View source for Signal Representation/Calculating with Complex Numbers

{{Header
|Untermenü=Basic Terms of Communications Engineering
|Vorherige Seite=Signal Classification
|Nächste Seite=General Description
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==The set of real numbers==
<br>	 
In the following chapters of this book,&nbsp; complex quantities always play an important role.&nbsp; Although calculating with complex numbers is already treated and practiced in school mathematics,&nbsp; our experience has shown that even students of natural sciences and technical subjects have problems with it.&nbsp; Perhaps these difficulties are also related to the fact that&nbsp; "complex"&nbsp; is often used as a synonym for&nbsp; "complicated"&nbsp; in everyday life,&nbsp; while&nbsp; "real"&nbsp; stands for&nbsp; "reliable, honest and truthful".

Therefore,&nbsp; the calculation rules for complex numbers are briefly summarized here at the end of this first basic chapter.

First there are some remarks about real quantities of numbers,&nbsp; for which in the strict mathematical sense the term&nbsp; &raquo;number fields&laquo; would be more correct.&nbsp; These include:

{{BlaueBox|TEXT=
$\text{Definitions:}$&nbsp;
*&raquo;'''Natural Numbers'''&laquo;&nbsp; $\mathbb{N} = \{1, 2, 3, \text{...}\hspace{0.05cm} \}$. &nbsp; Using these numbers, for&nbsp; $n, \ k \in \mathbb{N}$&nbsp; the arithmetic operations&nbsp; &raquo;addition&laquo;&nbsp; $(m = n +k)$,&nbsp; &raquo;multiplication&laquo;&nbsp; $(m = n \cdot k)$&nbsp; and&nbsp; &raquo;potency formation&laquo;&nbsp; $(m = n^k)$&nbsp; are possible.&nbsp; The respective result of a calculation is again a natural number: &nbsp; $m \in \mathbb{N}$.


*&raquo;'''Integer Numbers'''&laquo;&nbsp; $\mathbb{Z} = \{\text{...}\hspace{0.05cm} , -3, -2, -1, \ 0, +1, +2, +3, \text{...}\hspace{0.05cm}\}$. &nbsp; This set of numbers is an extension of the natural numbers&nbsp; $\mathbb{N}$.&nbsp; The introduction of the set&nbsp; $\mathbb{Z}$&nbsp; was necessary to capture the result set of a subtraction&nbsp; $(m = n -k$,&nbsp; for example&nbsp; $5 - 7 = - 2)$.


*&raquo;'''Rational Numbers'''&laquo;&nbsp; $\mathbb{Q} = \{z/n\}$&nbsp; with&nbsp; $z \in \mathbb{Z}$&nbsp; and&nbsp; $n \in \mathbb{N}$. &nbsp; With this set of numbers, also known as fractions, there is a defined result for each division.&nbsp; If you write a rational number in decimal notation, only zeros appear after a certain decimal place&nbsp; $($Example:&nbsp; $-2/5 = -0.400\text{...}\hspace{0.05cm})$&nbsp; or periodicities&nbsp; $($Example:&nbsp; $2/7 = 0.285714285\text{...}\hspace{0.05cm})$.&nbsp; Since&nbsp; $n = 1$&nbsp; is allowed,&nbsp; the integers are a subset of the rational numbers: &nbsp; $\mathbb{Z} \subset \mathbb{Q}$.


*&raquo;'''Irrational Numbers'''&laquo;&nbsp; $\mathbb{I} \neq {z/n}$&nbsp; mit&nbsp; $z \in \mathbb{Z}$, $n \in \mathbb{N}$. &nbsp; Although there are infinite rational numbers, there are still infinite numbers which cannot be represented as a fraction.&nbsp; Examples are the number&nbsp;  $\pi = 3.141592654\text{...}\hspace{0.05cm}$&nbsp;  $($where there are no periods even with more decimal places$)$&nbsp; or the result of the equation &nbsp; $a^{2}=2 \,\,\Rightarrow \;\;a=\pm \sqrt{2}=\pm1.414213562\text{...}\hspace{0.05cm}$.&nbsp; This result is also irrational, which has already been proved by&nbsp; [https://en.wikipedia.org/wiki/Euclid $\text{Euclid}$]&nbsp; in antiquity.
[[File:EN_Sig_T_1_3_S1.png |right|frame|Real numbers on the number line]]


