CT111 Introduction to Communication Systems Lecture 4...

Signal Model Complex Numbers

CT111 Introduction to Communication SystemsLecture 4: Complex Numbers; Fourier Transform

Yash M. Vasavada

Associate Professor, DA-IICT, Gandhinagar

10th January 2018

Yash M. Vasavada (DA-IICT) CT111: Intro to Comm. Systems 10th January 2018 1 / 120

Overview of Today’s Talk1 Signal Model2 Complex Numbers

Square Root of Negative NumbersProperties of NumbersComplex Numbersnth Roots of UnityComplex Powers

Overview of Today’s Talk1 Signal Model2 Complex Numbers

Square Root of Negative NumbersProperties of NumbersComplex Numbersnth Roots of UnityComplex Powers

A Standard Form of Communication SignalElectromagnetic Radiation

We have talked about how the communication system in our brains ismore powerful than any technological communication system devisedso farHowever, there is one area where the technology has now outpacedthe humans: ability to generate the communication signals that travelfar awayThe key signal that makes the long distance communication possibleis given by Electromagnetics, and it has a wavy shape that you seewhen you drop a pebble in a pond

A Standard Form of Communication SignalElectromagnetic Radiation

The basic ingredient of a communication signal is a periodicsinusoidal waveform given as: sr (t) = A cos (Θ(t)) = A cos (2πft + θ)

A Standard Form of Communication SignalSinusoidal Electromagnetic Radiation

The basic ingredient of a communication signal is a periodicsinusoidal waveform, with a cycle duration of Tcycle given as:sr (t) = A cos (Θ(t)) = A cos (2πft + θ)A : is called the amplitude of the signalf : is the frequency of the signal, and it is the inverse of the cycleduration Tcycle . This is measured in HertzΘ(t) : is the phase angle of the signal, measured in radiansθ : is the initial phase angle (at time t = 0)

A Standard Form of Communication SignalComplex Exponential

Sinusoidal waveform can be thought of as a component of a moregeneral, complex-valued, waveform:

s(t) = A exp (iΘ(t)) = A exp (i(2πft + θ))

= sr (t) + isi (t)

→ s(t) : is called the complex phasor→ sr (t) : is the real part of this phasor→ si (t) : is the imaginary part of this phasor→ i : is the imaginary number

√−1

Several new concepts: complex phasor, its real and imaginary part,and i =

√−1

Square Root of Negative Numbers

A Real Tale of Imaginary NumbersTaking Square Roots

What is square-root of −1, i.e.,√−1?

Ancient mathematicians had to grapple with this, and they concludedthat such a number cannot be a real number. Why?

→ Square 1, and you get 1. Square −1, and you also get 1.→ Alternatively, there is not a way to take any real number, square it and

get −1 as the result.→ This surely means conclusively that no such number as

√−1 can really

exist.→ OR does it? Let’s make sure.

A Real Tale of Imaginary NumbersAn Algebraic View

y = x2 defines a parabola.This parabola intersects y = 1 line two times at ±1, however...it never meets y = −1 line!

A Real Tale of Imaginary NumbersAn Algebraic/Graphical View

A Tale of Imaginary NumbersAn Algebraic View

Equation for the non-horizontal lines: y = bx + c .The solutions to quadratic equation x2 − bx − c = 0 are given by thepoint(s) of intersection of parabola y = x2 with the line y = bx + c.We know that the solution to this quadratic equation is given as

x = −1

√b2 − 4ab

)When b2 − 4ab < 0, we again have the problem of taking square-rootof a negative number. This happens because the line never meets theparabola.

A Real Tale of Imaginary NumbersAn Algebraic/Graphical View

A Real Tale of Imaginary NumbersAn Algebraic View

Okay, all of these settles the issue.Absurd to talk/think about

√−1.

A Real Tale of Imaginary NumbersThird Order Polynomials

Let’s see the situation for the third-order polynomials:

x3 = bx + c

The line y = bx + c always meets the cubic x3

Okay, looks like for the cubic polynomials x3 = bx + c , we should nothave to worry about situations in which square-root of a negativenumber is required.In the year 1545, a mathematician in Italy, Gerolamo Cardano,provided a general formula for solving the cubic polynomialsx3 = bx + c

√q +

√q2 − p3 +

√q −

√q2 − p3,

where q = c/2 and p = b/3

A Real Tale of Imaginary NumbersBombelli’s Problem

Cardano’s formula in the year 1545 for solving the cubic polynomialsx3 = bx + c:

√q +

√q2 − p3 +

√q −

√q2 − p3,

where q = c/2 and p = b/3If we don’t use Cardano’s formula, the solution is clear, and asexpected, it is a real number x = 4.However, if we do use Cardano’s formula, we obtain

