lecture 3 review of fourier transform - site.uottawa.ca

20-Jan-15 Lecture 3, ELG3175 : Introduction to Communication Systems © S. Loyka

Review of Fourier Transform

• Fourier series works for periodic signals only. What’s

about aperiodic signals? This is very large & important

class of signals

• Aperiodic signal can be considered as periodic with

• Fourier series changes to Fourier transform, complex

exponents are infinitesimally close in frequency

• Discrete spectrum becomes a continuous one, also

known as spectral density

T →∞

Lecture 3

1(25)


Fourier Series -> Fourier Transform

14T T=

18T T=

116T T=

Periodic signal

Its spectrum

As T increases, spectral

components are getting closer

and closer, becoming the

continuous spectrum at the limit

A.V. Oppenheim, A.S. Willsky, Signals and Systems, 1997.

Lecture 3

2(25)


Fourier Transform

• Fourier transform (spectrum):

• Inverse Fourier transform:

• Dirichlet conditions (details on the next page):

– x(t) is absolutely integrable,

– x(t) has a finite number of maxima & minima within any finite

interval,

– x(t) has a finite number of discontinuities within any finite interval.

These discontinuities must be finite.

( ) ( ) 2j ftxS f x t e dt

+∞− π

−∞

= ∫

( ) ( ) 2j ftxx t S f e df

+∞π

−∞

= ∫ ( ) ( )1

2

j txx t S e d

+∞ω

−∞

= ω ω

π∫

( ) ( ) j txS x t e dt

+∞− ω

−∞

ω = ∫

radial frequency

radial frequency

Lecture 3

3(25)


Convergence of Fourier Transform

• Dirichlet conditions:– x(t) must be absolutely integrable (finite energy)

or

– x(t) has a finite number of maxima & minima within any finite

interval,

– x(t) has a finite number of discontinuities within any finite

interval. These discontinuities must be finite.

• Dirichlet conditions are only sufficient, but are not necessary.

• All physically-reasonable (practical) signals meet these conditions.

( )x t dt

∞

−∞

< ∞∫ ( )2

x t dt

∞

−∞

< ∞∫

Lecture 3

4(25)


Example: Rectangular Pulse

2

τ

−

2

τ

1

τ

1−

τ

τ

( )x

S f

f

t

signal

spectrum

( )1, / 2

( / )0, / 2

t

x t t

t

< τ= Π τ =

> τ

( )sin

sinc( )x

fS f f

f

π τ

= τ = τ ⋅ τ

π τ

Lecture 3

1

5(25)


Example: sinc(t)

1

1

2 f−

1

1

2 f

1f− 1f

2

1

2 f−

2

1

2 f

2f− 2f

signal signal

spectrum spectrum

Shortening pulse widens its spectrum!

1( )

xS f

2( )

xS f

Lecture 3

12 f

22 f

6(25)


Generalized (Singular) Functions

• Unit step function

• Dirac delta function (unit impulse function)

1, 0( )

0, 0

t

u t

t

>=

<

1

t

u(t)

( ) ( ) (0)t x t dt x

∞

−∞

δ =∫

, 0( ) & ( ) 1

0, 0

tt t dt

t

∞

−∞

∞ =δ = δ =

≠∫

t

( )tδ

Lecture 3

7(25)


Generalized Functions

• Dirac Delta Function as a limit:

• Useful properties of delta function

1/ , / 2( )

0,

t

t

t∆

∆ < ∆Π =

> ∆

t

( )t∆Π

2

∆−

2

∆

1

∆

0

( ) lim ( )t t∆∆→

δ = Π

0 0( ) ( ) ( )t t x t dt x t

∞

−∞

δ − =∫( )

( )du t

tdt

= δ( ) ( )

t

t dt u t

−∞

δ =∫

1( ) ( )at t

a

δ = δ ( ) ( )t tδ − = δ ( ) ( ) (0) ( )x t t x tδ = δ

Fourier transform?

Lecture 3

8(25)


Fourier Transform of Periodic Signal

• FT of a complex exponent:

• Important property:

• FT of a periodic signal:

• FT of ?