*&raquo;'''Real Numbers'''&laquo;&nbsp; $\mathbb{R} = \mathbb{Q}  \cup  \mathbb{I}$ as the sum of all rational and irrational numbers. 
:These can be ordered according to their numerical values and can be drawn on the so-called&nbsp; "number line"&nbsp; as shown in the adjacent graph.}}




==Imaginary and complex numbers==
<br>
With the introduction of the irrational numbers the solution of the equation&nbsp; $a^2-2=0$&nbsp; was possible,&nbsp; but not the solution of the equation&nbsp; $a^2+1=0$.&nbsp; The mathematician&nbsp; [https://en.wikipedia.org/wiki/Leonhard_Euler $\text{Leonhard Euler}$]&nbsp; solved this problem by extending the set of real numbers by the&nbsp; &raquo;imaginary numbers&laquo;&nbsp;. He defined the&nbsp; &raquo;'''imaginary unit'''&laquo;&nbsp; as follows:

:$${\rm j}=\sqrt{-1} \ \Rightarrow \ {\rm j}^{2}=-1.$$
 
It should be noted that Euler called this quantity&nbsp; "$\rm i$"&nbsp; and this is still common in mathematics today.&nbsp; In Electrical Engineering,&nbsp; on the other hand,&nbsp; the designation&nbsp; "$\rm j$"&nbsp; has become generally accepted since&nbsp; "$\rm i$"&nbsp; is already occupied by the time-dependent current.

{{BlaueBox|TEXT=
$\text{Definition:}$&nbsp;
The&nbsp; &raquo;'''complex number'''&laquo;&nbsp; $z$&nbsp; is generally the sum of a real number&nbsp; $x$&nbsp; and an imaginary number&nbsp; ${\rm j} \cdot y$:

:$$z=x+{\rm j}\cdot y.$$

*$x$&nbsp; and&nbsp; $y$&nbsp; are derived from the quantity&nbsp; $\mathbb{R}$&nbsp; from the real numbers. 

*The set of all possible complex numbers is called the body&nbsp; $\mathbb{C}$&nbsp; of the complex numbers.}}


The number line of real numbers now becomes the complex plane,&nbsp; which is spanned by two number lines twisted by &nbsp; $90^\circ$&nbsp; for real part and imaginary part.

{{GraueBox|TEXT=
$\text{Example 1:}$&nbsp;
[[File:P_ID823_Sig_T_1_3_S2_neu.png|right|frame|Numbers in the complex plane]]

*The complex number&nbsp; $z_1 = 2 \cdot {\rm j}$&nbsp; is one of two possible solutions of the equation&nbsp; $z^2+4=0$. The other solution is&nbsp; 
:$$z_2 = -2 \cdot {\rm j}.$$

*In contrast&nbsp; $z_3 = 2 + {\rm j}$&nbsp; and&nbsp; $z_4 = 2 -{\rm j}$&nbsp; give the two solutions to the following equation:&nbsp;

:$$(z-2- {\rm j})(z-2+ {\rm j}) = 0 \; \ \Rightarrow \;\ z^{2}-4 \cdot z+5=0.$$

: $z_4 = z_3^\ast$&nbsp; is also called the&nbsp; &raquo;complex conjugate&laquo;&nbsp; of&nbsp; $z_3$. 
*The sum&nbsp; $z_3 + z_4$&nbsp; is real:&nbsp;

:$$z_3 + z_4 = 2 \cdot {\rm Re}[z_3]=2 \cdot {\rm Re}[z_4].$$

*The difference&nbsp; $z_3 - z_4$&nbsp; is purely imaginary:&nbsp;

:$$z_3 - z_4 = {\rm j} \cdot \big [2 \cdot {\rm Im}[z_3] \big ] ={\rm j} \cdot \big [-2 \cdot {\rm Im}[z_4] \big ].$$}}


<u>Note</u>: &nbsp; In the literature,&nbsp; complex quantities are often marked by underlining.&nbsp; This is not used in the&nbsp; $\rm LNTwww$&nbsp; books.

