Digital Signal Processing Reference
In-Depth Information
In general, a complex number
s
can be expressed as
s = a + jb,
(D.2)
where both
a
and
b
are real. The real number
a
is known as the real part of
s
, which
is denoted as The real number
b
is known as the imaginary part of
s
,
which is denoted as Note that the imaginary part of
s
is
b
, not
jb
; the
imaginary part of a complex number is real. A complex number expressed as in
(D.2) is said to be in the
rectangular form
. We will define other forms for a complex
number later.
The following relationships are seen from the definition of
j
:
a = Re(s).
b = Im(s).
j
5
= j(j
4
) = j;
j
=
2
-1;
j
2
j
6
= j
2
(j
4
) =-1;
=-1;
j
3
= j(j
2
) =-j;
j
7
=-j;
(D.3)
j
4
= (j
2
)
2
j
8
= 1;
= 1;
j
9
= j
o
.
Also, the reciprocal of
j
is
-j;
that is,
j
j
j
-1
1
j
=
1
j
=
=-j.
(D.4)
All real arithmetic operations apply to complex-number arithmetic, but we must
remember the definition of
j
. First, the complex numbers
s
1
= a + jb
and
s
2
= c + jd
are equal, or
s
1
= s
2
‹ a + jb = c + jd,
(D.5)
if and only if and Hence, an equation relating complex numbers is in
fact two equations relating real numbers. The real parts of the numbers must be
equal, and the imaginary parts of the numbers must also be equal.
Let the complex number
a = c
b = d.
s
3
be the sum of
s
1
and
s
2
,
where
s
1
= 2 + j2
and
s
2
= 3 - j1.
Then
s
3
= s
1
+ s
2
= 2 + j2 + 3 - j1 = 5 + j1.
This addition can be represented in the complex plane as shown in Figure D.2(a). In
the summation of complex numbers, the real part of the sum is equal to the sum of
the real parts, and the imaginary part of the sum is equal to the sum of the imaginary
parts.
