4-fourier

Processamento Digital de SinaisEL66D - Engenharia Eletronica

Representacao Fourier de Sinais

Prof. Daniel R. [email protected]

wikipipa.org

2014/05/09 Prof. Daniel R. Pipa 1/30

mailto:[email protected]

http://wikipipa.org

Representacao Fourier de Sinais Sinais senoidais no tempo contınuo

Senoides no tempo contınuoUma senoide no tempo contınuo e dada por

x.t/ D A cos .2F0t C / , t 2 R; 0 D 2F0

DA

2e j e j0t

CA

2e j e j0t

135 4.1 Sinusoidal signals and their properties

4.1 Sinusoidal signals and their properties• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

The goal of Fourier analysis of signals is to break up all signals into summations of sinu-soidal components. Thus, we start our discussion with the definitions and properties ofcontinuous-time and discrete-time sinusoidal signals. Fourier analysis is like a glass prism,which splits a beam of light into frequency components corresponding to different colors.

4.1.1 Continuous-time sinusoids

A continuous-time sinusoidal signal may be represented as a function of time t by theequation

x(t) = A cos(2πF0t + θ), −∞ < t < ∞ (4.1)

where A is the amplitude, θ is the phase in radians, and F0 is the frequency. If we assumethat t is measured in seconds, the units of F0 are cycles per second or Hertz (Hz). Inanalytical manipulations it is more convenient to use the angular or radian frequency

#0 = 2πF0 (4.2)

measured in radians per second. The meaning of these quantities is illustrated in Figure 4.1.Using Euler’s identity, e± jφ = cos φ ± j sin φ, we can express every sinusoidal signal

in terms of two complex exponentials with the same frequency, as follows:

A cos(#0t + θ) = A2

ejθ ej#0t + A2

e− jθ e− j#0t. (4.3)

Therefore, we can study the properties of the sinusoidal signal (4.1) by studying theproperties of the complex exponential x(t) = ej#0t.

Frequency, viewed as the number of cycles completed per unit of time, is an inherentlypositive quantity. However, the use of negative frequencies is a convenient way to describesignals in terms of complex exponentials. The concept of negative frequencies is usedthroughout this book, mainly for mathematical convenience.

0

A

t

x(t)

T0 F0

1 2π= =Ω0

Acosφ

Figure 4.1 Continuous-time sinusoidal signal and its parameters.Frequencia positiva

x.t/ D e j0t

Frequencia negativa

x.t/ D e j0t


Representacao Fourier de Sinais Sinais senoidais no tempo discreto

Senoides no tempo discreto

Amostra-se senoide contınua com perıodo T ou freq. de amostragem Fs .138 Fourier representation of signals

–0.5 0 0.5 1 1.5 2 2.5 3

–1

1

Time (t)T

x(t)

Figure 4.4 Sampling of a continuous-time sinusoidal signal.

Although each sinusoidal signal on the right-hand side of (4.9) is periodic, there is noperiod T0 in which x2(t) repeats itself. The signal x2(t), which is shown in Figure 4.3(b), issaid to be “almost”-periodic or “quasi”-periodic. It turns out that we can create aperiodicfinite duration signals (“pulse-like”) by combining sinusoidal components with frequencieswithin a continuous frequency range through integration (see Section 4.2.2).

4.1.2 Discrete-time sinusoids

A discrete-time sinusoidal signal is conveniently obtained by sampling the continuous-time sinusoid (4.1) at equally spaced points t = nT as shown in Figure 4.4. The resultingsequence is

x[n] = x(nT) = A cos(2πF0nT + θ) = A cos!