 	 
== Representation by magnitude and phase== 
<br>
A complex number&nbsp; $z$&nbsp; can be described not only by the real part&nbsp; $x$&nbsp; and the imaginary part&nbsp; $y$&nbsp; but also by its magnitude&nbsp; $|z|$&nbsp; and the phase&nbsp; $\phi$&nbsp;. 

[[File:P_ID1246__Sig_T_1_3_S3_neu.png|right|frame|Complex conjugate <br>of a number]] 
*The following conversions apply:

:$$\left | z \right | = \sqrt{x^{2}+y^{2}},  \hspace{0.6cm}\phi = \arctan ({y}/{x}),$$

:$$x = |z| \cdot \cos(\phi), \hspace{0.6cm} y = |z| \cdot \sin(\phi ).$$
 
*Thus the complex quantity&nbsp; $z$&nbsp; can also be displayed in the following form:
 
:$$z = |z| \cdot \cos (\phi) + {\rm j} \cdot |z| \cdot \sin (\phi) = |z| \cdot {\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi}.$$

*Here,&nbsp; the&nbsp; $\text{Euler's theorem}$&nbsp; was used,&nbsp; which is proved below. &nbsp;<br>This states that the complex quantity&nbsp; $ {\rm e}^{{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \phi}$&nbsp;exhibits the real part&nbsp; $\cos(\phi)$&nbsp; and the imaginary part&nbsp; $\sin(\phi)$&nbsp;.

*Further one recognizes from the diagram that for the&nbsp; &raquo;complex conjugates&laquo;&nbsp; of&nbsp; $z = x + {\rm j}\cdot y$&nbsp; applies:

:$$z^{\star} = x - {\rm j} \cdot y = |z| \cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}\phi}.$$

{{BlaueBox|TEXT=
$\text{Proof of the Euler theorem:}$&nbsp; This is based on the comparison of power series developments. 
*The series development of the exponential function is:&nbsp;
 
:$${\rm e}^{x} = 1 +  \frac{x}{1!}+  \frac{x^2}{2!}+  \frac{x^3}{3!}
+  \frac{x^4}{4!} +\text{ ...} \hspace{0.15cm}.$$

*With an imaginary argument you can also write:&nbsp;
 
:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = 1 +  {\rm j} \cdot \frac{x}{1!}+ {\rm j}^2 \cdot \frac{x^2}{2!}+ {\rm j}^3 \cdot \frac{x^3}{3!}
+ {\rm j}^4 \cdot \frac{x^4}{4!} + \text{ ...}  \hspace{0.15cm}.$$

*Considering&nbsp; <math>{\rm j}^{2}=-1, \ \ {\rm j}^{3} = -{\rm j},\ \ {\rm j}^{4} = 1, \ \ {\rm j}^{5} = {\rm j},  \text{ ...} \hspace{0.15cm}</math>&nbsp; and combining the real and the imaginary terms,&nbsp; one obtains

:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = A(x) +  {\rm j}\cdot B(x).$$

* The following applies to both series:
:$$A(x) = 1 - \frac{x^2}{2!}
+  \frac{x^4}{4!} -  \frac{x^6}{6!}+
\text{ ...} \hspace{0.1cm}=
\cos(x),\hspace{0.5cm}
B(x) =  \frac{x}{1!}-  \frac{x^3}{3!}
+  \frac{x^5}{5!} -  \frac{x^7}{7!}+
\text{ ...}=
\sin(x).$$
 
*From this the&nbsp; $\text{Euler Theorem}$&nbsp; follows directly: 
 
:$${\rm e}^{ {\rm j}\hspace{0.03cm} \cdot \hspace{0.03cm}x} = \cos (x) +  {\rm j} \cdot \sin (x) \hspace{2cm}                                            
\rm q.e.d.$$}}


==Calculation laws for complex numbers==
<br>
The arithmetic laws for two complex numbers
 
:$$z_1 = x_1 + {\rm j} \cdot y_1 = |z_1| \cdot {\rm e}^{{\rm j}\hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_1},$$ 
:$$z_2 = x_2 + {\rm j} \cdot y_2 = |z_2| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_2}$$

are defined in such a way,&nbsp; that for the special case of a vanishing imaginary part,&nbsp; the calculation rules for real numbers are given.&nbsp; This is called&nbsp; "principle of permanence".&nbsp; The following rules apply to the basic arithmetic operations:&nbsp;
*The sum of two complex numbers&nbsp; $($resp. their difference$)$&nbsp; is made by adding&nbsp; $($resp. subtracting$)$&nbsp; their real and imaginary parts :

