eed1004- introduction to signals instructor: dr. gülden köktürk copyright 2012 | instructor: dr....

EED1004-Introduction to Signals

Instructor: Dr. Gülden Köktürk

Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING

COMPLEX NUMBERS

•


Rectangular Notation for Complex Numbers

where and • Ordered pair can be interpreted as a

point in the two-dimensional plane.• Rectangular notation is also called Cartesian notation.

Examples


Polar Notation for Complex Numbers

As you see in the picture, complex vector is sometimes defined by its length (r), and angle (ϴ).

Examples

Note that always



Conversion between Rectangular and Polar Notations

• Both polar and rectangular forms are commonly used to represent complex numbers.

• For representing sinusoidal signals, polar form is especially useful. But, at some other times, rectangular form is preferred. Thus, we have to know how to convert between both forms.


Examples


Exercise: Convert the following rectangular notation complex numbers into polar form.

4-j3, 2+j5, 0+j3, -3-j3, -5+j0


EULER’S FORMULA

is called complex exponential, which is equivalent to (a vector of length 1 at angle ϴ)


Examples


Conversion between Degrees and Radians

Example: If ϴ is radians, then


Inverse Euler Formulas

Proof:

Exercise: Prove

İn a similar way.


ALGEBRAIC RULES FOR COMPLEX NUMBERS

Addition:

Subtracion:

Multiplication:


Conjugate:

Division:

All these are done in rectangular notation. Multiplication, conjugate and division are easy in polar form.


Multiplication:

Conjugate:

Division:

Exercises: 1) Add the following complex numbers and then plot the result.

2) Multiply the following complex numbers and then plot the result.


3)

4)

5)

Prove that the following identites are true.

6)

7) Im

8)


GEOMETRIC VIEWS OF COMPLEX OPERATIONS

Addition:

(4-j3)+(2+j5)=6+j2

What is the addition of following four complex numbers?

(1+j)+(-1+j)+(-1-j)+(1-j)=?


Subtraction:

z1= -1-j2

z2=5+j1

-z1=1+j2

z2-z1=z2+(-z1)=5+j1+1+j2=z3=6+j3

Multiplication: Multiplication can be viewed best in polar form.


Division:

Very similar to multiplication. Instead of adding, we now subtract angles, and instead of multiplication, we divide the lengths.

Exercise: Two complex numbers z1 and z2 are given. The difference between the angles of z1 and z2 is 90° (ϴ2-ϴ1=90°). Also , length of z2 is twice the length of z1 (r2=2r1). Evaluate .


Rotation:

Rotation is a special case of multiplicaiton. Assume the length of z2 is equal to 1 (. Then, we can write z2 as . If we multiply z2 with another complex number, we obtain

Thus, length of z1 does not change and remains as r1. But, its angle changes and becomes ϴ1+ϴ2.

For example, if z2=j then and and . Then, multiplication by becomes a rotation by or .


Conjugate:

Inverse: Inverse is a special case of division when z1=1. Because, in that case

Thus, angle is made negative (-ϴ) and length is inverted .


POWERS AND ROOTS

Integer powers of a complex number can be defined in the following manner:

length is raised to the Nth power angle is multiplied by N.

Note that if , successive powers spiral towards the origin.

If , all powers lie on the unit circle.


De Moivre’s Formula

How do you prove this?

Example: Let be three consecutive members of a sequence such as in the example above. If and N=11, plot the three numbers .

Roots of Unity: In many problems related to signals, we have to solve the following equation:

=1 where N is an integer. One solution is obviously z=1.


It can be shown that all solutions are given by

Example: Solve the equation =1.

This time N=7.

First note that (l is an integer) Why?

Let’s write z in polar form .


Thus, 7th roots of unity are given as

eed1004- introduction to signals instructor: dr. gülden köktürk copyright 2012 | instructor: dr....

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