Digital Signal Processing Reference
In-Depth Information
multiplication from Section B.2.3; i.e.
x
1
x
2
=
a
1
+
j
b
1
=
(
a
1
+
j
b
1
)
(
a
2
−
j
b
2
)
a
2
+
j
b
2
(
a
2
+
j
b
2
)
(
a
2
−
j
b
2
)
=
a
1
a
2
+
b
1
b
2
+
j
a
2
b
1
−
a
1
b
2
,
(B.11)
a
2
+
b
2
a
2
+
b
2
where the final expression is obtained by noting that j
2
=−
1. We illustrate
these concepts with an example.
Example B.1
Two complex numbers are given by
x
=
5
+
j7 and
y
=
2
−
j4. Calculate
∗
∗
(i)
ℜ
(
x
),
ℑ
(
x
),
ℜ
(
y
),
ℑ
(
y
); (ii)
x
+
y
; (iii)
x
−
y
;(iv)
xy
; (v)
x
,
y
; (vi)
x
,
y
; and (vii)
x
/
y
.
Solution
(1) The real and imaginary components of the complex number
x
are
ℜ
(
x
)
=
5 and
ℑ
(
x
)
=
7. Likewise, the real and imaginary components of
y
are
ℜ
(
y
)
=
2 and
ℑ
(
y
)
=−
4.
(2) Adding
x
and
y
yields
x
+
y
=
(5
+
j7)
+
(2
−
j4)
=
(5
+
2)
+
j(7
−
4)
=
7
+
j3
.
Since addition is commutative, the order of the operands does not matter,
i.e.
x
x
.
(3) Subtracting
y
from
x
yields
+
y
=
y
+
x
−
y
=
(5
+
j7)
−
(2
−
j4)
=
(5
−
2)
+
j(7
−
(
−
4))
=
3
+
j11
.
Subtraction is not commutative. In fact,
x
−
y
=−
(
y
−
x
).
(4) Multiplication of
x
and
y
is performed as follows:
j
2
28
xy
=
(5
+
j7)(2
−
j4)
=
10
+
j14
−
j20
−
=
+
+
−
=
−
j6
.
(10
28)
j(14
20)
38
yx.
(5) The complex conjugate of the complex number
x
Multiplication is commutative, therefore
xy
=
∗
=
=
5
+
j7 is
x
5
−
j7.
∗
=
Likewise, the complex conjugate of
y
=
2
−
j4 is
y
2
+
j4.
=
√
=
√
(6) The modulus of
x
=
5
+
j7 is given b
y
x
5
2
+
7
2
74. Likewise,
=
√
2
2
+
4)
2
the modulus of
y
=
2
−
j4 is
y
=
(
−
20.
(7) Dividing
x
by
y
yields
+
+
+
x
y
=
5
j7
=
(5
j7)
(2
j4)
2
−
j4
(2
−
j4)
(2
+
j4)
=
(5)(2)
−
(7)(4)
j
(7)(2)
+
(5)(4)
=−
18
20
j
34
+
+
20
.
2
2
+
4
2
2
2
+
4
2
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