Chapter 4 The Fourier Series and Fourier Transform

Chapter 4The Fourier Series and

Fourier Transform

Chapter 4The Fourier Series and

Fourier Transform

• Let x(t) be a CT periodic signal with period T, i.e.,

• Example: the rectangular pulse trainrectangular pulse train

Fourier Series Representation of Periodic Signals

Fourier Series Representation of Periodic Signals

( ) ( ),x t T x t t R

• Then, x(t) can be expressed as

where is the fundamental fundamental frequencyfrequency (rad/sec) of the signal and

The Fourier SeriesThe Fourier Series

0( ) ,jk tk

k

x t c e t

/ 2

/ 2

1( ) , 0, 1, 2,o

Tjk t

k

T

c x t e dt kT

0 2 /T

is called the constant or dc component of x(t)0c

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

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

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

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

Dirichlet ConditionsDirichlet Conditions

| ( ) | ,a T

a

x t dt a

• From figure , so

• Clearly x(t) satisfies the Dirichlet conditions and thus has a Fourier series representation

Example: The Rectangular Pulse TrainExample: The Rectangular Pulse Train

2T 0 2 / 2

Example: The Rectangular Pulse Train – Cont’d

Example: The Rectangular Pulse Train – Cont’d

x(t) =12

+1k

k=−∞k≠0

∞

∑ sink2

⎛

⎝⎜⎞

⎠⎟ejkt, t∈°

• By using Euler’s formula, we can rewrite

as

as long as x(t) is real

• This expression is called the trigonometric trigonometric Fourier seriesFourier series of x(t)

Trigonometric Fourier SeriesTrigonometric Fourier Series

0( ) ,jk tk

k

x t c e t

0 01

( ) 2 | |cos( ),k kk

x t c c k t c t

dc componentdc component kk-th harmonic-th harmonic

• The expression

can be rewritten as

Example: Trigonometric Fourier Series of the Rectangular Pulse Train

Example: Trigonometric Fourier Series of the Rectangular Pulse Train

x(t) =12

+1k

k=−∞k≠0

∞

∑ sink2

⎛

⎝⎜⎞

⎠⎟ejkt, t∈°

( 1) / 2

1

1 2( ) cos ( 1) 1 ,

2 2k

k

k odd

x t k t tk

• Given an odd positive integer N, define the N-th partial sum of the previous series

• According to Fourier’s theoremFourier’s theorem, it should be

Gibbs PhenomenonGibbs Phenomenon

( 1) / 2

1

1 2( ) cos ( 1) 1 ,

2 2

Nk

Nk

k odd

x t k t tk

lim | ( ) ( ) | 0NN

x t x t

Gibbs Phenomenon – Cont’dGibbs Phenomenon – Cont’d

3( )x t9 ( )x t

Gibbs Phenomenon – Cont’dGibbs Phenomenon – Cont’d

21( )x t 45( )x t

overshootovershoot: about 9 % of the signal magnitude (present even if )N

• Let x(t) be a periodic signal with period T

• The average poweraverage power P of the signal is defined as

• Expressing the signal as

it is also

Parseval’s TheoremParseval’s Theorem

/ 22

/ 2

1( )

T

T

P x t dtT

0( ) ,jk t

kk

x t c e t

2| |kk

P c

• We have seen that periodic signals can be represented with the Fourier series

• Can aperiodic signalsaperiodic signals be analyzed in terms of frequency components?

• Yes, and the Fourier transform provides the tool for this analysis

• The major difference w.r.t. the line spectra of periodic signals is that the spectra of spectra of aperiodic signalsaperiodic signals are defined for all real values of the frequency variable not just for a discrete set of values

Fourier TransformFourier Transform

Frequency Content of the Rectangular Pulse

Frequency Content of the Rectangular Pulse

( )Tx t

( )x t

( ) lim ( )TT

x t x t

• Since is periodic with period T, we can write

Frequency Content of the Rectangular Pulse – Cont’dFrequency Content of the

Rectangular Pulse – Cont’d

( )Tx t

0( ) ,jk tT k

k

x t c e t

where

/ 2

/ 2

1( ) , 0, 1, 2,o

Tjk t

k

T

c x t e dt kT

• What happens to the frequency components of as ?

