A.3. ARITHMETIC OPERATIONS OF COMPLEX
NUMBERS
A.3.1 Addition and Subtraction of
Complex Numbers
Which of the above forms for C in Eq. (A1) is
the best to use? It depends on the arithmetic operation we want to
perform. For example, if we're adding two complex numbers, the
rectangular form in Eq. (A1) is
the easiest to use. The addition of two complex numbers, C_{1} = R_{1} + jI_{1} and C_{2} = R_{2} + jI_{2}, is merely the sum of the
real parts plus j times the sum of
the imaginary parts as
Figure
A3 is a graphical depiction of the sum of two complex numbers
using the concept of phasors. Here the sum phasor C_{1} + C_{2} in Figure A3(a) is the new phasor from the
beginning of phasor C_{1}
to the end of phasor C_{2}
in Figure A3(b).
Remember, the Rs and the Is can be either positive or negative
numbers. Subtracting one complex number from the other is
straightforward as long as we find the differences between the two
real parts and the two imaginary parts separately. Thus
Equation A11
Figure A3. Geometrical representation
of the sum of two complex numbers.
An example of complex number addition is
discussed in Section
11.3, where we covered the topic of averaging fast Fourier
transform outputs.
A.3.2 Multiplication of Complex
Numbers
We can use the rectangular form to multiply two
complex numbers as
Equation A12
However, if we represent the two complex numbers
in exponential form, their product takes the simpler form
Equation A13
because multiplication results in the addition
of the exponents.
As a special case of multiplication of two
complex numbers, scaling is
multiplying a complex number by another complex number whose
imaginary part is zero. We can use the rectangular or exponential
forms with equal ease as follows:
Equation A14
or in exponential form,
Equation A15
A.3.3 Conjugation of a Complex
Number
The complex conjugate of a complex number is
obtained merely by changing the sign of the number's imaginary
part. So, if we denote C^{*} as the complex conjugate
of the number C = R + jI
= Me^{j}^{ø},
then C^{*} is expressed
as
Equation A16
There are two characteristics of conjugates that
occasionally come in handy. First, the conjugate of a product is
equal to the product of the conjugates. That is, if C = C_{1}C_{2}, then from Eq. (A13),
Equation A17
Second, the product of a complex number and its
conjugate is the complex number's magnitude squared. It's easy to
show this in exponential form as
Equation A18
(This property is often used in digital signal
processing to determine the relative power of a complex sinusoidal
phasor represented by Me^{j}^{w}^{t}.)
A.3.4 Division of Complex Numbers
The division of two complex numbers is also
convenient using the exponential and magnitude and angle forms,
such as
Equation A19
and
Equation A19'
Although not nearly so handy, we can perform
complex division in rectangular notation by multiplying the
numerator and the denominator by the complex conjugate of the
denominator as
Equation A20
A.3.5 Inverse of a Complex Number
A special form of division is the inverse, or
reciprocal, of a complex number. If C = Me^{j}^{ø}, its
inverse is given by
Equation A21
In rectangular form, the inverse of C = R +
jI is given by
Equation A22
We get Eq.
(A22) by substituting R_{1} = 1, I_{1} = 0, R_{2} = R, and I_{2} = I in Eq. (A20).
A.3.6 Complex Numbers Raised to a
Power
Raising a complex number to some power is easily
done in the exponential form. If C
= Me^{j}^{ø}
, then
Equation A23
For example, if C = 3e^{j}^{125°}, then
C cubed is
Equation A24
We conclude this appendix with four complex
arithmetic operations that are not very common in digital signal
processing—but you may need them sometime.
A.3.7 Roots of a Complex Number
The kth root of
a complex number C is the number
that, multiplied by itself k
times, results in C. The
exponential form of C is the best
way to explore this process. When a complex number is represented
by C = Me^{j}^{ø},
remember that it can also be represented by
Equation A25
In this case, the variable ø in Eq. (A25) is in degrees.
There are k distinct roots when
we're finding the kth root of
C. By distinct, we mean roots
whose exponents are less than 360°. We find those roots by
using the following:
Equation A26
Next, we assign the values 0, 1, 2, 3, . . .,
k–1 to n in Eq. (A26) to get the k roots of C. OK, we need an example here! Let's
say we're looking for the cube (third) root of C = 125e^{j}^{(75}^{°}^{)}.
We proceed as follows:
Equation A27
Next we assign the values n = 0, n = 1, and n = 2 to Eq. (A27) to get the three roots of C. So the three distinct roots are
and
A.3.8 Natural Logarithms of a Complex
Number
Taking the natural logarithm of a complex number
C = Me^{j}^{ø} is
straightforward using exponential notation; that is
Equation A28
where 0 ø < 2p . By way of
example, if C = 12e^{j}^{p/4}, the natural logarithm of C is
Equation A29
This means that e^{(2.485 +} ^{j}^{0.785)} = e^{2.485} · e^{j}^{0.785} = 12e^{j}^{p}^{/4}.
A.3.9 Logarithm to the Base 10 of a
Complex Number
We can calculate the base 10 logarithm of the
complex number C = Me^{j}^{ø}
using
Equation A30
Of course e is
the irrational number, approximately equal to 2.71828, whose log to
the base 10 is approximately 0.43429. Keeping this in mind, we can
simplify Eq. (A30)
as
Equation A31
Repeating the above example with C = 12e^{j}^{p}^{/4} and using the Eq. (A31) approximation,
the base 10 logarithm of C is
For the second term of the result in Eq. (A30) we used log_{a}(x^{n}) = n·log_{a}x according to the law of
logarithms.
Equation A32
The result from Eq. (A32) means that
Equation A33
A.3.10 Log to the Base 10 of a Complex
Number Using Natural Logarithms
Unfortunately, some software mathematics
packages have no base 10 logarithmic function and can calculate
only natural logarithms. In this situation, we just use
Equation A34
to calculate the base 10 logarithm of x. Using this change of base formula, we can find the
base 10 logarithm of a complex number C = Me^{j}^{ø}; that
is,
Equation A35
Because log_{10}(e) is approximately equal to 0.43429, we
use Eq. (A35) to state
that
Equation A36
Repeating, again, the example above of C = 12e^{j}^{p}^{/4}, the Eq. (A36) approximation allows us to take
the base 10 logarithm of C using
natural logs as
Equation A37
giving us the same result as Eq. (A32).
