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Topic 4: Continuous-Time Fourier Transform (CTFT) o Introduction to - - PowerPoint PPT Presentation

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ELEC361: Signals And Systems Topic 4: Continuous-Time Fourier Transform (CTFT) o Introduction to Fourier Transform o Fourier transform of CT aperiodic signals o CT Fourier transform examples o Convergence of the CT Fourier Transform o Convergence

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SLIDE 1
  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • CT Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • CT Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix:

Transition from CT Fourier Series to CT Fourier Transform

ELEC361: Signals And Systems

Topic 4: Continuous-Time Fourier Transform (CTFT)

  • Dr. Aishy Amer

Concordia University Electrical and Computer Engineering

Figures and examples in these course slides are taken from the following sources:

  • A. Oppenheim, A.S. Willsky and S.H. Nawab, Signals and Systems, 2nd Edition, Prentice-Hall, 1997
  • M.J. Roberts, Signals and Systems, McGraw Hill, 2004
  • J. McClellan, R. Schafer, M. Yoder, Signal Processing First, Prentice Hall, 2003
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SLIDE 2

2

Fourier Series versus Fourier Transform

Fourier Series (FS): a representation of periodic

signals as a linear combination of complex exponentials

Fourier Transform (FT): apply to signals that are not

periodic

Aperiodic signals can be viewed as a periodic signal

with an infinite period

The CT Fourier Series is a good analysis tool for

systems with periodic excitation but the CT Fourier Series cannot represent an aperiodic signal for all time

The CT Fourier transform can represent an aperiodic

signal for all time

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3

Fourier Series versus Fourier Transform: Types of signals

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SLIDE 4

4

Fourier Series versus Fourier Transform: Types of signals

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5

Fourier Series versus Fourier Transform

Four distinct Fourier representations: Each applicable to a different class of

signals

Determined by the periodicity properties of

the signal and whether the signal is discrete

  • r continuous in time

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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SLIDE 6

6

Fourier Series versus Fourier Transform

In FS representation of periodic signals:

  • As the period increases T↑, ω0↓

the harmonically related components become closer in frequency

As the period becomes infinite

the frequency components form a continuum and the FS sum becomes an integral

FT for aperiodic signals: T e a t x dt e t x T a k X

t jk k k T t jk k

/ 2 ; ) ( ; ) ( 1 ] [ π ω

ω ω

= = = =

∑ ∫

∞ −∞ = −

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SLIDE 7

7

Fourier Series versus Fourier Transform

Periodic in Time Discrete in Frequency Aperiodic in Time Continuous in Frequency Continuous in Time Aperiodic in Frequency Discrete in Time Periodic in Frequency

∑ ∫

∞ −∞ = −

= ⇒ ⊗ = ⇒ ⊗

k t jk k T t jk k

e a t x dt e t x T a ) ( P

  • CT

DT : Series Fourier Inverse CT ) ( 1 DT P

  • CT

: Series Fourier CT

T T ω ω

∫ ∑

= ⇒ + ⊗ = + ⇒ ⊗

∞ −∞ = − π ω ω π ω ω π

ω π 2

2 2

) ( 2 1 ] [ DT P CT : Transform Fourier DT Inverse ] [ ) ( P CT DT : Transform Fourier DT d e e X n x e n x e X

n j j n n j j

∑ ∑

− = − = −

= ⇒ ⊗ = ⇒ ⊗

1 N N 1 N N

] [ 1 ] [ P

  • DT

P

  • DT

Series Fourier DT Inverse ] [ ] [ P

  • DT

P

  • DT

Series Fourier DT

N k kn j N n kn j

e k X N n x e n x k X

ω ω

∫ ∫

∞ ∞ − ∞ ∞ − −

= ⇒ ⊗ = ⇒ ⊗ ω ω π ω

ω ω

d e j X t x dt e t x j X

t j t j

) ( 2 1 ) ( CT CT : Transform Fourier CT Inverse ) ( ) ( CT CT : Transform Fourier CT

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SLIDE 8

8

Outline

  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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SLIDE 9

9

Fourier Transform of CT aperiodic signals

Consider the CT aperiodic signal

given below:

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Fourier Transform of CT aperiodic signals

We have: Define: This means that:

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11

As , approaches

approaches zero, and the right-hand side changes to an integral

The pair of equations:

are referred to as a Fourier Transform pair

Fourier Transform of CT aperiodic signals

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CT Fourier transform for aperiodic signals

Spectrum) Magnitude (Symmetric ) ( ) ( signals, real For ) ( ) ( : ) transform (forward analysis Fourier ) ( 2 1 ) ( : ) transform (inverse Synthesis Fourier ω ω ω ω ω π

ω ω

j X j X dt e t x j X d e j X t x

t j t j

− = = =

− ∞ ∞ − ∞ ∞ −

∫ ∫

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13

CT Fourier transform for aperiodic signals

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14

CT Fourier Transform of aperiodic signal

The CT Fourier Transform expresses a finite-

amplitude, real-valued, aperiodic signal x(t)

x(t) can also, in general, be time-limited

as a summation (an integral) of an infinite continuum of weighted, infinitesimal- amplitude, complex sinusoids, each of which is unlimited in time

Time limited means “having non-zero values

  • nly for a finite time”
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15

CTFT: Pulse (rectangular) Function Time Domain

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CTFT: Pulse (rectangular) Function Spectrum

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CTFT: Exponential Decay Time Domain