• For

• For



( )Tx t T 0 :k

0 1/c T1, 2, :k

0 0

0

2 1sin sin

2 2k

k kc

k T k

0 2 /T



plots of vs. for

| |kT c

0k 2,5,10T

• It can be easily shown that

where



lim sinc ,2k

TTc

sin( )sinc( )

sin( )

sinc( )

• The Fourier transform of the rectangular pulse x(t) is defined to be the limit of as , i.e.,

Fourier Transform of the Rectangular Pulse

Fourier Transform of the Rectangular Pulse

( ) lim sinc ,2k

TX Tc

kTcT

| ( ) |X | ( ) |X arg( ( ))X arg( ( ))X

• Given a signal x(t), its Fourier transform Fourier transform is defined as

• A signal x(t) is said to have a Fourier transform in the ordinary sense if the above integral converges

The Fourier Transform in the General Case

The Fourier Transform in the General Case

( )X

( ) ( ) ,j tX x t e dt

• The integral does converge if1. the signal x(t) is “well-behavedwell-behaved” 2. and x(t) is absolutely integrableabsolutely integrable, namely,

1. Note: well behavedwell behaved means that the signal has a finite number of discontinuities, maxima, and minima within any finite time interval

The Fourier Transform in the General Case – Cont’d

The Fourier Transform in the General Case – Cont’d

| ( ) |x t dt

• Consider the signal • Clearly x(t) does not satisfy the first

requirement since

• Therefore, the constant signal does not have a Fourier transform in the ordinary senseFourier transform in the ordinary sense

• Later on, we’ll see that it has however a Fourier transform in a generalized senseFourier transform in a generalized sense

Example: The DC or Constant SignalExample: The DC or Constant Signal

( ) 1,x t t

| ( ) |x t dt dt

• Consider the signal

• Its Fourier transform is given by

Example: The Exponential SignalExample: The Exponential Signal

( ) ( ),btx t e u t b

( ) ( )

0 0

( ) ( )

1

bt j t

t

b j t b j t

t

X e u t e dt

e dt eb j

• If , does not exist

• If , and does not exist either in the ordinary sense

• If , it is

Example: The Exponential Signal – Cont’d

Example: The Exponential Signal – Cont’d

0b ( )X 0b ( ) ( )x t u t ( )X

0b 1

( )Xb j

2 2

1| ( ) |X

b

amplitude spectrumamplitude spectrum

arg( ( )) arctanXb

phase spectrumphase spectrum

Example: Amplitude and Phase Spectra of the Exponential SignalExample: Amplitude and Phase

Spectra of the Exponential Signal10( ) ( )tx t e u t 10( ) ( )tx t e u t

• Consider

• Since in general is a complex function, by using Euler’s formula

Rectangular Form of the Fourier Transform

Rectangular Form of the Fourier Transform

( ) ( ) ,j tX x t e dt

( )X

( ) ( )

( ) ( )cos( ) ( )sin( )

R I

X x t t dt j x t t dt

( ) ( ) ( )X R jI

• can be expressed in a polar form as

where

Polar Form of the Fourier TransformPolar Form of the Fourier Transform

( ) | ( ) | exp( arg( ( )))X X j X

( ) ( ) ( )X R jI

2 2| ( ) | ( ) ( )X R I

( )arg( ( )) arctan

( )

IX

R

• If x(t) is real-valued, it is

• Moreover

whence

Fourier Transform of Real-Valued Signals

Fourier Transform of Real-Valued Signals

( ) ( )X X

( ) | ( ) | exp( arg( ( )))X X j X

| ( ) | | ( ) | and

arg( ( )) arg( ( ))

X X

X X

Hermitian Hermitian symmetrysymmetry

• Consider the even signal

• It is

Example: Fourier Transform of the Rectangular Pulse

Example: Fourier Transform of the Rectangular Pulse

/ 2

/ 2

00

2 2( ) 2 (1)cos( ) sin( ) sin

2

sinc2

t

tX t dt t

Example: Fourier Transform of the Rectangular Pulse – Cont’d


( ) sinc2

X



amplitude amplitude spectrumspectrum

phase phase spectrumspectrum

• A signal x(t) is said to be bandlimitedbandlimited if its Fourier transform is zero for all where BB is some positive number, called the bandwidth of the signalbandwidth of the signal

• It turns out that any bandlimited signal must have an infinite duration in time, i.e., bandlimited signals cannot be time limited

Bandlimited SignalsBandlimited Signals

( )X B

• If a signal x(t) is not bandlimited, it is said to have infinite bandwidthinfinite bandwidth or an infinite infinite spectrumspectrum

• Time-limited signals cannot be bandlimited and thus all time-limited signals have infinite bandwidth

• However, for any well-behaved signal x(t) it can be proven that whence it can be assumed that

Bandlimited Signals – Cont’dBandlimited Signals – Cont’d

lim ( ) 0X

| ( ) | 0X B B being a convenient large number

• Given a signal x(t) with Fourier transform , x(t) can be recomputed from by applying the inverse Fourier transforminverse Fourier transform given by

