Remember ``factoring polynomials''? Consider the second-order polynomial
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
The general second-order (real) polynomial is
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
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
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
As a simple example, let , , and , i.e.,
It can be checked that all algebraic operations for real numbers2.2 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 ).
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:
As mentioned above, for any negative number , we have
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.)
Note that the real numbers are the subset of the complex numbers having a zero imaginary part ().
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 .
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.
We also have special notation and various names for the polar coordinates of a complex number :
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:
Since is the algebraic expression of in terms of its rectangular coordinates, the corresponding expression in terms of its polar coordinates is
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:
A corollary of Euler's identity is obtained by setting to get
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:
We can now easily add a fourth line to that set of examples:
Euler's identity can be used to derive formulas for sine and cosine in terms of :
Similarly, , and we obtain the following classic identities:
As a more complicated example of the value of the polar form, we'll prove De Moivre's theorem:
To explore further the mathematics of complex variables, see any textbook such as Churchill  or LePage . Topics not covered here, but which are important elsewhere in signal processing, include analytic functions, contour integration, analytic continuation, residue calculus, and conformal mapping.
Proof of Euler's Identity
Introduction to the DFT