Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE-TIME SIGNALS 6.1 Introduction 6.2 Fourier Transform Revisited

c 2005- by Andreas Antoniou Copyright  Victoria, BC, Canada Email: [email protected] March 7, 2008

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction t Digital filters are often used to process discrete-time

signals that have been generated by sampling continuous-time signals.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction t Digital filters are often used to process discrete-time

signals that have been generated by sampling continuous-time signals. t Frequently digital filters are designed indirectly through the

use of analog filters.

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Slide # 3

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction t Digital filters are often used to process discrete-time

signals that have been generated by sampling continuous-time signals. t Frequently digital filters are designed indirectly through the

use of analog filters. t In order to understand the basis of these techniques, the

spectral relationships among continuous-time, impulse-modulated, and discrete-time signals must be understood.

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Slide # 4

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction t Digital filters are often used to process discrete-time

signals that have been generated by sampling continuous-time signals. t Frequently digital filters are designed indirectly through the

use of analog filters. t In order to understand the basis of these techniques, the

spectral relationships among continuous-time, impulse-modulated, and discrete-time signals must be understood. t These relationships are derived by using the Fourier

transform, the Fourier series, the z transform, and Poisson’s summation formula.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction Cont’d t Impulse-modulated signals comprise sequences of

continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson’s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction Cont’d t Impulse-modulated signals comprise sequences of

continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson’s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. t This presentation begins with a review of the Fourier

transform.

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Slide # 7

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction Cont’d t Impulse-modulated signals comprise sequences of

continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson’s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. t This presentation begins with a review of the Fourier

transform. t Then impulse functions are defined and their properties

are examined.

Frame # 3

Slide # 8

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Introduction Cont’d t Impulse-modulated signals comprise sequences of

continuous-time impulse functions and to understand their significance, the properties of impulse functions must be understood. On the other hand, Poisson’s summation formula is based on a relationship between the Fourier series and the Fourier transform of periodic signals. t This presentation begins with a review of the Fourier

transform. t Then impulse functions are defined and their properties

are examined. t Subsequently, the application of the Fourier transform to

impulse functions and periodic signals is investigated. Frame # 3

Slide # 9

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Review of Fourier Transform t The Fourier transform of a continuous-time signal x (t) is defined as  ∞ X (jω) =

Frame # 4 Slide # 10

A. Antoniou

−∞

x (t)e−jωt dt

Digital Signal Processing – Secs. 6.1, 6.2

(A)

Review of Fourier Transform t The Fourier transform of a continuous-time signal x (t) is defined as  ∞ X (jω) =

−∞

x (t)e−jωt dt

t In general, X (jω) is complex and can be written as X (jω) = A(ω)ejφ(ω) where A(ω) = |X (jω)|

Frame # 4 Slide # 11

A. Antoniou

and

φ(ω) = arg X (jω)

Digital Signal Processing – Secs. 6.1, 6.2

(A)

Review of Fourier Transform t The Fourier transform of a continuous-time signal x (t) is defined as  ∞ X (jω) =

−∞

x (t)e−jωt dt

(A)

t In general, X (jω) is complex and can be written as X (jω) = A(ω)ejφ(ω) where A(ω) = |X (jω)|

and

φ(ω) = arg X (jω)

t Functions A(ω) and φ(ω) are the amplitude spectrum and phase spectrum of the signal, respectively.

Frame # 4 Slide # 12

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Review of Fourier Transform t The Fourier transform of a continuous-time signal x (t) is defined as  ∞ X (jω) =

−∞

x (t)e−jωt dt

(A)

t In general, X (jω) is complex and can be written as X (jω) = A(ω)ejφ(ω) where A(ω) = |X (jω)|

and

φ(ω) = arg X (jω)

t Functions A(ω) and φ(ω) are the amplitude spectrum and phase spectrum of the signal, respectively. t Together, the amplitude and phase spectrums or constitute the frequency spectrum.

