p. 1

TELE3113 Analogue & Digital Communications

Review of Fourier Transform

p. 2

Signal Representation

A: Amplitude

f : Frequency (Hz) (ω=2πf)

φ : Phase (radian or degrees)

s(t) = A sin(2π fo t +φo ) or A sin(ωo t +φo )

Time (seconds)

Period (seconds)

Time-domain: waveform

Frequency-domain: spectrum

Frequency (Hz)fo

S(f)

p. 3

Energy and Power of Signals

For an arbitrary signal f(t), the total energy normalized to unit resistance is defined as

joules, )(lim 2 dttfET

TT ∫−∞→

∆

=

and the average power normalized to unit resistance is defined as

, watts )(21lim 2 dttfT

PT

TT ∫−∞→

∆

=

• Note: if 0 < E < ∞ (finite) P = 0.• When will 0 < P < ∞ happen?

p. 4

ttfTtf allfor )()( 0 =+ (*)

where the constant T0 is the period.

The smallest value of T0 such that equation (*) is satisfied is referred to as the fundamental period, and is hereafter simply referred to as the period.

Any signal not satisfying equation (*) is called aperiodic.

A signal f(t) is periodic if and only if

Periodic Signal

p. 5

Deterministic signal can be modeled as a completely specified function of time.

Example)cos()( 0 θ+ω= tAtf

Random signal cannot be completely specified as a function of time and must be modeled probabilistically.

Deterministic & Random Signals

p. 6

Mathematically, a system is a rule used for assigning a function g(t)(the output) to a function f(t) (the input); that is,

where h{•} is the rule or we call the impulse response.

For two systems connected in cascade, the output of the first system forms the input to second, thus forming a new overall system:

System

h(t)

g(t) = h{ f(t) }

f(t) g(t)

g(t) = h2 { h1 [ f(t) ] } = h{ f(t) }

p. 7

If a system is linear then superposition applies; that is, if

then

where a1, a2 are constants. A system is linear if it satisfiesEq. (*); any system not meeting these requirement is nonlinear.

h{ a1 f1(t) + a2 f2(t) } = a1 g1(t) + a2 g2(t) (*)

Linear System

g1(t) = h{ f1(t) }, and g2(t) = h{ f2(t) }

p. 8

A system is time-invariant if a time shift in the input resultsin a corresponding time shift in the output so that

.any for )}({)( 000 tttfhttg −=−

The output of a time-invariant system depends on time differences and not on absolute values of time.

Any system not meeting this requirement is said to be time-varying.

Time-Invariant and Time-Varying

p. 9

A periodic function of time s(t) with a fundamental period of T0 can be represented as an infinite sum of sinusoidal waveforms. Such summation, a Fourier series, may be written as:

∑∑∞

=

∞

=

π+

π+=

1 01 00 ,2sin2cos)(

nn

nn T

ntBTntAAts

where the average value of s(t), A0 is given by

∫−= 2

0

20

,)(1

00

T

T dttsT

A

while

.2sin)(2 20

20

00∫−

π=

T

T dtTntts

TBn

∫−

π= 2

0

20

,2cos)(2

00

T

T dtTntts

TAn

and

(1)

(2)

(3)

(4)

Fourier Series

p. 10

An alternative form of representing the Fourier series is

∑∞

=

φ−

π+=

1 00

2cos)(n

nn TntCCts

where ,00 AC =

,22nnn BAC +=

.tan 1

n

nn A

B−=φ

The Fourier series of a periodic function is thus seen to consist of a summation of harmonics of a fundamental frequency f0 = 1/T0.

The coefficients Cn are called spectral amplitudes, which represent the amplitude of the spectral component Cn cos(2πnf0t − φn) at frequency nf0.

(5)

(6)

(7)

(8)

Fourier Series

p. 11

The exponential form of the Fourier series is used extensively in communication theory. This form is given by

∑∞

−∞=

π

=n

nTntj

eSts ,)( 02

where

∫−

−= 2

0

20

0

2

)(1

0

T

TTntj

dtetsT

S nπ

Note that Sn and S−n are complex conjugate of one another, that is

.*nn SS −=

These are related to the Cn by

,00 CS = .2

njnn eCS φ−=

(9)

(10)

(11)

(12)

Fourier Series

p. 12

Amplitude Spectra (Line Spectra)

Note that except S0 = C0, each spectral line in Fig. (a) at frequency fis replaced by the two spectral lines in Fig. (b), each with half amplitude, one at frequency f and one at frequency - f.