2πF0

Fsn + θ

". (4.10)

If we define the normalized frequency variable

f ! FFs

= FT , (4.11)

and the normalized angular frequency variable

ω ! 2π f = 2πFFs

= $T , (4.12)

we can express the discrete-time sinusoid (4.10) as

x[n] = A cos(2π f0n + θ) = A cos(ω0n + θ), −∞ < n < ∞ (4.13)

where A is the amplitude, f0 (or ω0) the frequency, and θ the phase (see Figure 4.5).If the input to a discrete-time LTI system is a complex exponential sequence, its output

is a complex exponential with the same frequency. Indeed, we have

x[n] = ejωn H#−→ y[n] = H (ejω)ejωn, for all n, (4.14)

which is obtained from (3.6) by setting z = ejω. Thus, the complex exponentials ejωn areeigenfunctions of discrete-time LTI systems with eigenvalues given by the system functionH(z) evaluated at z = ejω. As we will see in the next chapter, this property is of majorimportance in signal and system analysis.

xŒn D x.nT / D A cos.2F0nT C / D A cos

2F0

FsnC

.

Defini-se frequencia normali-zada ou frequencia digital

f DF

FsD F T

Frequencia angular normalizadaou frequencia angular digital

! D 2f D 2F

FsD T



Frequencia tradicional versus frequencia digital

Sinais no tempo contınuo Sinais no tempo discreto

D 2F ! D 2f

radianossegundos Hz radianos

amostraciclos

amostra

! D T

f D F=Fs

)

1 < <1 < ! <

1 < F <1 12

< f < 12

Importante!Sinais no tempo discreto tem frequencia maxima limitada pelafrequencia de amostragem. A freq. digital e normalizada!



Senoides discretasxŒn D cos.!n/, para ! D 0 e ! D =10.


–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

Figure 4.6 The signal x[n] = cos ω0n for different values of ω0. The rate of oscillationincreases as ω0 increases from 0 to π and decreases again as ω0 increases from π to 2π .

Frequency variables and units After studying continuous- and discrete-time sinusoidal(or complex exponential) signals it is quite obvious that we are dealing with different (butrelated) frequency variables. To keep these variables in perspective and to avoid confusionit is important to be careful and consistent in using units to express them.



Senoides discretasxŒn D cos.!n/, para ! D =2 e ! D .


–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

–10 –5 0 5 10 15 20 25 30–1

–0.5

0

0.5

1

Time (n)

x[n]

Figure 4.6 The signal x[n] = cos ω0n for different values of ω0. The rate of oscillationincreases as ω0 increases from 0 to π and decreases again as ω0 increases from π to 2π .

Frequency variables and units After studying continuous- and discrete-time sinusoidal(or complex exponential) signals it is quite obvious that we are dealing with different (butrelated) frequency variables. To keep these variables in perspective and to avoid confusionit is important to be careful and consistent in using units to express them.



Periodicidade no tempo

Seja uma senoide discreta xŒn D A cos .!0nC /

I Periodicidade no tempo: xŒn D xŒnCN

xŒnCN D A cos .!0nC !0N C / D A cos .!0nC / D xŒn

dado que !0N D 2k.

Uma senoide discreta so e periodica se sua frequencia f0 D k=N eracional ou !0 D 2k=N e racional em 2 . Caso contrario, nao harepeticao (a cada ciclo valores sao diferentes).

ExemplosI Periodicas: ! D , ! D =3, ! D =30

I Nao periodicas: ! D 1, ! D 1=2, ! D 2:3



Periodicidade na frequencia

Seja uma senoide discreta xŒn D A cos .!0nC /

I Periodicidade na frequencia

A cos Œ.!0 C k2/ nC D A cos .!0nC kn2 C /

D A cos .!0nC /

Uma senoide discreta e periodica na frequencia, com perıodofundamental de 2 .

Ou seja, se aumentarmos a frequencia em k2 teremos a mesmasenoide! Por isso a frequencia digital tem limitacao < ! < .



Senoides discretas como autofuncoes

Sabe-se que exponenciais complexas ń sao autofuncoes de sistemasLTI, pois

xŒn D ń H7! yŒn D H.´/ń

Se fizermos ´ D e j! , temos

xŒn D

e j!nD e j!n H

7! yŒn D H. e j!/ e j!n

Autofuncoes de sistemas LTI

Exponenciais complexas do tipo e j!n tambem sao autofuncoes desistemas LTI. Ou seja, o sistema LTI modifica apenas sua amplitude efase pela quantidade H. e j!/.