Frame # 4 Slide # 13

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Review of Fourier Transform Cont’d

···

 X (jω) =

∞

−∞

x (t)e−jωt dt

(A)

t Function x (t) is the inverse Fourier transform of X (jω) and is given by  ∞ 1 x (t) = X (jω)ejωt dω (B) 2π −∞

Frame # 5 Slide # 14

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Review of Fourier Transform Cont’d

···

 X (jω) =

∞

−∞

x (t)e−jωt dt

(A)

t Function x (t) is the inverse Fourier transform of X (jω) and is given by  ∞ 1 x (t) = X (jω)ejωt dω (B) 2π −∞ t Eqs. (A) and (B) can be written in operator format as X (jω) = F x (t) and

x (t) = F −1 X (jω)

respectively.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Review of Fourier Transform Cont’d

···

 X (jω) =

∞

−∞

x (t)e−jωt dt

(A)

t Function x (t) is the inverse Fourier transform of X (jω) and is given by  ∞ 1 x (t) = X (jω)ejωt dω (B) 2π −∞ t Eqs. (A) and (B) can be written in operator format as X (jω) = F x (t) and

x (t) = F −1 X (jω)

respectively. t An alternative shorthand notation is x (t)↔X (jω)

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Convergence Theorem t The convergence theorem of the Fourier transform states that if  lim

T

T →∞ −T

|x (t)| dt < ∞

then the Fourier transform of x (t), X (jω), exists and its inverse can be obtained by using the equation  ∞ 1 X (jω)ejωt dω x (t) = 2π −∞

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Convergence Theorem t The convergence theorem of the Fourier transform states that if  lim

T

T →∞ −T

|x (t)| dt < ∞

then the Fourier transform of x (t), X (jω), exists and its inverse can be obtained by using the equation  ∞ 1 X (jω)ejωt dω x (t) = 2π −∞ t Many signals that are of considerable interest in practice violate the above condition, for example, impulse functions, impulse-modulated signals, and periodic signals.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Convergence Theorem t The convergence theorem of the Fourier transform states that if  lim

T

T →∞ −T

|x (t)| dt < ∞

then the Fourier transform of x (t), X (jω), exists and its inverse can be obtained by using the equation  ∞ 1 X (jω)ejωt dω x (t) = 2π −∞ t Many signals that are of considerable interest in practice violate the above condition, for example, impulse functions, impulse-modulated signals, and periodic signals. t However, convergence problems can be circumvented by paying particular attention to the definition of impulse functions.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions t The unit impulse function has been defined in the past as

 δ(t) = lim p¯ τ (t) = lim τ →0

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A. Antoniou

τ →0

1 τ

0

for |t| ≤ τ/2 otherwise

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions t The unit impulse function has been defined in the past as

 δ(t) = lim p¯ τ (t) = lim τ →0

τ →0

1 τ

0

for |t| ≤ τ/2 otherwise

t Obviously, this is an infinitesimally thin, infinitely tall pulse

whose area is equal to unity for any finite value of τ .

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d Pulse function p¯ τ (t) for three values of τ : 25

τ = 0.50 τ = 0.10 τ = 0.05

20

p-τ(t)

15

10

5

0 −1

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0 (a)

A. Antoniou

t

1

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem t The Fourier transform of the unit impulse function as

defined in the past should be given by the integral  ∞ x(t)e−jωt dt X (jω) = −∞ ∞ lim [p¯ τ (t)]e−jωt dt = −∞ τ →0

where

 p¯ τ (t) =

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A. Antoniou

1 τ

0

for |t| ≤ τ/2 otherwise

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem t The Fourier transform of the unit impulse function as

defined in the past should be given by the integral  ∞ x(t)e−jωt dt X (jω) = −∞ ∞ lim [p¯ τ (t)]e−jωt dt = −∞ τ →0

where

 p¯ τ (t) =

1 τ

0

for |t| ≤ τ/2 otherwise

t If we now attempt to evaluate the function p¯ τ (t)e−jωt at

τ = 0, we find that it becomes infinite and, therefore, the above integral cannot be evaluated.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t We can write

τ →0

Frame # 10 Slide # 25



∞

lim [p¯ τ (t)]e−jωt dt   1 dt lim ≈ −τ/2 τ →0 τ

F lim p¯ τ (t) =

−∞ τ →0  τ/2

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t We can write

τ →0



∞

lim [p¯ τ (t)]e−jωt dt   1 dt lim ≈ −τ/2 τ →0 τ

F lim p¯ τ (t) =

−∞ τ →0  τ/2

t Since the area of the pulse function p¯ τ (t) is unity for any

finite value of τ , we might be tempted to assume that the area is equal to unity even for τ = 0, i.e., F lim p¯ τ (t) = 1 τ →0

