Complex Numbers
This chapter introduces complex numbers, beginning with
factoring
polynomials, and proceeding on to the
complex plane and
Euler's
identity.

Factoring a Polynomial
Remember ``
factoring polynomials''? Consider the second-order polynomial
It is second-order because the highest power of

is

(only
non-negative integer powers of

are allowed in this context). The
polynomial is also
monic
because its leading coefficient, the
coefficient of

, is

. By the
fundamental theorem of algebra
(discussed further in §
2.4), there are exactly two
roots
(or
zeros) of any
second order polynomial. These roots may be real or complex (to be defined).
For now, let's assume they are both real and denote them by

and

. Then we have

and

, and we can write
This is the
factored form of the monic polynomial

.
(For a non-monic polynomial, we may simply divide all coefficients
by the first to make it monic, and this doesn't affect the zeros.)
Multiplying out the symbolic factored form gives
Comparing with the original polynomial, we find we must have
This is a system of two equations in two unknowns. Unfortunately, it is a
nonlinear system of two equations in two
unknowns.
2.1 Nevertheless, because it is so small,
the equations are easily solved. In beginning algebra, we did them by
hand. However, nowadays we can use a software tool such as
Matlab or
Octave to solve very large systems of linear equations.
The factored form of this simple example is
Note that polynomial factorization rewrites a monic

th-order
polynomial as the product of
first-order monic polynomials,
each of which contributes one zero (root) to the product. This
factoring business is often used when working with
digital
filters [
68].
The Quadratic Formula
The general second-order (real) polynomial is
 |
(2.1) |
where the coefficients

are any
real numbers, and we assume

since otherwise
it would not be second order. Some experiments plotting

for different
values of the coefficients leads one to guess that the curve is always a
scaled and translated
parabola. The canonical parabola centered
at

is given by
 |
(2.2) |
where the magnitude of

determines the width of the parabola, and

provides an arbitrary vertical offset. If

, the parabola has
the minimum value

at

; when

, the parabola reaches a
maximum at

(also equal to

). If we can find

in
terms of

for any quadratic polynomial, then we can easily
factor the polynomial. This is called
completing the square.
Multiplying out the right-hand side of Eq.

(
2.2) above, we get
 |
(2.3) |
Equating coefficients of like powers of

to the general second-order
polynomial in Eq.

(
2.1) gives
Using these answers, any second-order polynomial

can be rewritten as a scaled, translated parabola
In this form, the roots are easily found by solving

to get
This is the general
quadratic formula. It was obtained by simple
algebraic manipulation of the original polynomial. There is only one
``catch.'' What happens when

is negative? This introduces the
square root of a negative number which we could insist ``does not exist.''
Alternatively, we could invent
complex numbers to accommodate it.
Complex Roots
Figure 2.1:
An example parabola defined by
.
![\includegraphics[scale=0.5]{eps/parabola}](http://www.dsprelated.com/josimages_new/mdft/img139.png) |
As a simple example, let

,

, and

,
i.e.,
As shown in Fig.
2.1, this is a parabola centered at

(where

) and reaching upward to positive infinity, never going below

.
It has no real zeros. On the other hand, the
quadratic formula says that the
``roots'' are given formally by

. The
square root of any negative number

can be expressed as

, so the only new algebraic object is

.
Let's give it a name:
Then, formally, the roots of

are

, and we can formally
express the polynomial in terms of its roots as
We can think of these as ``imaginary roots'' in the sense that square roots
of negative numbers don't really exist, or we can extend the concept of
``roots'' to allow for
complex numbers, that is, numbers of the form
where

and

are
real numbers, and

.
It can be checked that all algebraic operations for real
numbers
2.2 apply equally well to complex numbers. Both real numbers
and complex numbers are examples of a
mathematical
field.
2.3 Fields are
closed with respect to multiplication and addition, and all the rules
of algebra we use in manipulating polynomials with real coefficients (and
roots) carry over unchanged to polynomials with complex coefficients and
roots. In fact, the rules of algebra become simpler for complex numbers
because, as discussed in the next section, we can
always factor
polynomials completely over the field of complex numbers while we cannot do
this over the reals (as we saw in the example

).
Fundamental Theorem of Algebra
This is a very powerful algebraic tool.
2.4 It says that given any polynomial
we can
always rewrite it as
where the points

are the polynomial roots, and they may be real or
complex.
This section introduces various notation and terms associated with
complex
numbers. As discussed above, complex numbers arise by introducing
the square-root of

as a primitive new algebraic object among
real
numbers and manipulating it symbolically as if it were a real number
itself:
Mathematicians and physicists often use

instead of

as

.
The use of

is common in engineering where

is more often used for
electrical current.
As mentioned above, for any negative number

, we have
where

denotes the absolute value of

. Thus, every square
root of a negative number can be expressed as

times the square
root of a positive number.
By definition, we have
and so on. Thus, the sequence

,

is a
periodic sequence with
period 
, since

. (We'll
learn later that the sequence

is a sampled complex
sinusoid having
frequency equal to one fourth the
sampling rate.)
Every
complex number 
can be written as
where

and

are real numbers.
We call

the
real part and

the
imaginary part.
We may also use the notation
Note that the real numbers are the subset of the complex numbers having
a zero imaginary part (

).
The rule for
complex multiplication follows directly from the definition
of the imaginary unit

:
In some mathematics texts, complex numbers

are defined as ordered pairs
of real numbers

, and algebraic operations such as multiplication
are defined more formally as operations on ordered pairs,
e.g.,

. However, such
formality tends to obscure the underlying simplicity of complex numbers as
a straightforward extension of real numbers to include