• Transform pairTransform pair

Inverse Fourier TransformInverse Fourier Transform

( )X ( )X

1( ) ( ) ,

2j tx t X e d t

( ) ( )x t X

Properties of the Fourier Transform Properties of the Fourier Transform

• Linearity:Linearity:

• Left or Right Shift in Time:Left or Right Shift in Time:

• Time Scaling:Time Scaling:

( ) ( )x t X ( ) ( )y t Y

( ) ( ) ( ) ( )x t y t X Y

00( ) ( ) j tx t t X e

1( )x at X

a a


• Time Reversal:Time Reversal:

• Multiplication by a Power of t:Multiplication by a Power of t:

• Multiplication by a Complex Exponential:Multiplication by a Complex Exponential:

( ) ( )x t X

( ) ( ) ( )n

n nn

dt x t j X

d

00( ) ( )j tx t e X


• Multiplication by a Sinusoid (Modulation):Multiplication by a Sinusoid (Modulation):

• Differentiation in the Time Domain:Differentiation in the Time Domain:

0 0 0( )sin( ) ( ) ( )2

jx t t X X

0 0 0

1( )cos( ) ( ) ( )

2x t t X X

( ) ( ) ( )n

nn

dx t j X

dt


• Integration in the Time Domain:Integration in the Time Domain:

• Convolution in the Time Domain:Convolution in the Time Domain:

• Multiplication in the Time Domain:Multiplication in the Time Domain:

1( ) ( ) (0) ( )

t

x d X Xj

( ) ( ) ( ) ( )x t y t X Y

( ) ( ) ( ) ( )x t y t X Y


• Parseval’s Theorem:Parseval’s Theorem:

• Duality:Duality:

1( ) ( ) ( ) ( )

2x t y t dt X Y d

2 21| ( ) | | ( ) |

2x t dt X d

( ) ( )y t x tif

( ) 2 ( )X t x

Properties of the Fourier Transform - Summary

Properties of the Fourier Transform - Summary

Example: LinearityExample: Linearity

4 2( ) ( ) ( )x t p t p t

2( ) 4sinc 2sincX

Example: Time ShiftExample: Time Shift

2( ) ( 1)x t p t

( ) 2sinc jX e

Example: Time ScalingExample: Time Scaling

2 ( )p t

2 (2 )p t

2sinc

sinc2

time compression frequency expansiontime expansion frequency compression

1a 0 1a

Example: Multiplication in TimeExample: Multiplication in Time

2( ) ( )x t tp t

2

sin cos sin( ) 2sinc 2 2

d dX j j j

d d

Example: Multiplication in Time – Cont’d

Example: Multiplication in Time – Cont’d

2

cos sin( ) 2X j

Example: Multiplication by a SinusoidExample: Multiplication by a Sinusoid

0( ) ( )cos( )x t p t t sinusoidal burst

0 01 ( ) ( )( ) sinc sinc

2 2 2X

Example: Multiplication by a Sinusoid – Cont’d

Example: Multiplication by a Sinusoid – Cont’d

0 01 ( ) ( )( ) sinc sinc

2 2 2X

0 60 / sec

0.5

rad

Example: Integration in the Time Domain

Example: Integration in the Time Domain

2 | |( ) 1 ( )

tv t p t

( )( )

dv tx t

dt

Example: Integration in the Time Domain – Cont’d


• The Fourier transform of x(t) can be easily found to be

• Now, by using the integration property, it is

( ) sinc 2sin4 4

X j

21( ) ( ) (0) ( ) sinc

2 4V X X

j



2( ) sinc2 4

V

• Fourier transform of

• Applying the duality property

Generalized Fourier TransformGeneralized Fourier Transform

( )t

( ) 1j tt e dt

( ) 1t

( ) 1, 2 ( )x t t

generalized Fourier transformgeneralized Fourier transform of the constant signal ( ) 1,x t t

Generalized Fourier Transform of Sinusoidal Signals

Generalized Fourier Transform of Sinusoidal Signals

0 0 0cos( ) ( ) ( )t

0 0 0sin( ) ( ) ( )t j

Fourier Transform of Periodic SignalsFourier Transform of Periodic Signals

• Let x(t) be a periodic signal with period T; as such, it can be represented with its Fourier transform

• Since , it is002 ( )j te

0( ) jk tk

k

x t c e

0 2 /T

0( ) 2 ( )kk

X c k

• Since

using the integration property, it is

Fourier Transform of the Unit-Step FunctionFourier Transform of

the Unit-Step Function

( ) ( )t

u t d

1( ) ( ) ( )

t

u t dj

Common Fourier Transform PairsCommon Fourier Transform Pairs

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Chapter 4 The Fourier Series and Fourier Transform