The Continuous-time Fourier Transform

The Continuous-time Fourier TransformTransform

Rui Wang, Assistant professorDept. of Information and Communication

T ji U i itTongji University

Email: [email protected]

OutlineOutline

Representation of Aperiodic signals The Representation of Aperiodic signals: The continuous-time Fourier Transform

The Fourier transform for periodic signals Properties of the continuous-time Fourier

transform The convolution/multiplication propertyThe convolution/multiplication property System characterization by linear constant-

coefficientcoefficient

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4.1 The continuous-time Fourier t ftransform Extend the Fourier series representation to Aperiodic Extend the Fourier series representation to Aperiodic

signals Almost all of the signals including all signals with finite Almost all of the signals, including all signals with finite

energy can be represented by a linear combination of complex exponentialsp p

The representation of in terms of a linear combination takes a form of an integral (rather than a sum)

Fourier transform: the resulting spectrum of coefficients in the representation

Inverse Fourier transform: use these coefficients to represent the signal as a linear combination of

l ti lcomplex exponentials 3

4.1 The continuous-time Fourier t ftransform Development of Fourier transform representation of an Development of Fourier transform representation of an

aperiodic signal Consider a periodic square wave Consider a periodic square wave

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4.1 The continuous-time Fourier t ftransform Alternative interpreting Alternative interpreting

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4.1 The continuous-time Fourier t ftransform As T > infinite As T -> infinite

The original periodic signals becomes a rectangular pulse The Fourier series coefficients become more and more The Fourier series coefficients become more and more

closely spaced samples approached by a continuous envelop function

For aperiodic signal We think of an aperiodic signal as the periodic signal with We think of an aperiodic signal as the periodic signal with

arbitrary large period

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4.1 The continuous-time Fourier t ftransform An example: An example:

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4.1 The continuous-time Fourier t ftransform The Fourier series representation: The Fourier series representation:

Therefore, define the envelop as

We have

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4.1 The continuous-time Fourier t ftransform Combining them together we have Combining them together, we have

A h As , we have

Fourier Transform

Inverse Fourier Transform

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Transform

4.1 The continuous-time Fourier t ftransform We call X(jw) as spectrum of x(t) We call X(jw) as spectrum of x(t)

It provides us with the information needed for describing x(t) as a linear combination of the exponential signals with p gdifferent frequencies

Some conclusions: For a periodic signal , the Fourier coefficients

can be expressed in terms of equally spaced samples of the Fourier transform of one periodFourier transform of one period

Let be a finite-duration signal in one period

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4.1 The continuous-time Fourier t ftransform Convergence of Fourier transform: following the Convergence of Fourier transform: following the

Dirichlet conditions X(t) is absolutely integrable X(t) is absolutely integrable

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

X(t) has a finite number of discontinuities within any finite X(t) has a finite number of discontinuities within any finite interval and each of these discontinuities is finite

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4.1 The continuous-time Fourier t ftransform Example 1: Consider the signal Example 1: Consider the signal

Determine the Fourier transform

Solution:

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Solution:

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Solution:

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Solution:

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4.1 The continuous-time Fourier t ftransform

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4.1 The continuous-time Fourier t ftransform Example 5: Consider the signal x(t) whose Fourier Example 5: Consider the signal x(t) whose Fourier

transform is

Determine x(t)Determine x(t)

Solution:

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4.1 The continuous-time Fourier t ftransform Sinc function: Sinc function:

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4.1 The continuous-time Fourier t ftransform Some property for sinc function: Some property for sinc function:

As W increases, X(jw) becomes broader Main peak of x(t) becomes higher Main peak of x(t) becomes higher The width of the first lobe ( ) becomes narrower

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4.2 The Fourier transform for i di i lperiodic signal

The Fourier transform of periodic signal The Fourier transform of periodic signal Consists of a train of impulses in the frequency domain

So we have

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Example 1: Consider periodic square wave Example 1: Consider periodic square wave

Determine its Fourier transform Determine its Fourier transform

Solution:

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Example 2: Consider periodic signals Example 2: Consider periodic signals


Solution:

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Example 2: Consider the impulse train Example 2: Consider the impulse train


Solution:

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4.3 Properties4.3 Properties Linearity: Linearity:

Time shifting:

Proof:

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Example 1: Consider the signal Example 1: Consider the signal

Determine its Fourier transform Solution: Solution:

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4.3 Properties4.3 Properties Conjugation: Conjugation:

Proof:

For real signal:

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4.3 Properties4.3 Properties For a real and even signal is real and even For a real and even signal, is real and even

P fProof:

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4.3 Properties4.3 Properties By decompose the signal as even and odd parts By decompose the signal as even and odd parts

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Determine its Fourier transform

S l ti Solution:

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4.3 Properties4.3 Properties Differentiation and integration: Differentiation and integration:

Proof:

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S l ti Solution:

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S l ti Solution:

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4.3 Properties4.3 Properties Time and frequency scaling: Time and frequency scaling:

Proof:Proof:

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4.3 Properties4.3 Properties Duality: Duality:

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4.3 Properties4.3 Properties

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4.3 Properties4.3 Properties Example 5: Consider the signal Example 5: Consider the signal


S l ti Solution:

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4.3 Properties4.3 Properties Parseval’s relation: Parseval s relation:

Proof:Proof:

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4.3 Properties4.3 Properties Example 6: Evaluate the following time-domain Example 6: Evaluate the following time domain

expressions:

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4.3 Properties4.3 Properties

S l ti Solution:

2 5 / 81 | ( ) |E X j d

| ( ) |12

E X j db

1(0) ( )D g G j d

(0) ( )2

01

D g G j d

1 ( ) 1

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D j X j d

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4.4 The convolution property4.4 The convolution property Linear convolution in time domain product in Linear convolution in time domain product in

frequency domain

Proof:

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4.4 The convolution property4.4 The convolution property In Using Fourier analysis to study LTI system we In Using Fourier analysis to study LTI system, we

require The unit impulse function of the LTI system has Fourier

transform

Using transform techniques to examine unstable LTI tsystem

Use the Laplace transform (in Chapter 9)

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4.4 The convolution property4.4 The convolution property Example 1: Check the time-shifting with unit impulse Example 1: Check the time shifting with unit impulse

transform as follows by using Fourier transform

Solution: Solution:

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4.4 The convolution property4.4 The convolution property Example 2: Determine the Fourier transform of a Example 2: Determine the Fourier transform of a

differentiator

Solution: Solution:

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4.4 The convolution property4.4 The convolution property Example 3: Determine the Fourier transform of a Example 3: Determine the Fourier transform of a

integrator

Solution: the unit impulse response is a unit step Solution: the unit impulse response is a unit step function

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4.4 The convolution property4.4 The convolution property Example 4: Accomplish a frequency-selective filter Example 4: Accomplish a frequency selective filter

using LTI system

Solution: Solution:

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4.4 The convolution property4.4 The convolution property Example 5: Consider a system with input and unit Example 5: Consider a system with input and unit

impulse function given as follows. Determine the output by using Fourier transform.

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4.4 The convolution property4.4 The convolution propertySolution: for a != bSolution: for a ! b

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4.4 The convolution property4.4 The convolution propertySolution: for a = bSolution: for a b

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4.4 The convolution property4.4 The convolution property Example 6: Consider a system with input and unit Example 6: Consider a system with input and unit

impulse function given as follows. Determine the output by using Fourier transform.

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4.4 The convolution property4.4 The convolution propertySolution:Solution:

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4.5 The multiplication property4.5 The multiplication property The multiplication in the time domain corresponds to The multiplication in the time domain corresponds to

the convolution in the frequency domain

Amplitude modulation: multiplication of one signal by Amplitude modulation: multiplication of one signal by another Using one signal to scale or modulate the amplitude of the

other

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4.5 The multiplication property4.5 The multiplication property Example 1: Let s(t) be a signal with spectrum given as Example 1: Let s(t) be a signal with spectrum given as

Fig. (a). Also, consider the signal of

ThenThen

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4.5 The multiplication property4.5 The multiplication property

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4.5 The multiplication property4.5 The multiplication property Example 2: Determine the Fourier transform of the Example 2: Determine the Fourier transform of the

signal

Solution:

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4.5 The multiplication property4.5 The multiplication property Consider a system given as follows: Consider a system given as follows:

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4.5 The multiplication property4.5 The multiplication property Consider a system given as follows: Consider a system given as follows:

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4.6 Summary of the properties4.6 Summary of the properties

X(t) , y(t)

Linearity

Time shifting

Frequency shiftingq y g

Conjugation

Time reversale e e sa

Time Scaling (Period )Convolution

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Multiplication

Differentiation in timein timeIntegration

Differentiation inDifferentiation in frequencyConjugate Symmetry for Real Signals realfor Real Signals real

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S t f R l dSymmetry for Real and Even Signals X(t) real and even real and even

Real and Odd Signals X(t) real and odd imaginary and Real and Odd Signals X(t) real and odd oddEven-odd

Decomposition of Real pSignal [x(t) real]

Parseval’s Relation for Periodic SignalsPeriodic Signals

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4.7 System characterization by linear t t ffi i t diff ti l ticonstant-coefficient differential equations

Determine the frequency response of the LTI system Determine the frequency response of the LTI system of

For a LTI system, we have

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Taking Fourier transform on both sides we have Taking Fourier transform on both sides, we have

Apply linearity

Apply differentiation property, we have

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Or equivalently Or equivalently,

We thus have We thus have

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Example 1: consider a LTI system (a>0) Example 1: consider a LTI system (a>0)

The frequency response is

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The frequency response isThe frequency response is

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The frequency response isThe frequency response is

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Then we have Then we have

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The Continuous-time Fourier Transform

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