Digital Signal Processing Reference
In-Depth Information
as just defined:
s
5
= s
1
s
2
= (2 + j2)
(3 - j1) = 6 - j2 + j6 - (j)
2
2
= (6 + 2) + j(6 - 2) = 8 + j4.
This multiplication is performed in MATLAB by
s1 = 2 + j*2; s2 = 3
j ;
s5 = s1*s2
result: s5 = 8 + 4i
The rules of real division also apply to complex division; however, the pres-
ence of
j
in the denominator complicates the operation. We often wish to express
the quotient of two complex numbers as a complex number in rectangular form.
First, we define the
conjugate
of a complex number. The conjugate of
denoted by
s = a + jb,
s
*
,
is defined as
s
*
= (a + jb)
*
= a - jb.
We obtain the conjugate of a complex number by changing the sign of its imaginary
part. A property of complex numbers is that the product of a number with its conju-
gate is real; that is,
ss
*
= (a + jb)
(a - jb) = a
2
+ b
2
.
(D.6)
For division of two complex numbers, we multiply both the numerator and the
denominator by the conjugate of the denominator to express the quotient in rectan-
gular form; that is,
a
+
jb
c + jd
=
a
+
jb
c + jd
c
-
jd
ac
+
bd
c
2
+ d
2
+ j
bc
-
ad
c - jd
=
+ d
2
.
(D.7)
c
2
As an example, for
s
1
and
s
2
as defined earlier,
s
1
s
2
2
+
j2
3 - j1
=
2
+
j2
3 - j1
3
+
j1
6
-
2
9 + 1
+ j
6
+
2
=
3 + j1
=
9 + 1
= 0.4 + j0.8.
The following MATLAB statement performs this division:
s1 = 2+j*2; s2 = 3
j;
s6 = s1/s2
result: s6 = 0.4 + 0.8i
In general, the quotient of two complex numbers can be expressed as
s
1
s
2
s
1
s
*
s
2
s
*
=
Re(s
1
s
*
)
s
2
s
*
+ j
Im(s
1
s
*
)
s
2
s
*
=
.
(D.8)
