analysis of continuous signals

8/12/2019 analysis of continuous signals

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Fourier representations

A Fourier representation is unique, i.e., no two

same signals in time domain give the same

function in frequency domain


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Fourier Series versus

Fourier Transform

Fourier Series (FS): a discrete representation of a periodic signalas a linear combination of complex exponentials

The CT Fourier Series cannot represent an aperiodic signal for alltime

Fourier Transform (FT): a continuous representation of a notperiodic signal as a linear combination of complex exponentials

The CT Fourier transform represents an aperiodic signal for all time

A not-periodic signal can be viewed as a periodic signal with

an infinite period


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Fourier Series versus

Fourier Transform FS of periodic CT signals:

As the period increases T, 0

The harmonically related components kw0 become closer in

frequency

As T becomes infinite

the frequency components form a continuum and the FS

sum becomes an integral

TekXtxdtetxT

kX tjk

k

T

tjk/2;][)(;)(

1][ 0

0

00


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Fourier Series versus Fourier

Transform

k

tjk

k

T

tjk

k

eatx

dtetxT

a

0

0

)(

TPer-CTimeDFreq:SeriesFourierInverseCT

)(1

DFreqT

Per-CTime:SeriesFourierCT

0

dejXtx

dtetxjX

tj

tj

)(

2

1)(

CTimeCFreq:TransformFourierCTInverse

)()(

CFreqCTime:TransformFourierCT


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Outline

Introduction to Fourier Transform

Fourier transform of CT aperiodic signals

Fourier transform examples

Convergence of the CT Fourier Transform Convergence examples

Properties of CT Fourier Transform

Fourier transform of periodic signals

Summary

Appendix Transition: CT Fourier Series to CT Fourier Transform


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As , approaches

approaches zero

uncountable number of harmonics

Integral instead of

Fourier Transform of CT

aperiodic signals

0k

dtetxjX

dejXtx

tj

tj

)()(:)transform(forwardanalysisFourier

)(

2

1)(:)transform(inverseSynthesisFourier


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CT Fourier transform for

aperiodic signals


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CT Fourier Transform of

aperiodic signal

The CT FT expresses

a finite-amplitude aperiodic signal x(t)

as a linear combination (integral) of an infinite continuum of weighted, complex

sinusoids, each of which is unlimited in time

Time limited means having non-zero values

only for a finite time x(t) is, in general, time-limited but must not be


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Outline







Summary



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CTFT: Rectangular Pulse Function


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CTFT: Exponential Decay (Right-sided)


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Outline







Summary



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Convergence of CT Fourier

Transform

Dirichlets sufficient conditions for the

convergence of Fourier transform are similar to

the conditions for the CT-FS:

a) x(t) must be absolutely integrable

b) x(t) must have a finite number of maxima andminima within any finite interval

c) x(t) must have a finite number of discontinuities,

all of finite size, within any finite interval


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Transform


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Transform: Example 1


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The Fourier transform for this example is real at all frequencies

The time signal and its Fourier transform are


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discontinuities at t=T1 and t=-T1

C f CT F i


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Reducing the width of x(t)will have an oppositeeffect on X(j)

Using the inverse Fourier

transform, we get xrec(t)which is equal to x(t) atall points exceptdiscontinuities (t=T1 & t=-

T1), where the inverseFourier transform is equalto the average of thevalues of x(t) on bothsides of the discontinuity


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This example shows

the reveres effect in

the time and frequencydomains in terms of

the width of the time

signal and thecorresponding FT


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Convergence of the CT-FT:

Generalization


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Convergence of the CTFT:

Generalization


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Generalization


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Generalization


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Outline







Summary



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Properties of the CT Fourier

Transform

The properties are useful in determining the Fourier transform

or inverse Fourier transform

They help to represent a given signal in term of operations

(e.g., convolution, differentiation, shift) on another signal for

which the Fourier transform is known

Operations on {x(t)} Operations on {X(j)}

Help find analytical solutions to Fourier transform problems of

complex signals

Example:

tionmultiplicaanddelaytuatyFT t })5()({


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Properties of the CT Fourier