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The Fourier transform of p¯ τ (t) for a finite τ is given by

F p¯ τ (t) =

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1 2 sin ωτ/2 Fpτ (t) = τ ωτ

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The Fourier transform of p¯ τ (t) for a finite τ is given by

F p¯ τ (t) =

1 2 sin ωτ/2 Fpτ (t) = τ ωτ

t Obviously, this is well defined and, interestingly, it has the

limit

lim F p¯ τ (t) = 1

τ →0

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The Fourier transform of p¯ τ (t) for a finite τ is given by

F p¯ τ (t) =

1 2 sin ωτ/2 Fpτ (t) = τ ωτ

t Obviously, this is well defined and, interestingly, it has the

limit

lim F p¯ τ (t) = 1

τ →0

t So far so good!

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t If we now attempt to find the inverse Fourier transform of 1,

we run into certain mathematical difficulties.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t If we now attempt to find the inverse Fourier transform of 1,

we run into certain mathematical difficulties. t From the definition of the inverse Fourier transform, we

have F

Frame # 12 Slide # 31

−1

 ∞ 1 1= ejωt d ω 2π −∞  ∞   ∞ 1 cos ωt d ω + j sin ωt d ω = 2π −∞ −∞

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t If we now attempt to find the inverse Fourier transform of 1,

we run into certain mathematical difficulties. t From the definition of the inverse Fourier transform, we

have F

−1

 ∞ 1 1= ejωt d ω 2π −∞  ∞   ∞ 1 cos ωt d ω + j sin ωt d ω = 2π −∞ −∞

t However, mathematicians will tell us that these integrals do

not converge or do not exist!

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t Summarizing, by cheating a little bit we can get a more or

less meaningful Fourier transform for the unit impulse function.

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Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t Summarizing, by cheating a little bit we can get a more or

less meaningful Fourier transform for the unit impulse function. t Unfortunately, it is impossible to recover the impulse

function from its Fourier transform by applying the inverse Fourier transform.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The impulse-function problem can be circumvented in two

ways, a practical and a theoretical one:

Frame # 14 Slide # 35

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The impulse-function problem can be circumvented in two

ways, a practical and a theoretical one:

– The practical approach is easy to understand and apply but it lacks rigor.

Frame # 14 Slide # 36

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Mathematical Problem Cont’d t The impulse-function problem can be circumvented in two

ways, a practical and a theoretical one:

– The practical approach is easy to understand and apply but it lacks rigor. – The theoretical approach is rigorous but it is rather abstract and more difficult to understand or apply in practical situations.

Frame # 14 Slide # 37

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions t In the practical approach to impulse functions, a function γ (t) is said to be a unit impulse function if, for any continuous function x (t) over the range − < t < , the following relation is satisfied:  ∞ γ (t)x (t) dt  x (0) (C) −∞

Frame # 15 Slide # 38

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions t In the practical approach to impulse functions, a function γ (t) is said to be a unit impulse function if, for any continuous function x (t) over the range − < t < , the following relation is satisfied:  ∞ γ (t)x (t) dt  x (0) (C) −∞

t The special symbol  is used to signify that the two sides can be made to approach one another to any desired degree of precision but cannot be made exactly equal.

Frame # 15 Slide # 39

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions t In the practical approach to impulse functions, a function γ (t) is said to be a unit impulse function if, for any continuous function x (t) over the range − < t < , the following relation is satisfied:  ∞ γ (t)x (t) dt  x (0) (C) −∞

t The special symbol  is used to signify that the two sides can be made to approach one another to any desired degree of precision but cannot be made exactly equal. t Now consider the pulse function  lim p¯ τ (t) = p¯  (t) =

τ →

1 

0

for |t| ≤ /2 otherwise

where  is a small but finite constant.

