cosc-4317 lecture-4 frequency-01 fs

8/7/2019 COSC-4317 Lecture-4 Frequency-01 FS

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Goal

In this module, we will look at the frequency domain

representation of signals

Different models are used depending on the nature of the

independent/dependent variable (Continuous vs. Discrete)

Next module applies these concepts to filtering techniques.

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Time domain operations are often not very informative and/or

efficient in signal processing

Examples: nature of noise present (filtering); data that can be eliminated

(compression);

An alternative representation and characterization of signals

and systems can be made in transform domain

Essentially mathematical operators

The two domains or characterizations are complementary of

each other Most are invertible (You can get one from the other and vise versa)

Provide different insight to solving problems

Why Another Domain

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Jean Baptiste Joseph Fourier

Fourier was born in Auxerre, France in 1768 Most famous for his work La Thorie Analitique de

la Chaleur published in 1822

Translated into English in 1878: The AnalyticTheory of Heat

Nobody paid much attention when the work was firstpublished

Had crazy idea (1807): Any periodic function can berewritten as a weighted sum of sines and cosines ofdifferent frequencies.

Many didnt believe him including Lagrange, Laplace,Poisson.

Not translated into English until 1878!

One of the most important mathematical theories inmodern era.

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The Big Idea

Any function that periodically repeats itself can be expressed as a

an (infinite) sum of sines and cosines of different frequencies

each multiplied by a different coefficient a Fourier series

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=


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Example

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The Fourier theory

shows how most real

functions can be

represented in termsof a basis of sinusoids.

The building block:

A sin( x + )

Add enough of them to get any

signal you want.Notice how we get closer andcloser to the original function as weadd more and more frequencies


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Example

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The concept/model was later expanded to accommodate different types ofsignals:

Fourier Series

For Periodic continuous functions

Fourier Transform

For any continuous function

functions that are not periodic (but whose area under the curve is finite) can beexpressed as the integral of sines and cosines multiplied by a weighing function.

Discrete Fourier Transform

For sampled sequence of data (digital data)

Digital signal processing uses the discrete Fourier transform, DFT (1D and 2D) Today, the concept of composing a signal in terms of basis functions (delta

functions, polynomilas, sinusoidal functions, wavelets, etc.) is taking forgranted and forms the basis of many fields including compression andfiltering, among many others.

Fourier Types

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Frequency Family

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Type of Function (Signal) Fourier Model Used

Signals that are continuous

and aperiodic

Fourier Transform

Signals that are continuous

and periodic

Fourier Series

Signals that are discrete

and aperiodic

Discrete Time Fourier

Transform (infinite sum)

Signals that are discrete

and periodic

Discrete Fourier

Transform (finite sum-

discretized frequency)


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Math Background

Complex Numbers (we will use i or j interchangeably)

j2 = -1 or

C= R + j I

R is the real part and I is the imaginary part

C*= R j I

Complex conjugate (replacing each j with j)

Can be easily viewed in an Re-Im Plane (as 2-tuple vector)

The magnitude is given by

C can also be written as

22 IRC

)sin(cos CjCC

2//2-),/(tan 1 RI

10

1j


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Magnitude and Phase in the Complex

Plane

The graph show the

magnitude and phase of a

complex number z

11

22 IRz

2//2-),/(tan 1 RI


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Magnitude and Phase in the Complex

Plane

Another way (very

important way) to write

the complex number is as

an exponential

12

jezz


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Special case of the Unit Circle

13

je


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Multiplication

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When you multiply two complex numbers, their magnitudes

multiply:

and their phases add:

This can be easily seen in the exponential notation

)()()( yxxy

xxxy

111

jezz 222 jezz)(

2121212121 || jjj ezzezezzz


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For a complex number z:

z = a + bj

Its conjugates is given by :z* = a - bj

The complex conjugate z has

the same real part but opposite imaginary part, and

the same magnitude but opposite phase.