Decomposicao e filtragem

DecomposicaoPortanto, e interessante decompor um sinal qualquer em funcoese j!n, para diferentes !, pois essas sao funcoes especiais.

FiltragemA escolha de um sistema com resposta ao impulso hŒn tal queH. e j!/ 1 para uma faixa de frequencias e H. e j!/ 0 para outra,e a base para o projeto de filtros.

H. e j!1n/ 1

H. e j!2n/ 1

H. e j!3n/ 0

H. e j!4n/ 0:::

e j!1n

e j!2n

e j!3n

e j!4n

:::

e j!1n

e j!2n

0

0:::



Ortogonalidade de senoides discretasSejam duas senoides complexas e periodicas

x1Œn D e j!1nD e j 2

Nk1n e x2Œn D e j!2n

D e j 2N

k2n

com produto interno definidoXnD<N >

x1Œnx2 Œn D

XnD<N >

D e j 2N

.k1k2/nD

(N; k1 D k2

0; k1 ¤ k2

OrtogonalidadeSenoides complexas discretas de diferentes frequencias saoortogonais, ou seja, seu produto interno e zero.

Decomposicao em base ortogonalSenoides complexas discretas formam um base ortogonal e pode-sedecompor um sinal qualquer nessa base fazendo produtos internos.


Representacao Fourier de Sinais Transformada de Fourier no tempo discreto

Transformada de Fourier no tempo discreto

A DTFT (discrete-time Fourier transform - transformada de Fourierno tempo discreto) tem grande importancia em PDS.

xŒn D1

2

Z2

X. e j!/ e j!n d!DTFT ! X. e j!/ D

1XnD1

xŒn e j!n

CaracterısticasI Sinal discreto no tempo (em geral, real).I Espetro complexo contınuo na frequencia.

Por que um sinal discreto tem espectro contınuo?As amostras no tempo sao restritas a inteiros (xŒn; n 2 Z), porem asfrequencias ! admitem qualquer valor!



Periodicidade na frequenciaPela definicao

X. e j.!C2// D

1XnD1

xŒn e j.!C2/nD

1XnD1

xŒn e j!n: 1

e j2n

D

1XnD1

xŒn e j!nD X. e j!/

Repeticao de espectroA DTFT e periodica e o espectro completo e formado por repeticoesnos multiplos de 2 .

182 Fourier representation of signals

0

0

x n[ ] y[n]

Figure 4.30 The frequency shifting property of the DTFT for ωc > 0. For ωc < 0, thespectrum is shifted to the left (at lower frequencies).

0

0

x[n] y[n]

−

Figure 4.31 The modulation property of the DTFT using a real sinusoidal carrier.

or equivalentlyωm < ωc < π − ωm. (4.144)

Relation (4.142) follows from (4.140) using Euler’s identity 2 cos ωcn = ejωcn + e− jωcn

and the frequency shifting theorem.



ExemploCalcular a DTFT de xŒn D A; 0 n L 1

179 4.5 Properties of the discrete-time Fourier transform

Example 4.14 Rectangular pulse sequenceConsider the sequence

x[n] =!

A, 0 ≤ n ≤ L − 1

0. otherwise(4.133)

which is illustrated in Figure 4.29(a). Since x[n] is absolutely summable, its Fouriertransform exists. Using the geometric summation formula, we obtain

–10 –5 0 5 10 15 200

0.5

1

n

x[n]

(a)

−π −π/2 0 π/2 π0

5

10

ω2π 2π

LL

L

−

(b)

−π −π/2 0 π/2 π−5π

0

5π

ω(c)

−π −π/2 0 π/2 π−π

0

π

ω(d)

Figure 4.29 The rectangular pulse sequence and its DTFT X(ejω). (a) Sequence x[n], (b)magnitude |X(ejω)| from (4.135), and (c) phase ∠X(ejω) from (4.136). The plot in (d) showsthe phase function computed using MATLAB function angle(X).