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CTFT: Exponential Decay Spectrum

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19

Outline

  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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CT Fourier Transform: Example

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CT Fourier Transform: Example

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CT Fourier Transform: Example

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CT Fourier Transform: Example

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24

Outline

  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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SLIDE 25

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Convergence of CT Fourier Transform

  • Dirichlet’s sufficient conditions for the

convergence of Fourier transform are similar to the conditions for the FS:

1.

x(t) must be absolutely integrable

2.

x(t) must have a finite number of maxima and minima within any finite interval

3.

x(t) must have a finite number of discontinuities, all of finite size, within any finite interval

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26

Convergence of CT Fourier Transform:Example 5.4

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Convergence of CT Fourier Transform: Example 5.4

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Convergence of CT Fourier Transform:Example 5.5

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Convergence of CT Fourier Transform:Example 5.5

  • The Fourier transform for this example is real at all frequencies
  • The time signal and its Fourier transform are
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Convergence of CT Fourier Transform: Example 5.6

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Convergence of CT Fourier Transform: Example 5.7

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Convergence of CT Fourier Transform: Example 5.7

Reducing the width of x(t)

will have an opposite effect on X(jω)

Using the inverse Fourier

transform, we get a time signal which is equal to x(t) at all points except discontinuities (t=T1 and t=-T1), where the inverse Fourier transform is equal to the average of the values of x(t) on both sides of the discontinuity

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Convergence of CT Fourier Transform: Example 5.8

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Convergence of CT Fourier Transform: Example 5.8

This example shows the

reveres effect in the time and frequency domains in terms of the width of the time signal and the corresponding Fourier transform

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Convergence of CT Fourier Transform: Example 5.8

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Convergence of the CT-FT: Generalization

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Convergence of the CTFT: Generalization

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Convergence of the CTFT: Generalization

which is equal to A, independent of the value of σ

  • So, in the limit as σ approaches zero, the CT Fourier Transform

has an area of A and is zero unless f = 0

This exactly defines an impulse of strength, A. Therefore

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SLIDE 39

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Convergence of the CTFT: Generalization

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Convergence of the CTFT: Generalization

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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
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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42

Fourier transform for periodic signals

Consider a signal x(t) whose Fourier transform is given by Using the inverse Fourier transform, we will have: This implies that the time signal corresponding to the

following Fourier transform

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Fourier transform for periodic 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 frequency kω0 equal to the kth coefficient of the Fourier series representation ak times 2π

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Fourier Transform of Periodic Signal: Example 6.1

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Fourier Transform of Periodic Signal: Example 6.1

This example shows the

reveres effect in the time and frequency domains in terms of the width of the time signal and the corresponding Fourier transform

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Fourier Transform of Periodic Signal: Example 6.1

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Fourier transform for periodic signals: Example 6.2

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Fourier transform for periodic signals: Example 6.2

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Fourier transform for periodic signals: Example 6.2

The periodic impulse train and its Fourier transform are very useful in the analysis of sampling systems

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Fourier Transform of Periodic Signals: Example

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51

Outline

  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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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

  • perations (e.g., convolution, differentiation, shift) on

another signal for which the Fourier transform is known

Operations on {x(t)} Operations on {X(jω)} Help find analytical solutions to Fourier transform

problems of complex signals

Example:

⇔ tion multiplica and delay t u a t y FT

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(jω) and Y(jω), respectively (or X(f) and Y(f))

The following properties can easily been

shown using

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

The time and frequency scaling properties indicate that

if a signal is expanded in one domain it is compressed in the other domain. This is also called the “uncertainty principle” of Fourier analysis

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Properties of the CT Fourier Transform

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Properties of the CT Fourier: Time Shifting & Scaling: Example

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Properties of the CT Fourier: Time Shifting & Scaling: Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Example 6.4

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Properties of the CT Fourier Transform: Differentiation Property Example

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Properties of the CT Fourier Transform: Examples 6.5 & 6.6

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Properties of the CT Fourier Transform: Differentiation Property Example

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Properties of the CT Fourier Transform: Differentiation Property: Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

Multiplication Convolution Duality Proof

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Convolution Property Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Differential Equations

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Properties of the CT Fourier Transform: Differential Equation Example 6.8

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Properties of the CT Fourier Transform: Modulation Property: Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Integration Property

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Properties of the CT Fourier Transform: Integration Property: Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Area Property: Example

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Properties of the CT Fourier Transform: Area Property: Example

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Properties of the CT Fourier Transform: Area Property: Example

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform

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Properties of the CT Fourier Transform: Example 6.4

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Properties of the CT Fourier Transform: Duality Property: Example

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92

Outline

  • Introduction to Fourier Transform
  • Fourier transform of CT aperiodic signals
  • Fourier transform examples
  • Convergence of the CT Fourier Transform
  • Convergence examples
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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CTFT: Summary

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Summary of CTFT Properties

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Summary of CTFT Properties

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CTFT: Summary of Pairs

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CTFT: Summary of Pairs

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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
  • Fourier transform of periodic signals
  • Properties of CT Fourier Transform
  • Summary
  • Appendix
  • Transition: CT Fourier Series to CT Fourier Transform
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Transition: CT Fourier Series to CT Fourier Transform

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Transition: CT Fourier Series to CT Fourier Transform

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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Transition: CT Fourier Series to CT Fourier Transform

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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Transition: CT Fourier Series to CT Fourier Transform