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions Cont’d

···



∞ −∞

t If we let

γ (t)x (t) dt  x (0)

γ (t) = lim p¯ τ (t) τ →

in the left-hand side of Eq. (C), we obtain  /2  ∞ 1 x (t) dt lim [p¯ τ (t)]x (t) dt = τ → −∞ −/2   /2 1  x (0) dt  x (0)  −/2

Frame # 16 Slide # 41

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

(C)

Practical Approach to Impulse Functions Cont’d

···



∞ −∞

t If we let

γ (t)x (t) dt  x (0)

(C)

γ (t) = lim p¯ τ (t) τ →

in the left-hand side of Eq. (C), we obtain  /2  ∞ 1 x (t) dt lim [p¯ τ (t)]x (t) dt = τ → −∞ −/2   /2 1  x (0) dt  x (0)  −/2 t Thus we conclude that the very thin pulse function limτ → p¯ τ (t) behaves as an impulse function and, therefore, we can write δ(t) = lim p¯ τ (t) τ →

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A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions Cont’d

···

δ(t) = lim p¯ τ (t) τ →

t Now if we apply the Fourier transform to the impulse

function as defined, we get lim p¯ τ (t) ↔ lim

τ →

Frame # 17 Slide # 43

A. Antoniou

τ →

2 sin ωτ/2 ωτ

Digital Signal Processing – Secs. 6.1, 6.2

Practical Approach to Impulse Functions Cont’d

···

2 sin ωτ/2 ωτ t As τ is reduced, the pulse function at the left tends to become thinner and taller whereas the sinc function at the right tends to be flattened out. lim p¯ τ (t) ↔ lim

τ →

τ →

25

1.2

 = 0.50  = 0.10  = 0.05

δω∞

1.0

20 sin (ω/2)/(ω/2)

0.8

p-(t)

15

10

0.6 0.4 0.2 0

−

5 −0.2 0 −1

Frame # 18 Slide # 44

0 (a)

t

1

A. Antoniou

−0.4 −40

−30

ω∞ 2

−20

 = 0.50  = 0.10  = 0.05 −10

0 (b)

10

ω∞ 2

20 ω 30

Digital Signal Processing – Secs. 6.1, 6.2

40

Impulse Functions Cont’d t For some small but finite , the sinc function will be equal

to unity to within an error δω∞ over some frequency range −ω∞ /2 < ω < ω∞ /2. 1.2 δω∞

1.0

sin (ω/2)/(ω/2)

0.8 0.6 0.4 0.2 0

−

−0.2 −0.4 −40

Frame # 19 Slide # 45

−30

ω∞ 2

−20

 = 0.50  = 0.10  = 0.05 −10

A. Antoniou

0 (b)

10

ω∞ 2

20 ω 30

40

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d t Therefore, we can write

δ(t) = lim p¯ τ (t) ↔ lim τ →

τ →

2 sin ωτ/2 = i(ω) ωτ

where i(ω) may be referred to as a frequency-domain unity function.

Frame # 20 Slide # 46

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d t Summarizing, – the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and – the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function,

Frame # 21 Slide # 47

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d t Summarizing, – the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and – the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function, i.e.,

Frame # 21 Slide # 48

δ(t) ↔ i(ω)

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d t Summarizing, – the Fourier transform of a time-domain impulse function is a frequency-domain unity function, and – the inverse Fourier transform of a frequency-domain unity function is a time-domain impulse function, i.e.,

δ(t) ↔ i(ω)

t Since i(ω)  1 for the frequency range of interest, we can write δ(t)  1 where the wavy double arrow  signifies that the relation is approximate with the understanding that it can be made as exact as desired by making  sufficiently small.

Frame # 21 Slide # 49

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Impulse Functions Cont’d The impulse and unity functions can be represented by the idealized graphs: i(ω)

δ(t)

1

1

ω

t

(a)

Frame # 22 Slide # 50

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Properties of Impulse Functions t Assuming that x(t) is a continuous function of t over the

range − < t < , the following relations apply:  ∞  ∞ (a) δ(t − τ )x(t) dt = δ(−t + τ )x(t) dt  x(τ ) −∞

−∞

(b)

δ(t − τ )x(t) = δ(−t + τ )x(t)  δ(t − τ )x(τ )

(c)

δ(t)x(t) = δ(−t)x(t)  δ(t)x(0)

(See textbook for proofs.)