√2 + 11

√−1 +

√2− 11

√−1,

→ The square-root of −1 has reared its ugly head here→ This was first observed by another mathematician Rafeal Bombelli in

about 1575

√q +

√q2 − p3 +

√q −

√q2 − p3,

√2 + 11

√−1 +

√2− 11

√−1,

about 1575

√q +

√q2 − p3 +

√q −

√q2 − p3,

√2 + 11

√−1 +

√2− 11

√−1,

about 1575

√q +

√q2 − p3 +

√q −

√q2 − p3,

√2 + 11

√−1 +

√2− 11

√−1,

about 1575

√q +

√q2 − p3 +

√q −

√q2 − p3,

√2 + 11

√−1 +

√2− 11

√−1,

about 1575

How Bombelli would have likely reacted:

First reaction: blame Cardano, he messed up!Second reaction (when it turned out that there was nothing wrong inCardano’s formula): hit the head against a wallFinal solution (after cooling down and thinking through the problem):discovery of imaginary and complex numbers

Properties of Numbers

Definition of Numbers

What are the different (types of) numbers? One answer:1 N = {0, 1, 2, 3, . . .}2 Z = {. . . ,−2,−1, 0, 1, 2, . . .}3 Q = insert all the fractions4 R = complete the real number line

What are the different (types of) numbers? Another answer:

→ No one has seen a number 1 or number 5. This is because numbers arenot like the objects of the physical world. If there was indeed such athing as number 1 that was tangible or visible as the rocks andtable-chairs are, it would be in a musueum!

→ Numbers live inside our mental world, and are bound by certain rules

Numbers live inside our mental world, and they follow certain rules:1 Numbers collectively form a set2 There is a zero (additive identity)3 There is a one (multiplicative identity)4 Addition and multiplication are defined and are closed under the set

→ Pick any two numbers, their sum and product are also ensured to be inthe same set

5 Addition and multiplication follow several desirable properties:

→ Associative, distributive, commutative

6 Several other properties are optional, not all the number sets havethem:

→ Existence of additive inverse, multiplicative inverse, etc.

Nothing in these properties say that the numbers cannot be paired

→ In fact, the fractional number in the set Q is actually a pair of numbers

(m, n)def=

n, where the individual numbers are drawn from Z, and

where the order matters, i.e., the fractional number (m, n) is differentfrom (n,m)

Consider a new type of number x , which is also a pair of numbers

xdef= (xr , xi )

→ Individual numbers are drawn from R; and the order matters here aswell

Complex Numbers

Definition of Complex Numbers

Consider a new type of number, which is a pair of numbers

xdef= (xr , xi )

→ Individual numbers are drawn from R; the order matters

Important differences compared to Q: there is a geometrical aspect tothis number

→ xr and xi are from two real number lines that are perpendicular to eachother

→ We put xr on the line drawn horizontally, this is our standard real (R)number line

→ We put xi on the line drawn vertically, this is the standard real numberline but it is to be thought of as getting multiplied by i =

√−1

everywhere!

Complex Numbers

Definition of Complex NumbersInterpretation of xr , xi

Why did we use the subscripts r and i to denote the Cartesiancoordinates?

→ xr : to be thought of as a number on the real number line→ xi : to be thought of as along the imaginary number line

x : to be thought of as a sum of real and imaginary numbers

x = xr +√−1 xi

√−1 is cumbersome to write: denote it by i (imaginary)

x = xr + i xi

(Matlab note: to avoid the confusion between a program variable iand the imaginary number i =

√−1, Matlab uses a specific notation

1i instead of i)

Complex Numbers

Definition of Complex NumbersExample 1

Complex Numbers

Definition of Complex Numbers

This new number is like a point on a plane

→ This generalizes all the other numbers, which are like a point on a line

xdef= (xr , xi ): coordinates of this point

→ Called the Cartesian Coordinates

A point can be specified either by its Cartesian or Polar coordinates

In the Polar coordinates, x is represented as xdef= (|x |, θx)

→ |x | is called the magnitude (or amplitude) of x (denotes the length ofthe line connecting x to the origin of the plane)

→ θx is called the phase angle of x

Both these specifications are completely equivalent; either can be usedAny one specification can be derived given the other

Complex Numbers

Definition of Complex NumbersGoing from Cartesian to Polar

If x = (xr , xi ) is specified in the Cartesian coordinates

→ its Polar coordinates (a, θ) can be determined as follows:

a =√x2r + x2i

θ = tan−1

)If, instead, x = (a, θ) is specified in its Polar coordinates

→ the Cartesian coordinates (xr , xi ) can be determined as follows:

xr = a cos θ

xi = a sin θ

Complex Numbers

Definition of Complex NumbersAdditions and Multiplications

Addition of two numbers a and b:

→ To obtain c = a + b, add the Cartesian coordinates of a and b→ cr = ar + br , ci = ai + bi

Multiplication of two numbers a and b:

→ To obtain d = a× b, multiply the magnitudes of a and b and add thephase

→ |c | = |a| × |b|, θc = θa + θb

Complex Numbers

Definition of Complex NumbersSubtractions and Divisions

Subtration of b from a:

→ To obtain c = a− b, subtract the Cartesian coordinates of b from a→ cr = ar − br , ci = ai − bi

Division of a by b:

→ To obtain d = a/b, divide the magnitudes of a and b and subtract thephase

→ |c | = |a|/|b|, θc = θa − θb

Complex Numbers

A Real Tale of Imaginary NumbersBombelli’s Solution

Bombelli applied Cardano’s formula to solve x3 = 15x + 4:

√2 + 11

√−1 +

√2− 11

√−1,

Bombelli’s ingenious approach was to show:

2 + 11√−1 =

(2 +√−1)3

2− 11√−1 =

(2−√−1)3

Therefore,

√2 + 11

√−1 +

√2− 11

√−1 = 2 +

√−1 + 2−

√−1 = 4

Problem solved, and the imaginary number got its birth

Complex Numbers

√2 + 11

√−1 +

√2− 11

√−1,

2 + 11√−1 =

(2 +√−1)3

2− 11√−1 =

(2−√−1)3

Therefore,

√2 + 11

√−1 +

√2− 11

√−1 = 2 +

√−1 + 2−

√−1 = 4

Complex Numbers

√2 + 11

√−1 +

√2− 11

√−1,

2 + 11√−1 =

(2 +√−1)3

2− 11√−1 =

(2−√−1)3

Therefore,

√2 + 11

√−1 +

√2− 11

√−1 = 2 +

√−1 + 2−

√−1 = 4

Complex Numbers

√2 + 11

√−1 +

√2− 11

√−1,

2 + 11√−1 =

(2 +√−1)3

2− 11√−1 =

(2−√−1)3

Therefore,

√2 + 11

√−1 +

√2− 11

√−1 = 2 +

√−1 + 2−

√−1 = 4

nth Roots of Unity

Definition of Complex NumbersInterpretation of xr , xi

Why did we use the subscripts r and i to denote the Cartesiancoordinates?

→ xr : to be thought of as a number on the real number line→ xi : to be thought of as along the imaginary number line

x : to be thought of as a sum of real and imaginary numbers

x = xr +√−1 xi

√−1 is cumbersome to write: denote it by i (imaginary)

x = xr + i xi

(Matlab note: to avoid the confusion between a program variable iand the imaginary number i =

√−1, Matlab uses a specific notation

1i instead of i)

nth Roots of Unity

Definition of Complex NumbersInterpretation of i =

√−1

nth Roots of Unity

√−1

nth Roots of Unity

√−1

nth Roots of Unity

√−1

nth Roots of Unity

Solving nth-order Equation

We’ve seen that:

→ i is a number for which i4 = 1.→ Equivalently, i is the solution of an equation x4 = 1→ Equivalently, i is the root of a polynomial p4(x) : x4 − 1 = 0.

An nth-order polynomial has a total of n roots. Where are the otherthree roots of p4(x).

→ These are i2 = −1, i3 = −i and i4 = 1.→ The 4 roots of p4(x) are along the unit circle in the complex-number

plane and they are at an angle of n × 90o , where n = 0, 1, 2 and 3,from the positive direction of the real number line

nth Roots of Unity

We’ve seen that:

nth Roots of Unity

We’ve seen that:

nth Roots of Unity

What are k roots of pk(x) : xk − 1 = 0.

→ The k roots of pk(x) are along the unit circle in the complex-numberplane and they are at an angle of n × 360o/k , wheren = 0, 1, 2, . . . , k−1, from the positive direction of the real number line

nth Roots of Unity

What are k roots of pk(x) : xk − 1 = 0.

→ The k roots of pk(x) are along the unit circle in the complex-numberplane and they are at an angle of n × 360o/k , wheren = 0, 1, 2, . . . , k−1, from the positive direction of the real number line

Complex Powers

Complex Exponentials

How to make sense of y = exp (iθ)?It can be shown (see, for example, http://math2.org/math/oddsends/complexity/e%5Eitheta.htm) that

exp (iθ) = cos (θ) + i sin (θ)

This is known as Euler’s Formula (often called the most beautifulequation of the math!

→ Leads to Euler’s Identity: e iπ = −1.

Complex Powers

exp (iθ) = cos (theta) + i sin (θ) , when θ = 60o

Complex Powers

Sinusoidal waveform can be thought of as a component of a moregeneral, complex-valued, waveform:

s(t) = A exp (iΘ(t)) = A exp (i(2πft + θ))

= sr (t) + isi (t)

→ s(t) : is called the complex phasor→ sr (t) : is the real part of this phasor→ si (t) : is the imaginary part of this phasor→ i : is the imaginary number

√−1

Several new concepts: complex phasor, its real and imaginary part,and i =

√−1

Complex Powers

Let us watch a movie of the complex phasor s(t) when f is positive valued.

Complex Powers

Next is a movie of the complex phasor s(t) when f is negative valued.

Complex Powers

Cosine Waveform

Cosine waveform can be thought of comprised of two complexphasors, one with frequency f and another with frequency −f

exp (i 2πft) + exp (i 2π(−f )t) = 2 cos (2πft)

Similarly, Sine waveform can also be thought of as comprised of thesame two phasors

exp (i 2πft)− exp (i 2π(−f )t) = i × 2 sin (2πft)

Complex Powers

Cosine and Sine Waveforms

Complex Powers

Cosine and Sine Waveforms

CT111 Introduction to Communication Systems Lecture 4...

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