EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr....
-
Upload
samson-fletcher -
Category
Documents
-
view
223 -
download
2
Transcript of EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr....
![Page 1: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/1.jpg)
EED1004-Introduction to Signals
Instructor: Dr. Gülden Köktürk
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
![Page 2: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/2.jpg)
COMPLEX NUMBERS
•
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
![Page 3: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/3.jpg)
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
![Page 4: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/4.jpg)
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
![Page 5: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/5.jpg)
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.
![Page 6: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/6.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Examples
![Page 7: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/7.jpg)
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
![Page 8: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/8.jpg)
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)
![Page 9: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/9.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Examples
![Page 10: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/10.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Conversion between Degrees and Radians
Example: If ϴ is radians, then
![Page 11: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/11.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Inverse Euler Formulas
Proof:
Exercise: Prove
İn a similar way.
![Page 12: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/12.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
ALGEBRAIC RULES FOR COMPLEX NUMBERS
Addition:
Subtracion:
Multiplication:
![Page 13: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/13.jpg)
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.
![Page 14: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/14.jpg)
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.
![Page 15: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/15.jpg)
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)
![Page 16: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/16.jpg)
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)=?
![Page 17: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/17.jpg)
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.
![Page 18: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/18.jpg)
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 .
![Page 19: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/19.jpg)
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 .
![Page 20: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/20.jpg)
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 .
![Page 21: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/21.jpg)
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.
![Page 22: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/22.jpg)
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.
![Page 23: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/23.jpg)
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 .
![Page 24: EED1004- Introduction to Signals Instructor: Dr. Gülden Köktürk Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING.](https://reader036.fdocuments.net/reader036/viewer/2022062408/56649e8f5503460f94b93d98/html5/thumbnails/24.jpg)
Copyright 2012 | Instructor: Dr. Gülden Köktürk | EED1004-INTRODUCTION TO SIGNAL PROCESSING
Thus, 7th roots of unity are given as