Frame # 23 Slide # 51

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Frequency-Domain Impulse Functions t Given a transform pair δ(t) ↔ i(ω) where

δ(t) = lim p¯ τ (t) τ →

i(ω) = lim

τ →

2 sin ωτ/2  1 for |ω| < ω∞ ωτ

the corresponding transform pair i(t) ↔ 2πδ(ω) where

2 sin t/2 1 t δ(ω) = p¯  (ω) i(t) =

for |t| < t∞

can be generated by applying the symmetry theorem of the Fourier transform. Frame # 24 Slide # 52

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Frequency-Domain Impulse Functions Cont’d

···

i(t) ↔ 2π δ(ω)

t Function i(t) is a time-domain unity function whereas δ(ω) is a

frequency-domain unit impulse function.

Frame # 25 Slide # 53

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Frequency-Domain Impulse Functions Cont’d

···

i(t) ↔ 2π δ(ω)

t Function i(t) is a time-domain unity function whereas δ(ω) is a

frequency-domain unit impulse function.

t Since i(t)  1, we have

1  2π δ(ω)

Frame # 25 Slide # 54

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Frequency-Domain Impulse Functions Cont’d

···

i(t) ↔ 2π δ(ω) or 1  2π δ(ω)

This transform pair can be represented by the idealized graphs shown.

i(t)

δ(ω)

1

2π ω

t

(b)

Frame # 26 Slide # 55

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Properties of Frequency-Domain Impulse Functions t Assuming that X (jω) is a continuous function of ω over the

range − < ω < , the following relations apply:  ∞ (a) δ(ω − )X (jω) dt −∞  ∞ δ(−ω + )X (jω) dt  X (j ) = −∞

(b) δ(ω − )X (jω) = δ(−ω + )X (jω)  δ(t − )X (j ) (c)

δ(ω)X (jω) = δ(−ω)X (jω)  δ(t)X (0)

(See textbook for details.)

Frame # 27 Slide # 56

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Exponentials t Since

δ(t) ↔ i(ω) the application of the time-shifting theorem gives δ(t − t0 ) ↔ i(ω)e−jωt0 and since i(ω)  1, we get δ(t − t0 )  e−jωt0

Frame # 28 Slide # 57

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Exponentials t Since

δ(t) ↔ i(ω) the application of the time-shifting theorem gives δ(t − t0 ) ↔ i(ω)e−jωt0 and since i(ω)  1, we get δ(t − t0 )  e−jωt0 t Now applying the frequency-shifting theorem to the

frequency-domain impulse function, we obtain i(t)ejω0 t ↔ 2π δ(ω − ω0 ) and since i(t)  1, we get ejω0 t  2π δ(ω − ω0 )

Frame # 28 Slide # 58

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Sinusoidal Signals t We know that

and

Frame # 29 Slide # 59

i(t)ejω0 t ↔ 2π δ(ω − ω0 ) i(t)e−jω0 t ↔ 2π δ(ω + ω0 )

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Sinusoidal Signals t We know that

and

i(t)ejω0 t ↔ 2π δ(ω − ω0 ) i(t)e−jω0 t ↔ 2π δ(ω + ω0 )

t If we add the two equations, we get

i(t)(ejω0 t + e−jω0 t ) = 2i(t) · cos ω0 t ↔ 2π [δ(ω + ω0 ) + δ(ω − ω0 )] and since i(t)  1, we have cos ω0 t  π [δ(ω + ω0 ) + δ(ω − ω0 )]

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Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Sinusoidal Signals Cont’d

···

cos ω0 t  π [δ(ω + ω0 ) + δ(ω − ω0 )] X( jω)

x(t)

π

t

Frame # 30 Slide # 61

A. Antoniou

−ω0

ω0

Digital Signal Processing – Secs. 6.1, 6.2

ω

Fourier Transforms of Sinusoidal Signals Cont’d t As before,

and

Frame # 31 Slide # 62

i(t)ejω0 t ↔ 2π δ(ω − ω0 ) i(t)e−jω0 t ↔ 2π δ(ω + ω0 )

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Sinusoidal Signals Cont’d t As before,

and

i(t)ejω0 t ↔ 2π δ(ω − ω0 ) i(t)e−jω0 t ↔ 2π δ(ω + ω0 )

t If we subtract the top equation from the bottom one, we have

i(t)(e−jω0 t − ejω0 t ) = −2ji(t) · sin ω0 t ↔ 2π [δ(ω + ω0 ) − δ(ω − ω0 )] and since i(t)  1, we can write sin ω0 t  jπ [δ(ω + ω0 ) − δ(ω − ω0 )]