( ) 0

0 02 ( ) ( )

j tx t e f fω

= ↔ πδ ω− ω = δ −

( ) 0

0 02 ( ) ( )

FTjn t

n n n

n n n

x t c e c n c f nf+∞ +∞ +∞

ω

=−∞ =−∞ =−∞

= ↔ π δ ω − ω = δ −∑ ∑ ∑

( ) 2j ftf e dt

+∞± π

−∞

δ = ∫

0cos( )tω

Lecture 3

� prove this property

9(25)


Properties of Fourier Transform*

• Very similar to those of Fourier series!

• Linearity:

• Time shifting:

• Time reversal:

• Time scaling:

1 21 2( ) ( ) ( ) ( )

F

x xx t x t S f S fα +β ↔α +β

0

0( ) ( ) ( ) ( )j t

x xx t S x t t e S− ω

↔ ω ⇒ − ↔ ω

( ) ( ) ( ) ( )x x

x t S x t S↔ ω ⇒ − ↔ −ω

1( )

xx at S

a a

ω ↔

Lecture 3

*properties are useful for evaluating Fourier transform in a simple way

Prove these

properties.

10(25)


Properties of Fourier Transform

• Conjugation:

• Differentiation:

• Integration:

• Multiplication:

• Frequency shift (modulation):

*( ) ( ) * ( ) ( )

x xx t S x t S↔ ω ⇒ ↔ −ω

( )( ) ( ) ( )

x x

dx tx t S j S

dt↔ ω ⇒ ↔ ω ω

1( ) ( ) (0) ( )

t

x xx t dt S S

j−∞

↔ ω + π δ ωω

∫

1( ) ( ) ( ) ( )

2

( )* ( )

x y

x y

x t y t S S d

S S

∞

−∞

′ ′ ′↔ ω ω−ω ω =π

= ω ω

∫

0

0( ) ( )j t

x t e Sω

↔ ω−ω

Lecture 3

Pro

ve t

hese p

ropert

ies

11(25)


Duality of Fourier Transform

( ) ( ) ( ) 2 ( )x x

x t S S t x↔ ω ⇒ ↔ π −ω ( ) ( ) ( ) ( )x x

x t S f S t x f↔ ⇒ ↔ −


Lecture 3

12(25)


Convolution Property

• This property is of great importance

( ) ( ) ( ) ( ) ( ) ( )x yy t x h t d S H S

∞

−∞

= τ − τ τ↔ ω ω = ω∫

x(t) h(t)

y(t)

( )xS ω ( )H ω ( )yS ω

Cascade connection

of LTI blocks

Lecture 3


• Example: FT of a triangular pulse by convolution

13(25)


Parseval Theorem• Total energy in time domain is the same as the total

energy in frequency domain:

• - energy spectral density (ESD) of x(t).

Represents the amount of energy per Hz of bandwidth

• Counterpart of Parseval theorem for periodic signals

• Autocorrelation property:

2 22 1( ) ( ) ( )

2x x

E x t dt S f df S d

∞ ∞ ∞

−∞ −∞ −∞

= = = ω ω

π∫ ∫ ∫

2( ) ( )

xE f S f=

( ) ( ) ( )2

( )x x

R x t x t dt S+∞

∗

−∞

τ = − τ ↔ ω∫ ( )0x

R E=

Lecture 3

14(25)


Parseval Theorem: Example

2sin

?t

dtt

+∞

−∞

=

∫

Lecture 3

15(25)


Fourier Transform of Real Signal

• if x(t) is real,

• Fourier transform can be presented as

{ } *Im ( ) 0 ( ) ( )

x xx t S S= ⇒ −ω = ω

( )

[ ][ ]

0

1

( ) 2 ( ) cos 2 ( ) ,

Im ( )( ) tan

Re ( )

x

x

x

x t S f f f df

S ff

S f

∞

−

= π + ϕ

ϕ =

∫No negative

frequencies!

Lecture 3

16(25)


Signal Bandwidth & Negative Frequencies

• What negative frequency means?

• It means that there is term in signal spectrum

• Convenient mathematical tool. Do not exist in practice

(i.e., cannot be measured on spectrum analyzer)

• What is the signal bandwidth? There are many definitions.

• Defined for positive frequencies only.

• Determines the frequency band over which a substantial

part of the signal power/energy is concentrated.