.
It is important to realize that complex numbers can be treated
algebraically just like real numbers. That is, they can be added,
subtracted, multiplied, divided, etc., using exactly the same rules of
algebra (since both real and complex numbers are mathematical
fields). It is often preferable to think of complex numbers as
being the true and proper setting for algebraic operations, with real
numbers being the limited subset for which

.
The Complex Plane
Figure 2.2:
Plotting a complex number as a point in the complex plane.
![\includegraphics[scale=0.5]{eps/ComplexPlane}](http://www.dsprelated.com/josimages_new/mdft/img176.png) |
We can plot any complex number

in a plane as an ordered pair

, as shown in Fig.
2.2. A
complex plane (or
Argand diagram) is any 2D graph in which the horizontal axis is
the
real part and the vertical axis is the
imaginary
part of a complex number or function. As an example, the number

has coordinates

in the complex plane while the number

has
coordinates

.
Plotting

as the point

in the complex plane can be
viewed as a plot in
Cartesian or
rectilinear coordinates. We can
also express complex numbers in terms of
polar coordinates as
an ordered pair

, where

is the distance from the
origin

to the number being plotted, and

is the angle
of the number relative to the positive real coordinate axis (the line
defined by

and

). (See Fig.
2.2.)
Using elementary
geometry, it is quick to show that conversion from
rectangular to polar coordinates is accomplished by the formulas
where

denotes the arctangent of

(the angle

in radians whose tangent is

), taking the
quadrant of the vector

into account. We will take

in
the range

to

(although we could choose any interval of
length

radians, such as 0 to

, etc.).
In
Matlab and Octave,
atan2(y,x) performs the
``quadrant-sensitive'' arctangent function. On the other hand,
atan(y/x), like the more traditional mathematical notation

does not ``know'' the quadrant of

, so it maps
the entire real line to the interval

. As a specific
example, the angle of the vector

(in quadrant I) has the
same tangent as the angle of

(in quadrant III).
Similarly,

(quadrant II) yields the same tangent as

(quadrant IV).
The formula

for converting rectangular
coordinates to radius

, follows immediately from the
Pythagorean theorem, while the

follows from the definition of the tangent
function itself.
Similarly, conversion from polar to rectangular coordinates is simply
These follow immediately from the definitions of cosine and sine,
respectively.
More Notation and Terminology
It's already been mentioned that the rectilinear coordinates of a
complex
number 
in the
complex plane are called the
real part and
imaginary part, respectively.
We also have special notation and various names for the
polar
coordinates

of a complex number

:
The
complex conjugate of

is denoted

(or

) and is defined by
where, of course,

.
In general, you can always obtain the complex conjugate of any expression
by simply replacing

with

. In the complex plane, this is a
vertical flip about the real axis;
i.e., complex conjugation
replaces each point in the complex plane by its
mirror image on the
other side of the

axis.
From the above definitions, one can quickly verify
Let's verify the third relationship which states that a
complex number
multiplied by its conjugate is equal to its magnitude squared:
 |
(2.4) |
Since

is the algebraic expression of

in terms of its
rectangular coordinates, the corresponding expression in terms of its polar
coordinates is
There is another, more powerful representation of

in terms of its
polar coordinates. In order to define it, we must introduce
Euler's
identity:
 |
(2.5) |
A proof of Euler's identity is given in the next chapter.
Before, the only algebraic representation of a
complex number we had was

, which fundamentally uses Cartesian (rectilinear) coordinates in
the
complex plane. Euler's identity gives us an alternative
representation in terms of polar coordinates in the complex plane:
We'll call

the
polar form of the complex number

, in contrast with the
rectangular form 
. Polar
form often simplifies algebraic manipulations of complex numbers,
especially when they are multiplied together. Simple rules of
exponents can often be used in place of messier trigonometric
identities. In the case of two complex numbers being multiplied, we
have
A corollary of Euler's identity is obtained by setting

to get
This has been called the ``most beautiful formula in mathematics'' due
to the extremely simple form in which the fundamental constants

, and 0, together with the elementary operations of addition,
multiplication, exponentiation, and equality, all appear exactly once.
For another example of manipulating the polar form of a complex number,
let's again verify

, as we did above in
Eq.

(
2.4), but this time using polar form:
As mentioned in §
2.7, any complex expression can be conjugated
by replacing

by

wherever it occurs. This implies

,
as used above. The same result can be obtained by using Euler's
identity to expand

into

and negating the imaginary part
to obtain

,
where we used also the fact that cosine is an
even function
(

) while sine is
odd
(

).
We can now easily add a fourth line to that set of examples:
Thus,

for every

.
Euler's identity can be used to derive formulas for sine and cosine in
terms of

:
Similarly,

, and
we obtain the following classic identities:
De Moivre's Theorem
As a more complicated example of the value of the polar form, we'll prove
De Moivre's theorem:
Working this out using
sum-of-angle identities from
trigonometry is
laborious (see §
3.13 for details). However, using
Euler's identity, De Moivre's theorem simply ``falls out'':
Moreover, by the power of the method used to show the result,

can be any
real number, not just an integer.
This chapter has covered just enough about
complex numbers to enable
us to talk about the discrete
Fourier transform.
Manipulations of complex numbers in
Matlab and Octave are illustrated
in §
I.1.
To explore further the mathematics of
complex variables, see any
textbook such as Churchill [
15] or LePage [
37].
Topics not covered here, but which are important elsewhere in
signal
processing, include analytic functions, contour integration, analytic
continuation, residue
calculus, and
conformal mapping.
See
http://ccrma.stanford.edu/~jos/mdftp/Complex_Number_Problems.html
Next Section: Proof of Euler's IdentityPrevious Section: Introduction to the DFT