Complex Conjugates


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Adding z + z*, cancels the imaginary parts to leave a real

number:

(a + bj) + (a bj) = 2a

Multiplying z . Z* gives the real number equal to |z|2:

(a + bj)(a bj) = a2 (bj)2 = a2 + b2

Complex Conjugates


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Eulers Formula

Eulers formula uses exponential notation to encode complex

numbersuses j in the exponent to differentiate from real

numbers

Eulers formula:

Eulers formula allows us to rewrite C (generic complex number) as

Adding and subtracting these formulas, Euler obtained thefollowing expressions for cos and sin:

)sin()cos( jej

jeCC

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)sin()cos( je j


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Eulers Formula: Graphical

Interpretation

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)(zjezz


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What is (2 + 2j)(3 + 3j)?

Suppose that we already have these numbers in magnitude-

phase notation:

Eulers Formula: Application

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224422 j 239933 j

4/)2

2(tan)22(1

j 4/3)3

3(tan)33( 1j

/4je2222 j/4j3e2333 j

12e)12(e)12(

ee)2322(

e23.e22)33)(22(

)j(/4)3/4j(

/4j3/4j

/4j3/4jjj


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Suppose that we take a complex number

and raise it to some power n:

zn has magnitude |z|n and phase

Powers of Complex Numbers

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)(zjezz

jnn

njn

ez

ezz

)(zn


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What is jn for various n?

Powers of Complex Numbers:

Example

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...

1

1

1

2/44

2/33

2/22

2/

0

2/

j

j

j

j

j

ej

jej

ej

jejj

ej


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Series Representation

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Math Background

Even & Odd functions

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Math Background

Any function can be decomposed as a sum of the even

and odd part

f(t) = fe(t) + fo(t), where:

One of the functions used extensively in Fourier

transforms is the Sinc function, defined as:

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The term used to find the frequency content/components of a

signal is called Analysis or Decomposition

The term used to create a signal (in time) from its frequency

content/components is called Synthesis or Composition

Transform Terminology

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Fourier Series

Frequency domain representation of periodic signals

There are many forms for the Fourier Series including the

Trigonometric and complex representations

We will emphasize the complex representation and then relate it to the

trigonometric representation.

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A periodic signal x(t), has a Fourier series if it satisfies the

following conditions:

1. x(t) is absolutely integrable over any period, namely

2. x(t) has only a finite number of maxima and minima over

any period

3. x(t) has only a finite number of discontinuities over anyperiod

Dirichlet Conditions

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0

)(T

dttx


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Fourier Series

With f0=1/T0 and0=2f0

Notation: x(t) akwhere the double arrow signifies the invertibility

of one form to the other

Note that each complex exponential that makes up the sum isan integer multiple of0, the fundamental frequency. Hence, the complex exponentials are harmonically related

The coefficients ck, aka Fourier (series) coefficients, are possiblycomplex Fourier series (and all other types of Fourier transforms) are complex

valued! That is, there is a magnitude and phase (angle) term to the Fouriertransform!

This is the only unknown that is to be calculated from the waveform x(t)

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k

tjk

kectx0)(

tjke 0


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Fourier Series

Synthesis Part:

k0: kth integer multiple kth harmonic of the fundamental

frequency 0

ck: Fourier coefficients how much of kth harmonic exists in the

signal

|ck|: Magnitude of the kth harmonic (magnitude spectrum of

x(t))

k: Phase of the kth harmonic (phase spectrum of x(t))

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k

tjk

kectx0)(


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Fourier Series

Analysis Part:

The limits of the integral can be chosen to cover any interval of T0 (in

many book written as meaning integrate over any interval of lengthT)

Note that, while x(t) is a sum, ckare obtained through an integral of

complex values.

If x(t) is real, then the coefficients satisfy c-k=c*k, that is |c-k|=|ck|

c0 is the DC/Average value of the signal

c1 is the fundamental frequency

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00

0

0)(1

0

Ttt

tt

tjk

k dtetxT

c

0T


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Proof of how to get Fourier

coefficients

Graduate Student

In the proof, our Fourier coefficient is ak

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Example-01

33.2||0,2

1

,2

1

),2

11(),

2

11(,1

-:aretscoefficienseriesFourier

)

2

1)

2

1)

2

11()

2

11(1)(

][2

1][][

2j

11x(t)