• For band-limited signals

2j fte− π

max min max min, , 0f f f f f∆ = − ≥

Lecture 3

17(25)


Power and Energy

• Power Px& energy E

xof signal x(t) are:

• Energy-type signals:

• Power-type signals:

• Signal cannot be both energy & power type!

• Signal energy: if x(t) is voltage or current, Exis the

energy dissipated in 1 Ohm resistor

• Signal power: if x(t) is voltage or current, Pxis the power

dissipated in 1 Ohm resistor.

2( )

xE x t dt

∞

−∞

= ∫21

lim ( )2

T

xT

T

P x t dtT→∞

−

= ∫

xE < ∞

0xP< < ∞

Lecture 3

18(25)


Energy-Type Signals (summary)

• Signal energy in time & frequency domains:

• Energy spectral density (energy per Hz of bandwidth):

• ESD is FT of autocorrelation function:

2 22 1( ) ( ) ( )

2x x x

E x t dt S f df S d

∞ ∞ ∞

−∞ −∞ −∞

= = = ω ω

π∫ ∫ ∫

2( ) ( )

x xE f S f=

( ) ( ) ( ) ( )x x

R x t x t dt E f+∞

∗

−∞

τ = − τ ↔∫ ( )0x x

R E=

Lecture 3

19(25)


Power-Type Signals: PSD

• Definition of the power spectral density (PSD) (power per

Hz of bandwidth):

• where is the truncated signal (to [-T/2,T/2]),

• and is its spectrum (FT),

( ) ( )2

( )lim

T

x x xT

S fP f P P f df

T

∞

−∞→∞

= ⇒ = < ∞∫

Lecture 3

( ), / 2 / 2 ( ) ( )

0, otherwiseT

x t T t Ttx t x t

T

− ≤ ≤ = Π =

( )Tx t

{ }( ) ( )T TS f FT x t=

( )TS f

20(25)


Power-Type Signals

• Time-average autocorrelation function:

• Power of the signal:

• Wiener-Khintchine theorem :

( ) ( ) ( )1

lim2

T

xTT

R x t x t dtT

∗

−→∞

τ = − τ∫

( ) ( ) { }( )x x x xP P f df P f FT R

∞

−∞

= ⇒ = τ∫

( ) ( )21

lim 02

T

x xTT

P x t dt RT −→∞

= =∫

Lecture 3

21(25)


Periodic Signals

• Power of a periodic signal:

• Autocorrelation function:

• Power spectral density (PSD):

221( ) (0)

x n xT

n

P x t dt c RT

∞

=−∞

= = =∑∫

( ) ( ) ( ) 021 jn

x nTn

R x t x t dt c eT

∞

ω τ∗

=−∞

τ = − τ = ∑∫

( ) { }2

( )x x n

n

nP f FT R c f

T

∞

=−∞

= τ = δ −

∑

Lecture 3

( ) 0jn t

n

n

x t c e

+∞

ω

=−∞

← = ∑

Prove these

properties!

22(25)


Relation Between Fourier Transform &

Series

• consider a periodic signal x(t)=x(t+T)

• truncate it (one period only):

• find FT of the truncated signal xT(t):

• Fourier series of the original periodic signal x(t) is

• Continuous spectrum is the envelope of discrete

spectrum (see slide 2)!

( ), / 2 / 2( )

0, otherwiseT

x t T t Tx t

− < ≤=

( ) ( )T

T xx t S↔ ω

0

1( )T

n xc S n

T= ω

Lecture 3

� prove this

23(25)


Fourier Transform of Single Pulse ~ Envelope of

Fourier Series of Pulse Train


Lecture 3

FT/S

24(25)


Summary

• Definition of Fourier Transform

• Properties of Fourier Transform

• Signal bandwidth

• Signal power & energy. Energy and power-type signals

• Fourier transform of periodic signals

• Relation between Fourier series & transform

• Reading: Couch, 2.1-2.6; Oppenheim & Willsky, Ch. 1, 3 & 4. Study

carefully all the examples (including end-of-chapter study-aid examples),

make sure you understand and can solve them with the book closed.

• Do some end-of-chapter problems. Students’ solution manual provides

solutions for many of them.

Lecture 3

25(25)

lecture 3 review of fourier transform - site.uottawa.ca

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