In order to propely discuss the concept of vector spaces in linear algebra, it is necessary to develop the notion of a set of “scalars” by which we allow a vector to be multiplied. A framework within which our concept of real numbers would fit is desireable. Thus, we would like a set with two associative, commutative operations (like standard addition and multiplication) and a notion of their inverse operations (like subtraction and division). The mathematical algebraic construct that addresses this idea is the field. A field $(S,+,*)$ is a set $S$ together with two binary operations $+$ and $*$ such that the following properties are satisfied.
While this definition is quite general, the two fields used most often in signal processing, at least within the scope of this course, are the real numbers and the complex numbers, each with their typical addition and multiplication operations.
The reader is undoubtedly already sufficiently familiar with the real numbers with the typical addition and multiplication operations. However, the field of complex numbers with the typical addition and multiplication operations may be unfamiliar to some. For that reason and its importance to signal processing, it merits a brief explanation here.
The notion of the square root of $-1$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $\sqrt{-1}$ could be defined. Euler first used $i$ for the imaginary unit but that notation did not take hold until roughly Ampère's time. Ampère used the symbol $i$ to denote current (intensité de current). It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using $i$ for current was entrenched and electrical engineers now choose $j$ for writing complex numbers.
An imaginary number has the form $jb=\sqrt{-{b}^{2}}$. A complex number, $z$, consists of the ordered pair ($a$,$b$), $a$ is the real component and $b$ is the imaginary component (the $j$ is suppressed because the imaginary component of the pair is always in the second position). The imaginary number $jb$ equals ($0$,$b$). Note that $a$ and $b$ are real-valued numbers.
Figure 1 shows that we can locate a complex number in what we call the complex plane. Here, $a$, the real part, is the $x$-coordinate and $b$, the imaginary part, is the $y$-coordinate.
The real part of the complex number $z=a+jb$, written as $\Re \left(z\right)$, equals $a$. We consider the real part as a function that works by selecting that component of a complex number not multiplied by $j$. The imaginary part of $z$, $\Im \left(z\right)$, equals $b$: that part of a complex number that is multiplied by $j$. Again, both the real and imaginary parts of a complex number are real-valued.
The complex conjugate of $z$, written as $\overline{z}$, has the same real part as $z$ but an imaginary part of the opposite sign.
Using Cartesian notation, the following properties easily follow.
Use the definition of addition to show that the real and imaginary parts can be expressed as a sum/difference of a complex number and its conjugate. $\Re \left(z\right)=\frac{z+\overline{z}}{2}$ and $\Im \left(z\right)=\frac{z-\overline{z}}{2j}$.
$z+\overline{z}=a+jb+a-jb=2a=2\Re \left(z\right)$. Similarly, $z-\overline{z}=a+jb-(a-jb)=2jb=2\left(j,\Im \left(z\right)\right)$
Complex numbers can also be expressed in an alternate form, polar form, which we will find quite useful. Polar form arises arises from the geometric interpretation of complex numbers. The Cartesian form of a complex number can be re-written as $$a+jb=\sqrt{{a}^{2}+{b}^{2}}(\frac{a}{\sqrt{{a}^{2}+{b}^{2}}}+j\frac{b}{\sqrt{{a}^{2}+{b}^{2}}})$$ By forming a right triangle having sides $a$ and $b$, we see that the real and imaginary parts correspond to the cosine and sine of the triangle's base angle. We thus obtain the polar form for complex numbers. $$\begin{array}{l}z=a+jb=r\mathrm{\angle}\theta \\ r=\left|z\right|=\sqrt{{a}^{2}+{b}^{2}}\\ a=r\mathrm{cos}\left(\theta \right)\\ b=r\mathrm{sin}\left(\theta \right)\\ \theta =\mathrm{arctan}\left(\frac{b}{a}\right)\end{array}$$ The quantity $r$ is known as the magnitude of the complex number $z$, and is frequently written as $\left|z\right|$. The quantity $\theta $ is the complex number's angle. In using the arc-tangent formula to find the angle, we must take into account the quadrant in which the complex number lies.
Convert $3-2j$ to polar form.
To convert $3-2j$ to polar form, we first locate the number in the complex plane in the fourth quadrant. The distance from the origin to the complex number is the magnitude $r$, which equals $\sqrt{13}=\sqrt{{3}^{2}+{\left(-2\right)}^{2}}$. The angle equals $-\mathrm{arctan}\left(\frac{2}{3}\right)$ or $-0.588$ radians ($-33.7$ degrees). The final answer is $\sqrt{13}\angle (-33.7)$ degrees.
Surprisingly, the polar form of a complex number $z$ can be expressed mathematically as
Adding and subtracting complex numbers expressed in Cartesian form is quite easy: You add (subtract) the real parts and imaginary parts separately.
What is the product of a complex number and its conjugate?
$z\overline{z}=(a+jb)(a-jb)={a}^{2}+{b}^{2}$. Thus, $z\overline{z}={r}^{2}={\left(\left|z\right|\right)}^{2}$.
Division requires mathematical manipulation. We convert the division problem into a multiplication problem by multiplying both the numerator and denominator by the conjugate of the denominator.
The properties of the exponential make calculating the product and ratio of two complex numbers much simpler when the numbers are expressed in polar form.