Solucao

X. e j!/ D

1XnD1

xŒn e j!nD

L1XnD0

A e j!nD A

1 e j!L

1 e j!

que pode ser reduzido a

X. e j!/ D A e j!.L1/=2 sin.!L=2/

sin.!=2/



Exemplo cont.A magnitude e fase das componentes sao dadas porˇ

X. e j!/ˇD jAj

ˇsin.!L=2/

sin.!=2/

ˇ†X. e j!/ D

!

2.L 1/


Example 4.14 Rectangular pulse sequenceConsider the sequence

x[n] =!

A, 0 ≤ n ≤ L − 1

0. otherwise(4.133)

which is illustrated in Figure 4.29(a). Since x[n] is absolutely summable, its Fouriertransform exists. Using the geometric summation formula, we obtain

–10 –5 0 5 10 15 200

0.5

1

n

x[n]

(a)

−π −π/2 0 π/2 π0

5

10

ω2π 2π

LL

L

−

(b)

−π −π/2 0 π/2 π−5π

0

5π

ω(c)

−π −π/2 0 π/2 π−π

0

π

ω(d)

Figure 4.29 The rectangular pulse sequence and its DTFT X(ejω). (a) Sequence x[n], (b)magnitude |X(ejω)| from (4.135), and (c) phase ∠X(ejω) from (4.136). The plot in (d) showsthe phase function computed using MATLAB function angle(X).



ExemploSeja um sinal com espectro X. e j!/ D 1;!c < ! < !c . Calcular aDTFT inversa.


(a)

πωc

ωπ

πω

c

c

−

–20 –15 –10 –5 0 5 10 15 20–0.1

0

0.1

0.2

0.3

nx[n]

−π −π/2 0 π/2 π0

0.5

1

ωωc−ωc

(b)

Figure 4.28 The “sinc” sequence (a) and its Fourier transform (b).

The sequence x[n] can be obtained using the synthesis formula (4.88)

x[n] = 12π

! ωc

−ωc

ejωndω = 12π jn

ejωn"""ωc

−ωc= sin(ωcn)

πn. n = 0 (4.130)

For n = 0 we obtain x[0] = 0/0, which is undefined. Since n is integer, we cannot takethe limit n → 0 to determine x[0] using l’Hôpital’s rule. However, if we use the definitiondirectly, we obtain

x[0] = 12π

! ωc

−ωc

dω = ωc

π. (4.131)

For convenience, we usually combine (4.130) and (4.131) into the single equation

x[n] = ωc

π

sin(ωcn)

ωcn= sin(ωcn)

πn, −∞ < n < ∞ (4.132)

with the understanding that at n = 0, x[n] = ωc/π . As we explained in Section 4.3.2, theDTFT of x[n] exists in the mean square sense. The sequence x[n] and its DTFT X(ejω) areshown in Figure 4.28. !

Solucao

xŒn D1

2

Z2

X. e j!/ e j!n d! D1

2

Z !c

!c

e j!n d!

D1

2 jne j!n

ˇ!c

!c

Dsin.!cn/

n



Exemplo cont.

O sinal xŒn Dsin.!cn/

ne conhecido como “sinc”.178 Fourier representation of signals

(a)

πωc

ωπ

πω

c

c

−

–20 –15 –10 –5 0 5 10 15 20–0.1

0

0.1

0.2

0.3

n

x[n]

−π −π/2 0 π/2 π0

0.5

1

ωωc−ωc

(b)

Figure 4.28 The “sinc” sequence (a) and its Fourier transform (b).