Frame # 31 Slide # 63

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Arbitrary Periodic Signals t An arbitrary periodic signal can be represented by the Fourier

series ˜ x(t) =

∞ 

Xk e−jk ω0 t

k =−∞

Frame # 32 Slide # 64

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Arbitrary Periodic Signals t An arbitrary periodic signal can be represented by the Fourier

series ˜ x(t) =

∞ 

Xk e−jk ω0 t

k =−∞

t Hence

˜ F x(t) =

∞ 

2π Xk Fe−jk ω0 t 

k =−∞

or

∞ 

2π Xk δ(ω − kω0 )

k =−∞

˜ x(t)  2π

∞ 

Xk δ(ω − kω0 )

k =−∞

Frame # 32 Slide # 65

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Periodic Signals Cont’d

···

˜ x(t)  2π

∞ 

Xk δ(ω − kω0 )

k =−∞

t Summarizing, the frequency spectrum of a periodic signal can

be represented by an infinite sequence of numbers Xk for −∞ < k < ∞, i.e., the Fourier-series coefficients as shown in Chap. 2

Frame # 33 Slide # 66

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Periodic Signals Cont’d

···

˜ x(t)  2π

∞ 

Xk δ(ω − kω0 )

k =−∞

t Summarizing, the frequency spectrum of a periodic signal can

be represented by an infinite sequence of numbers Xk for −∞ < k < ∞, i.e., the Fourier-series coefficients as shown in Chap. 2 or

Frame # 33 Slide # 67

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Fourier Transforms of Periodic Signals Cont’d

···

˜ x(t)  2π

∞ 

Xk δ(ω − kω0 )

k =−∞

t Summarizing, the frequency spectrum of a periodic signal can

be represented by an infinite sequence of numbers Xk for −∞ < k < ∞, i.e., the Fourier-series coefficients as shown in Chap. 2 or t by an infinite sequence of frequency-domain impulse functions

of strength 2π Xk for −∞ < k < ∞ as shown in the previous slide.

Frame # 33 Slide # 68

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Theoretical Approach to Impulse Functions t The Fourier transform pairs generated through the practical

approach are approximate since the pulse width  cannot be reduced to absolute zero.

Frame # 34 Slide # 69

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Theoretical Approach to Impulse Functions t The Fourier transform pairs generated through the practical

approach are approximate since the pulse width  cannot be reduced to absolute zero. t However, by defining impulse functions in terms of generalized

functions, analogous, but exact, Fourier transform pairs can be generated.

Frame # 34 Slide # 70

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Theoretical Approach to Impulse Functions t The Fourier transform pairs generated through the practical

approach are approximate since the pulse width  cannot be reduced to absolute zero. t However, by defining impulse functions in terms of generalized

functions, analogous, but exact, Fourier transform pairs can be generated. t Unfortunately, impulse functions so defined are rather

impractical and difficult to implement in a laboratory.

Frame # 34 Slide # 71

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Theoretical Approach to Impulse Functions t The Fourier transform pairs generated through the practical

approach are approximate since the pulse width  cannot be reduced to absolute zero. t However, by defining impulse functions in terms of generalized

functions, analogous, but exact, Fourier transform pairs can be generated. t Unfortunately, impulse functions so defined are rather

impractical and difficult to implement in a laboratory. t See textbook for more details and references on generalized

functions.

Frame # 34 Slide # 72

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

Summary of Fourier Transforms Derived

Frame # 35 Slide # 73

x(t)

X (jω)

δ(t)

1

1

2π δ(ω)

δ(t − t0 )

e−jωt0

ejω0 t

2π δ(ω − ω0 )

cos ω0 t

π [δ(ω + ω0 ) + δ(ω − ω0 )]

sin ω0 t

jπ [δ(ω + ω0 ) − δ(ω − ω0 )]

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2

This slide concludes the presentation. Thank you for your attention.

Frame # 36 Slide # 74

A. Antoniou

Digital Signal Processing – Secs. 6.1, 6.2