of the complex number
x
are obtained
from the rectangular format, i.e.
ℜ
(
x
)
=
5 and
ℑ
(
x
)
=
7. Likewise, for
y
the components are
ℜ
(
y
)
=
2 and
ℑ
(
y
)
=−
4.
(2) Addition of
x
and
y
is performed in the rectangular format as follows:
x
+
y
=
(5
+
j7)
+
(2
−
j4)
=
(5
+
2)
+
j(7
−
4)
=
7
−
j3
.
If polar format is required, we
can
express the above answer for (
x
+
y
)in
the polar format as
x
+
y
=
√
−
j0
.
13
π
.
(3) Subtraction is also performed in the rectangular format as follows:
58e
j tan
−
1
(
−
3
/
7)
=
7
.
62e
x
−
y
=
(5
+
j7)
−
(2
−
j4)
=
(5
−
2)
+
j(7
−
(
−
4))
=
3
−
j11
.
C
onve
rting
the
above
answer
into
polar
form,
we
obtain
x
−
y
=
√
−
j0
.
415
π
.
(4) Multiplication of
x
and
y
is performed in the polar format as follows:
130e
j tan
−
1
(
−
11
/
3)
=
11
.
40e
√
√
74e
j0
.
3026
π
−
j0
.
3524
π
xy
=
20e
√
−
j0
.
0498
π
.
=
1480e
√
The rectangular format is
xy
=
1480(cos(0
.
0498
π
)
+
j sin(
−
0
.
0498
π
))
=
38
−
j6.
(5) In rectangular format, the complex conjugate of the complex number
x
=
∗
=
5
−
j7. Likewise, the complex conjugate of
y
5
+
j7 is
x
=
2
−
j4 is
∗
=
2
+
j
4 in
rectangular format. T
he
complex conjugates in polar format
are
x
y
√
√
20e
j0
.
3524
π
.
(6) The mo
du
li of
x
and
y
a
re obtained directly from the polar format as
x
=
∗
=
−
j0
.
3026
π
∗
=
74e
and
y
√
√
20.
(7) Dividing
x
by
y
is performed in polar format, yielding
74 and
y
=
√
√
74e
j0
.
3026
π
x
y
3
.
7e
j0
.
655
π
,
=
√
=
20e
−
j0
.
3524
π
√
which, in rectangular format, is
3
.
7 (cos(0
.
655
π
)
+
j sin(0
.
655
π
))
=
−
0
.
9
+
j1
.
7
.
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