Fourier Series

0 fo 2fo 3fo 4fo 5fo 6fo (n-1) fo nfo

CnFig.(a)

|Sn|

-nfo -(n-1)fo ••• - 6fo0-5fo -4fo -3fo -2fo -fo 0 fo 2fo 3fo 4fo 5fo 6fo ••• (n-1) fo nfo

••• •••

Fig.(b)

p. 13

Consider a unitary square wave defined by

<<−<<

= 150 ,15.00 ,1

)(t.t

tx

and periodically extended outside this interval. The average value is zero, so

.00 =A

Recall that

( )

( ) ( )

( ) ( )

0

22sin

22sin2

2cos22cos2

2cos)(2

2cos)(2

1

5.0

5.0

0

1

5.0

5.0

0

1

0

00

20

20

=

ππ−

ππ=

π−+π=

π=

π=

∫∫∫

∫−

nnt

nnt

dtntdtnt

dtnttx

dtTnttx

TA

T

Tn

Thus all An coefficients are zero.

The Bn coefficients are given by

( )

( ) ( )

( ) ( )

( )π−π

=

ππ+

ππ−=

π−+π=

π=

π=

∫∫∫

∫−

nn

nnt

nnt

dtntdtnt

dtnttx

dtTnttx

TB

T

Tn

cos12

22cos

22cos2

2sin22sin2

2sin)(2

2sin)(2

1

5.0

5.0

0

1

5.0

5.0

0

1

0

00

20

20

which results in

π=even is ,0

odd is ,4

n

nnBn

Fourier Series : Example

p. 14

The Fourier series of a square wave of unitary amplitude with odd symmetry is therefore

Fourier Series : Example

)10sin516sin

312(sin4)( K+π+π+π

π= ttttx

1st term 1st + 2nd terms 1st + 2nd + 3rd terms

Sum up to the 6th term

p. 15

Representation of an Aperiodic Function

)()(lim tftfTT=

∞→(13)

Consider an aperiodic function f(t)

To represent this function as a sum of exponential functions overthe entire interval (-∞, ∞), we construct a new periodic functionfT(t) with period T.

By letting T→∞,

Fourier Transform

p. 16

The new function fT(t) can be represented by an exponential Fourier series, which is written as

∑∞

−∞=

ω=n

tjnnT eFtf ,)( 0

where

∫−

ω−=2/

2/0)(1 T

T

tjnTn dtetf

TF

and ./20 Tπ=ω

(14)

(15)

Fourier Transform

p. 17

For the sake of clear presentation, we set

,0ω=ω∆

nn ,)( nn TFF∆

=ω

Thus, Eq.(14) and (15) become

∑∞

−∞=

ωω=n

tjnT

neFT

tf ,)(1)(

∫−

ω−=ω2/

2/.)()(

T

T

tjTn dtetfF n

The spacing between adjacent lines in the line stream of fT(t)is

./2 Tπ=ω∆

(16)

(17)

(18)

(19)

Fourier Transform

p. 18

Using this relation for T, we get

∑∞

−∞=

ω

πω∆

ω=n

tjnT

neFtf .2

)()(

As T becomes very large, ∆ω becomes smaller and the spectrumbecomes denser.

In the limit T → ∞, the discrete lines in the spectrum of fT(t) mergeand the frequency spectrum becomes continuous.

Therefore,∑

∞

−∞=

ω

∞→∞→ω∆ω

π=

n

tjnTTT

neFtf )(21lim)(lim

becomes ∫∞

∞−

ω ωωπ

= deFtf tj)(21)(

(20)

(21)

(22)

Fourier Transform

p. 19

In a similar way, Eq. (18) becomes

.)()( ∫∞

∞−

ω−=ω dtetfF tj

Eq. (22) and (23) are commonly referred to as the Fourier transform pair.

(23)

Fourier Transform

Inverse Fourier Transform

∫∞

∞−

−= dtetfF tjωω )()(

∫∞

∞−

ω ωωπ

= deFtf tj)(21)(

Fourier Transform

p. 20

F(ω): The spectral density function of f(t).