Transform The properties of the CT Fourier transform are

very similar to those of the CT Fourier series

Consider two signals x(t) and y(t) with Fourier

transforms X(j) and Y(j), respectively (orX(f) and Y(f))

The following properties can easily been

shown using


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Properties of the CTFT:

Linearity

P ti f th CTFT


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Time shift


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Freq. shift


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Frequency shift property:

Example

)(2 0


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Properties of the CTFT

The time and frequency scaling properties indicate that ifa signal is expanded in one domain it is compressed in

the other domain This is also called the uncertainty

principle of Fourier analysis

P ti f th CTFT S li


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Properties of the CTFT: Scaling

Proofs



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Proofs

P ti f th CTFT Ti Shifti & S li E l


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Properties of the CTFT: Time Shifting & Scaling: Example

Properties of the CTFT: Time Shifting & Scaling


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Properties of the CTFT: Time Shifting & Scaling

Example

P i f h CTFT


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Properties of the CTFT: average power


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Conjugation



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Convolution & Multiplication

x(t)y(t)

Multiplication of x(t) and y(t)

Modulation of x(t) by y(t)

Example: x(t)cos(w0t)

w0 is much higher than the max. frequency of x(t)

Properties of the CTFT: Convolution &


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ope es o e C Co o u o &

Multiplication



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Modulation Property Example



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Convolution

o h(t) is the impulse response of the LTI system

)(

)()(

jX

jYjH

Properties of the CTFT: Convolution


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Properties of the CTFT: Convolution

Property Example

Properties of the CTFT: Multiplication-


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p p

Convolution Duality Proof


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Properties of the CTFT: Integration


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Properties of the CTFT: Integration

Property Example 1

o Recall: Relation unit step and impulse:

Properties of the CTFT: Integration Property


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Properties of the CTFT: Integration Property

Example 2


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Properties of the CTFT: Differentiation Property Example 2


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Properties of the CTFT: Differentiation Property Example 2



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Differentiation Property Example 2

Proof of Integration Property


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Proof of Integration Property

using convolution property

Properties of the CTFT: Systems Characterized by


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p y y

linear constant-coefficient differential equations

(LCCDE)

P ti f th CTFT S t Ch t i d b li


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Properties of the CTFT: Systems Characterized by linear

constant-coefficient differential equations

M

kk

k

k

N

kk

k

kdt

txdb

dt

tyda

00

)()(

)(

)()(

jX

jYjH

N

k

k

k

M

k

kk

ja

jb

jX

jYjH

0

0

)(

)(

)(

)()(


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Properties of the CTFT: Systems Characterized by

LCCDE Example 1

)()()(

txtaydt

tyd a>0

ajja

jb

jH

h(t)

N

k

kk

M

k

k

k

1

)(

)(

)(

systemthisofresponseimpulsetheFind

0

0

)()(

2

1)( tuedejHth

attj


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Properties of the CTFT: Systems

Characterized by LCCDE Example 2a



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Characterized by LCCDE Example 2b

)(2)(

)(3)(

4)(

2

2

txdt

txdty

dt

tyd

dt

tyd

313)(4)(2)(

)(

)(

)( 21

21

2

0

0

jjjj

j

ja

jb

jHN

k

k

k

M

k

k

k

)(][)(

:responseimpulsetheFind

3

21

21 tueeth tt



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Characterized by LCCDE Example 2c

)(Find;1

1)()(Let ty

jtuetx

FTt

)3()1(

2

1

1

)3)(1(

2)()()(

2

jj

j

jjj

jjXjHjY

3)1(1)( 4

1

2

21

41

jjjjY

)(][)( 341

21

41 tueteety ttt



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pDuality property

Properties of the CTFT: Duality


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p y

property


Duality Property Example


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Duality Property Example

P ti f th CTFT l ti


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Properties of the CTFT: relation

between duality & other properties

P ti f th CTFT


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Properties of the CTFT:Differentiation in frequency Example