::<math>z_3 = z_1 + z_2 = (x_1+x_2) + {\rm j}\cdot (y_1 + y_2),</math> 
 
::<math>z_4 = z_1 - z_2 = (x_1-x_2) + {\rm j}\cdot (y_1 - y_2).</math> 

*The product of two complex numbers can be formed in the real part and imaginary part description by multiplication considering&nbsp; <math>{\rm j}^{2}=-1</math>.&nbsp; <br>However,&nbsp; multiplication is simpler if&nbsp; <math>z_1</math>&nbsp; and&nbsp; <math>z_2</math>&nbsp; are written with magnitude and phase:&nbsp;
 
::<math>z_5 = z_1 \cdot z_2 = (x_1\cdot x_2 - y_1\cdot y_2) + {\rm j}\cdot (x_1\cdot y_2 + x_2\cdot y_1),</math>

::<math>z_5 =  |z_1| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_1}
 \cdot |z_2| \cdot {\rm e}^{{\rm j}\hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_2}=  |z_5| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_5}
 \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} |z_5|  = |z_1|  \cdot |z_2| , \hspace{0.3cm}\phi_5 = \phi_1 + \phi_2 .</math>
 
*The division is also more manageable in the exponential notation.&nbsp; The two magnitudes are divided and the phases are subtracted in the exponent:&nbsp;

::<math>z_6 =  \frac{z_1}{z_2} = |z_6| \cdot {\rm e}^{{\rm j} \hspace {0.05cm}\cdot
\hspace {0.05cm} \phi_6}
 \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} |z_6| =  \frac{|z_1|}{|z_2|}, \hspace{0.3cm}\phi_6 = \phi_1 - \phi_2 .</math>

{{GraueBox|TEXT=
[[File:P_ID825_Sig_T_1_3_S4_neu.png|right|frame|Some operations with complex numbers]]

$\text{Example 2:}$&nbsp;
In the graphic are shown as points within the complex plane:

*The complex number&nbsp; <math>z=0.75 + {\rm j} = 1.25 \cdot {\rm e}^{\hspace{0.03cm}{\rm j}\hspace{0.03cm} \cdot \hspace{0.05cm}53.1^{\circ} }</math>,

*its complex conjugate&nbsp; <math>z^{\ast} = 0.75 - {\rm j} = 1.25 \cdot {\rm e}^{ - {\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm}53.1^{\circ} }</math>,
 
*the sum&nbsp; <math>s=z+z^{\ast}=1.5</math>&nbsp; $($purely real$)$, 

*the difference&nbsp; <math>d=z-z^{\ast}=2 \cdot {\rm j}</math>&nbsp; $($purely imaginary$)$,

*the product&nbsp; <math>p=z \cdot z^{\ast} = 1.25^{2} \approx 1.5625</math>&nbsp; $($purely real$)$,

*the division&nbsp; <math>q= {z}/{z^{\ast} }={\rm e}^{\hspace{0.05cm} {\rm j} \hspace{0.03cm}\cdot \hspace{0.03cm}106.2^{\circ} }</math>&nbsp;with magnitude &nbsp; $1$&nbsp; and the double phase angle of&nbsp; $z$.}}


The following&nbsp;  $($German language$)$&nbsp; learning video summarizes the topic of this chapter in a compact way:<br> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;[[Rechnen_mit_komplexen_Zahlen_(Lernvideo)|&raquo;Rechnen mit komplexen Zahlen&laquo;]] &nbsp; &rArr; &nbsp; "Arithmetic operations involving complex numbers".


 	 
==Exercises for the chapter==
<br>
[[Aufgaben:Exercise_1.3:_Calculating_With_Complex_Numbers|Exercise 1.3: Calculating with Complex Numbers]]

[[Aufgaben:Exercise_1.3Z:_Calculating_with_Complex_Numbers_II|Exercise 1.3Z: Calculating with Complex Numbers II]]

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