Fig. 3.2

2/)2/sin()2/(Sa

ωω

=ω

A unit gate function Its spectral density graph

Spectral Density Function

p. 21

The energy delivered to a 1-ohm resistor is

∫∫∞

∞−

∞

∞−== .)()()( *2 dttftfdttfE

Using Eq. (22) in (24), we get

.)()(21

)()(21

)(21)(

*

*

*

∫

∫ ∫

∫ ∫

∞

∞−

∞

∞−

∞

∞−

ω−

∞

∞−

∞

∞−

ω−

ωωωπ

=

ω

ω

π=

ωω

π=

dFF

ddtetfF

dtdeFtfE

tj

tj

Parseval’s Theorem:

∫∫∞

∞−

∞

∞−ωω

π= .)(

21)( 22 dFdttf

(24)

(25)

(26)

Parseval’s Theorem

∫∞

∞−

ω ωωπ

= deFtf tj)(21)(

p. 22

The unit impulse function satisfies

,1)( =δ∫∞

∞−dxx

≠=∞

=.0 0,0

)(xx

xδ

Using the integral properties of the impulse function, the Fourier transform of a unit impulse, δ(t), is

{ } .1)()( 0 ==δ=δℑ ∫∞

∞−

ω− jtj edtett

If the impulse is time-shifted, we have

{ } .)()( 000

tjtj edtetttt ω−∞

∞−

ω− =−δ=−δℑ ∫

(27)

(28)

(29)

(30)

Fourier Transform: Impulse Function

p. 23

{ }

,21

)(21)(

0

001

tj

tj

e

de

ω±

∞

∞−

ω−

π=

ωωωδπ

=ωωδℑ ∫ mm

The spectral density of will be concentrated at ±ω0.tje 0ω±

Taking the Fourier transform of both sides, we have

{ } { }tje 0

21)( 0

1 ω±− ℑπ

=ωωδℑℑ m

which gives{ } )(2 0

0 ωωπδω m=ℑ ± tje

(31)

(32)

(33)

Fourier Transform: Complex Exponential Function

p. 24

The sinusoidal signals and can be written in terms ofthe complex exponentials.

t0cosω t0sin ω

Their Fourier transforms are given by

{ } { }),()(

cos

00

21

21

000

ω+ωπδ+ω−ωπδ=

+ℑ=ωℑ ω−ω tjtj eet

(34)

{ } { }.)()(

sin

00

21

21

000

j

eet tjj

tjj

ω+ωπδ−ω−ωπδ=

−ℑ=ωℑ ω−ω

(35)

Fourier Transform: Sinusoidal Function

p. 25

(37)

We can express a function f(t) that is periodic with period T by itsexponential Fourier series

∑∞

−∞=

ω=n

tjnnT eFtf 0)( where ω0 = 2π/T.

Taking the Fourier transform, we have

(36)

{ }

{ }

∑

∑

∑

∞

−∞=

∞

−∞=

ω

∞

−∞=

ω

ω−ωδπ=

ℑ=

ℑ=ℑ

nn

n

tjnn

n

tjnnT

nF

eF

eFtf

).(2

)(

0

0

0 e.g.

Line spectrum of f(t) with period T

Its spectral density graph

Its Fourier transformA unit gate function

Fourier Transform: Periodic Functions

p. 26

Time and Spectral Density Functions

p. 27

Selected Fourier Transform Pairs

p. 28

Linearity (Superposition)

)()()()( 22112211 ω+ω↔+ FaFatfatfa

Complex Conjugate)()( ** ω−↔ Ftf

Scaling

ω

↔a

Fa

atf 1)( for .0≠a

Time Shifting (Delay)0

0 )()( tjeFttf ωω −↔−

Frequency Shifting (Modulation)

)()( 00 ω−ω↔ω Fetf tj

Differentiation

)()()( ωω↔ Fjtfdtd nn

n

Duality).(2)( ω−π↔ ftF

Multiplication

)()()()( 2121 ω∗ω↔ FFtftf

Convolution

)()()()( 2121 ωω↔∗ FFtftf

Properties of Fourier Transform

p. 29

Scaling

ω

↔a

Fa

atf 1)( for .0≠a

Duality ).(2)( ω−π↔ ftF

Properties of Fourier Transform

p. 30

Frequency Shifting (Modulation)

)()( 00 ω−ω↔ω Fetf tj

Properties of Fourier Transform