The sequence x[n] can be obtained using the synthesis formula (4.88)

x[n] = 12π

! ωc

−ωc

ejωndω = 12π jn

ejωn"""ωc

−ωc= sin(ωcn)

πn. n = 0 (4.130)

For n = 0 we obtain x[0] = 0/0, which is undefined. Since n is integer, we cannot takethe limit n → 0 to determine x[0] using l’Hôpital’s rule. However, if we use the definitiondirectly, we obtain

x[0] = 12π

! ωc

−ωc

dω = ωc

π. (4.131)

For convenience, we usually combine (4.130) and (4.131) into the single equation

x[n] = ωc

π

sin(ωcn)

ωcn= sin(ωcn)

πn, −∞ < n < ∞ (4.132)

with the understanding that at n = 0, x[n] = ωc/π . As we explained in Section 4.3.2, theDTFT of x[n] exists in the mean square sense. The sequence x[n] and its DTFT X(ejω) areshown in Figure 4.28. !



Pares de transformada

Os pares abaixo sao validos de < ! < . Devido a propriedadede repeticao, deve-se considerar suas repeticoes nos multiplos de 2 .

xŒn X. e j!/

ıŒn 1

ıŒn n0 e j!n0

e j!0n 2ı.! !0/

cos.!0n/ Œı.! !0/C ı.! C !0/

sin.!0n/ j Œı.! !0/ ı.! C !0/

anuŒn; jaj < 11

1 a e j!


Representacao Fourier de Sinais Propriedades da DTFT

Relacao da DTFT com a transf. ´

Transf. ´)DTFT

X.´/j´D e j! D

1XnD1


A magnitude da DTFT e obtida nos valores de ´ sobre o cırculounitario, j´j D 1.

DTFT)Transf. ´

X.´/ D X.r e j!/

Z fxŒng D F frnxŒng

A transf. ´ de um sinal xŒn equivale a DTFT do mesmo sinal“atenuado” por rn. Logo, a transf. ´ pode existir para sinais que naotenham DTFT.



Relacao da DTFT com a transf. ´ cont.

Funcoes e j! tem modulo 1ˇe j! ˇD jcos.!/C j sin.!/j D

qcos2.!/C sin2.!/ D 1

Cırculo unitarioFazendo X.´/j´D e j! D X. e j!/,avalia-se a funcao X.´/ apenaspara valores de ´ com modulo 1,ou seja, valores no cırculounitario.

92 The z-transform

0

z-plane

ωr cos ω

r sin ω

(a)

z-plane

0 1

Unit circle

ω

(b)

r

Figure 3.1 (a) A point z = rejω in the complex plane can be specified by the distance r fromthe origin and the angle ω with the positive real axis (polar coordinates) or the rectangularcoordinates r cos(ω) and r sin(ω). (b) The unit circle, |z| = 1, in the complex plane.

Figure 3.2 The magnitude |X(z)| of the z-transform represents a surface in the z-plane. Thereare two zeros at z1 = 0, z2 = 1 and two poles at p1,2 = 0.9e± jπ/4.

Example 3.1 Unit sample sequenceThe z-transform of the unit sample sequence is given by

X(z) =∞!

n=−∞δ[n]z−n = z0 = 1. ROC: All z (3.10)

!



Relacao da DTFT com a transf. ´ cont.

X.´/ D´2 1

´2 1:27´C 0:81: zeros ´1;2 D ˙1; polos p1;2 D 0:9 e˙ j=4


some operations on a signal or between two signals result in different operations betweentheir DTFTs.

4.5.1 Relationship to the z -transform and periodicity

The z-transform of a sequence x[n] was defined in Section 3.2 by

X(z) =∞!

n=−∞x[n]z−n. (4.101)

If the ROC of X(z) includes the unit circle, defined by z = ejω or equivalently |z| = 1, weobtain

X(z)|z=ejω =∞!

n=−∞x[n]e− jωn = X(ejω), (4.102)

that is, the z-transform reduces to the Fourier transform. The magnitude of DTFT isobtained by intersecting the surface |H(z)| with a vertical cylinder of radius one, centeredat z = 0. This is illustrated in Figure 4.26, which provides a clear demonstration of theperiodicity of DTFT. The radiant frequency ω is measured with respect to the positive realaxis and the unit circle is mapped on the linear frequency axis as shown in Figure 4.26.Multiple rotations around the unit circle create an inherent periodicity, with period 2π

ω

ω=0

ω=2pω= |H(e

jω)|

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real Axis

Imag

inar

y A

xis

p

Figure 4.26 The relationship between the z-transform and the DTFT for a sequence with twocomplex-conjugate poles at z = 0.9ej±π/4 and two zeros at z = ±1.