-:)e(e2

1

sinand

)e(e2

1cosforidentityEuleruse

)4

2cos(cos2sin1)(

)4/(

2

)4/(

2

110

2)4/(2)4/(

4/24/2

j-j

j-j

000

0000

000000

kforcecec

jc

jcc

eeeee

j

e

j

tx

eeeeee

j

ttttx

k

jj

tjjtjjtjtj

tjtjtjtjtjtj


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Example-02

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Example-03

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Trigonometric Form of FS

Trigonometric Form 1:

Trigonometric Form 2:

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1

000 sincosn

nn tnbtnaatf

0

0

0

00

0

00

0

00

0

sin2

cos2

1

T

n

T

n

T

dttntfT

b

dttntfT

a

dttfT

a

n

nn

nnn

n

nn

a

b

bacac

tncctf

1

22

00

1

00

tan

and,,where

cos This form is obtained from thetrigonometric identity

a cos(x) + b sin(x) = c cos(x + )


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Example

Fundamental period

T0 =

Fundamental frequency

f0 = 1/T0 = 1/ Hz

0 = 2 /T0 = 2 rad/s

.asamplitudeindecreaseand161

8504.02sin

2

161

2504.02cos

2

504.0121

2sin2cos

20

2

20

2

2

0

20

1

0

nban

ndtnteb

ndtntea

edtea

ntbntaatf

nn

t

n

t

n

t

n

nn

0

1e-t/2

f(t)

12

2sin42cos161

21504.0

n

ntnntn

tf


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Compact Trigonometric

Fundamental period

T0 =

Fundamental frequency

f0 = 1/T0 = 1/ Hz

0 = 2 /T0 = 2 rad/s na

b

n

baC

aC

n

nb

na

a

ntCCtf

n

n

n

nnn

o

n

n

n

nn

4tantan

161

2504.0

504.0

161

8504.0

161

2504.0

504.0

2cos

11

2

22

0

2

2

0

1

0

0

1e-t/2

f(t)

1

1

2

4tan2cos

161

2504.0504.0

n

nnt

n

tf


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Relationship among the forms

39

00

5.0

5.0

acjbacc

jbac

nnnn

nnn

00

5.05.0

dc

eddc njnnnn


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f() and F() must contain the same information.

It is very important to know that the Fourier series is

completely reversible

It provides one-to-one transform of signals from/to a time-

domain representation f(t) to/from a frequency domain

representation FS().

It is a mathematical prism to separate a function into

various components

It allows a frequency content(spectral) analysis of a signal.

Final Notes on FS

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Pay close attention to these properties as many of them will

apply to the other transforms later on

Properties of Fourier Series


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FS R i f Di


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Given a discrete-time periodic signal with fundamental period

of N and fundamental frequency 0=2/N

FS Representation of Discrete

Time Periodic Signals

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1

0

1

0

)/2(0][N

k

N

k

nNjk

k

njk

k eaeanx

1

0

1

0

)/2(][][1

0

N

n

N

n

nNjknjk

k enxenxN

a

k

FS

anx ][

FS R i f Di


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ak: Fourier series coefficients or spectral coefficients

Differences to continuous-time case

Discrete-time Fourier series is finite

There are only N distinct discretetime complex exponential signals

ejk(2/N)n that are periodic with period N (harmonically related signals).)

No mathematical issues with convergencediscretetime Fourier series

representation always exists

ak= ak+N

FS Representation of Discrete

Time Periodic Signals

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FS R i f Di


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The set of coefficients

is commonly referred to as the N-point discrete Fourier

transform (DFT) of a finite duration signal x[n] with x[n] = 0

outside the interval 0


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P i f Di Ti F i


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Similar to the continuous case

Properties of Discrete Time Fourier

Series

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Matlab

To use the CT Fourier transform, you need to have thesymbolic toolbox for Matlab installed. If this is so, trytyping:

>> syms t;>> fourier(cos(t))

>> fourier(cos(2*t))

>> fourier(sin(t))

>> fourier(exp(-t^2))Note also that the ifourier() function exists so

>> ifourier(fourier(cos(t)))

Matlab

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Great site to learn about Signals, Convolution, and Linear

Systems.

http://www.jhu.edu/~signals/

John Hopkins Website

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http://www.jhu.edu/~signals/http://www.jhu.edu/~signals/

cosc-4317 lecture-4 frequency-01 fs

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