EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.
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EED1004-Introduction to Signals
Instructor: Dr. Gülden Köktürk
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
COMPLEX NUMBERS
•
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Examples
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Exercise: Convert the following rectangular notation complex numbers into polar form.
4-j3, 2+j5, 0+j3, -3-j3, -5+j0
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
EULER’S FORMULA
is called complex exponential, which is equivalent to (a vector of length 1 at angle ϴ)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Examples
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Conversion between Degrees and Radians
Example: If ϴ is radians, then
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Inverse Euler Formulas
Proof:
Exercise: Prove
İn a similar way.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
ALGEBRAIC RULES FOR COMPLEX NUMBERS
Addition:
Subtracion:
Multiplication:
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Conjugate:
Division:
All these are done in rectangular notation. Multiplication, conjugate and division are easy in polar form.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
3)
4)
5)
Prove that the following identites are true.
6)
7) Im
8)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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)=?
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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 .
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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 .
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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 .
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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.
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
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 .
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Thus, 7th roots of unity are given as
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