)(tute at



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p

Total area-Integral

P ti f th CTFT


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Area Property Illustration

P ti f th CTFT


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Area Property Example 1

Properties of the CTFT: Area Property Example 2


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Periodic signals


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Outline




Convergence of the CT Fourier Transform

Convergence examples



Summary



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Fourier transform for periodic


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signals

Note that

is the Fourier series representation of periodic signals

Fourier transform of a periodic signal is a train of

impulses with the area of the impulse at the frequencyk0equal to the k

th coefficient of the Fourier series

representation aktimes 2



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signals: Example 1



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p

signals: Example 1

Recall: FT of square signal



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p

signals: Example 2

FT of periodic time

impulse train is another

impulse train (in frequency)

Fourier Transform of Periodic


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Signals: Example 2 (details)


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Outline








Summary



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CTFT: Summary

Summary of CTFT Properties


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y p

Summary of CTFT Properties


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y p


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CTFT: Summary of Pairs


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O li


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Outline

Introduction to Fourier Transform Fourier transform of CT aperiodic signals






Summary

Appendix: Transition CT Fourier Series to CT Fourier Transform

Appendix: Applications

Transition: CT Fourier Series to


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CT Fourier Transform



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Below are plots of the magnitude of X[k] for 50% and

10% duty cycles

As the period increases the sinc function widens and its

magnitude falls

As the period approaches infinity, the CT Fourier Series

harmonic function becomes an infinitely-wide sinc

function with zero amplitude (since X(k) is divided by To)



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This infinity-and-zero problem can be solved by normalizing the

CT Fourier Series harmonic function

Define a new modified CT Fourier Series harmonic function



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O tli


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Outline








Summary

Appendix: Transition: CT Fourier Series to CT Fourier Transform Appendix: Applications

Applications of frequency- domain


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representation

Clearly shows the frequency composition a signal Can change the magnitude of any frequency

component arbitrarily by a filtering operation

Lowpass -> smoothing, noise removal

Highpass -> edge/transition detection

High emphasis -> edge enhancement

Processing of speech and music signals

Can shift the central frequency by modulation A core technique for communication, which uses

modulation to multiplex many signals into a single

composite signal, to be carried over the same physical

medium

T i l Filt


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Typical Filters

Lowpass -> smoothing, noise removal

Highpass -> edge/transition detection

Bandpass -> Retain only a certain frequency range

Low Pass Filtering

(R hi h f k i l


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(Remove high freq, make signal

smoother)


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Filtering in Temporal Domain


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g p

(Convolution)

Communication:


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Signal Bandwidth

Bandwidth of a signal is a critical feature whendealing with the transmission of this signal

A communication channel usually operates only at

certain frequency range (called channel bandwidth) The signal will be severely attenuated if it contains

frequencies outside the range of the channelbandwidth

To carry a signal in a channel, the signal needed to

be modulated from its baseband to the channelbandwidth

Multiple narrowband signals may be multiplexed touse a single wideband channel

Signal Bandwidth:


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Highest frequency estimation in a signal: Find

the shortest interval between peak and valleys

Estimation of Maximum


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Frequency


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Sample Speech Waveform 1


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Numerical Calculation of CT


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FT The original signal is digitized, and then a Fast

Fourier Transform (FFT) algorithm is applied, which

yields samples of the FT at equally spaced intervals

For a signal that is very long, e.g. a speech signal

or a music piece, spectrogram is used.

Fourier transforms over successive overlapping short

intervals


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Speech Spectrogram 2


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Speech Spectrogram 2

Sample Music Waveform


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Sample Music Waveform

Sample Music Spectrogram


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Sample Music Spectrogram

analysis of continuous signals

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