X. e j!/ D X.´/j´D e j! De j2! 1

e j2! 1:27 e j! C 0:81


some operations on a signal or between two signals result in different operations betweentheir DTFTs.

4.5.1 Relationship to the z -transform and periodicity

The z-transform of a sequence x[n] was defined in Section 3.2 by

X(z) =∞!

n=−∞x[n]z−n. (4.101)

If the ROC of X(z) includes the unit circle, defined by z = ejω or equivalently |z| = 1, weobtain

X(z)|z=ejω =∞!

n=−∞x[n]e− jωn = X(ejω), (4.102)

that is, the z-transform reduces to the Fourier transform. The magnitude of DTFT isobtained by intersecting the surface |H(z)| with a vertical cylinder of radius one, centeredat z = 0. This is illustrated in Figure 4.26, which provides a clear demonstration of theperiodicity of DTFT. The radiant frequency ω is measured with respect to the positive realaxis and the unit circle is mapped on the linear frequency axis as shown in Figure 4.26.Multiple rotations around the unit circle create an inherent periodicity, with period 2π

ω

ω=0

ω=2pω= |H(e

jω)|

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

Real Axis

Imag

inar

y A

xis

p

Figure 4.26 The relationship between the z-transform and the DTFT for a sequence with twocomplex-conjugate poles at z = 0.9ej±π/4 and two zeros at z = ±1.



Simetrias na frequencia

Supondo um sinal real xŒn e sua DTFT X. e j!/ D

1XnD1

xŒn e j!n

Aplicando-se o conjugado

X. e j!/ D

1X

nD1

xŒn e j!n

!

D

1XnD1


Supondo X. e j!/ D XR. e j!/C jXI . e j!/

X. e j!/ D XR. e j!/ jXI . e j!/ D XR. e j!/C jXI . e j!/ D X. e j!/

Parte real e imaginaria

XR. e j!/ D XR. e j!/

XI . e j!/ D XI . e j!/

A magnitude e faseˇX. e j!/

ˇDˇX. e j!/

ˇ†X. e j!/ D †X. e j!/



Simetrias no tempo

Supondo um sinal real e par xŒn D xŒn.

X. e j!/ D

1XnD1

xŒn e j!nD

1XmD1

xŒm e j!mD X. e j!/

Se xŒn e real e par, X. e j!/ e real. Portanto X. e j!/ tambem e par.

Supondo um sinal real e ımpar xŒn D xŒn.

X. e j!/ D

1XnD1

xŒn e j!nD

1XmD1

xŒm e j!mD X. e j!/

Se xŒn e real e ımpar, X. e j!/ e imaginario. Portanto X. e j!/

tambem e ımpar.



Exemplo

O sinal xŒn D cos.!0n/, que e real e par tem DTFT

X. e j!/ D Œı.! C !0/C ı.! !0/

onde

XR. e j!/ D Œı.! C !0/C ı.! !0/

XI . e j!/ D 0ˇX. e j!/

ˇD Œı.! C !0/C ı.! !0/

†X. e j!/ D 0

O sinal no tempo e real e par. A DTFT e real e par.



Exemplo

O sinal xŒn D sin.!0n/, que e real e ımpar tem DTFT

X. e j!/ D j Œı.! C !0/ ı.! !0/

onde

XR. e j!/ D 0

XI . e j!/ D Œı.! C !0/ ı.! !0/ˇX. e j!/

ˇD Œı.! C !0/C ı.! !0/

†X. e j!/ D

(; ! D !0

; ! D !0

O sinal no tempo e real e ımpar. A DTFT e imaginaria e ımpar.



Proprieadades da DTFT


Starting with the right hand side of (4.153) we have

12π

!

2πX1(ejω)X∗

2(ejω)dω = 12π

!

2π

" ∞#

n=−∞x1[n]e− jωn

$

X∗2(ejω)dω

=∞#

n=−∞x1[n]

%1

2π

!

2πX∗

2(ejω)e− jωndω

&

=∞#

n=−∞x1[n]

%1

2π

!

2πX∗

2(e− jω)ejωndω

&

=∞#

n=−∞x1[n]x∗

2[n]. (using (4.148))

For x1[n] = x2[n] = x[n], we obtain Parseval’s relation (4.94).

Summary of DTFT properties For easy reference, the operational properties of theDTFT are summarized in Table 4.4.

Table 4.4 Operational properties of the DTFT.

Property Sequence Transform

x[n] Fx[n]

1. Linearity a1x1[n] + a2x2[n] a1X1(ejω) + a2X2(ejω)

2. Time shifting x[n − k] e− jkωX(ejω)

3. Frequency shifting ejω0nx[n] X[ej(ω−ω0)]4. Modulation x[n] cos ω0n 1

2 X[ej(ω+ω0)] + 12 X[ej(ω−ω0)]

5. Folding x[−n] X(e− jω)

6. Conjugation x∗[n] X∗(e− jω)

7. Differentiation nx[n] − jdX(ejω)

dω8. Convolution x[n] ∗ h[n] X(ejω)H(ejω)

9. Windowing x[n]w[n] 12π

!

2πX(ejθ )W

'ej(ω−θ)(dθ

10. Parseval’s theorem∞#

n=−∞x1[n]x∗

2[n] = 12π

!

2πX1(ejω)X∗

2 (ejω)dω

11. Parseval’s relation∞#

n=−∞|x[n]|2 = 1

2π

!

2π|X(ejω)|2dω



ModulacaoNa modulacao de um sinal xŒn com DTFT X. e j!/ por uma portadorasenoidal cŒn D cos.!cn/, tem-se

xŒn cos.!cn/F !

1

2X. e j.!C!c//C

1

2X. e j.!!c//


0

0

x n[ ] y[n]

Figure 4.30 The frequency shifting property of the DTFT for ωc > 0. For ωc < 0, thespectrum is shifted to the left (at lower frequencies).

0

0

x[n] y[n]

−

Figure 4.31 The modulation property of the DTFT using a real sinusoidal carrier.

or equivalentlyωm < ωc < π − ωm. (4.144)

Relation (4.142) follows from (4.140) using Euler’s identity 2 cos ωcn = ejωcn + e− jωcn

and the frequency shifting theorem.


Representacao Fourier de Sinais Conclusao

Sumario

O que foi visto hoje?

I As representacoes de sinais no domınio do tempo e domınio dafrequencia contem a mesma informacao, porem de formasdiferentes.

I A representacao no domınio da frequencia (espectro) consistedas amplitudes e fases de componentes senoidais usadas para“construir” o sinal.

I A DTFT (discrete-time Fourier transform) e aplicada a sinais notempo discreto. No entanto, e uma funcao contınua, poisqualquer valor de frequencia e admitida.

I O projeto de filtros consiste em determinar hŒn tal que H. e j!/

seja proximo de 0 para frequencias que se deseja eliminar dosinal de entrada, e proximo de 1 para frequencia que se desejamanter.



Exercıcios

Questoes: 4.14, 4.20, 4.21Problemas: 4.33 (errata: X. e j!/ D 1=.1C 0:8 e j!/)



Respostas

Problemas:

I 4.33a: . e2 j.!=2//=.1C 0:8 e j.!=2//

I 4.33b:.1=2/=.1C 0:8 e j.!0:4//C .1=2/=.1C 0:8 e j.!C0:4//

I 4.33c: 1=.1:64C 1:6 cos.!//

I 4.33d: 1=.1C 0:8 e j!=2/

I 4.33e: 1=.1 0:82 e j2!/


4-fourier

Documents