Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE ...2.pdf · Introduction Digital ﬁlters are often used to process discrete-time signals that have been generated by sampling

Chapter 6CONTINUOUS-TIME,

IMPULSE-MODULATED, ANDDISCRETE-TIME SIGNALS

6.1 Introduction6.2 Fourier Transform Revisited

Copyright c© 2005- by Andreas AntoniouVictoria, BC, Canada

Email: [email protected]

March 7, 2008

Frame # 1 Slide # 1 A. Antoniou Digital Signal Processing – Secs. 6.1, 6.2

Introduction

� Digital filters are often used to process discrete-timesignals that have been generated by samplingcontinuous-time signals.


Introduction


� Frequently digital filters are designed indirectly through theuse of analog filters.


Introduction



� In order to understand the basis of these techniques, thespectral relationships among continuous-time,impulse-modulated, and discrete-time signals must beunderstood.


Introduction



� In order to understand the basis of these techniques, thespectral relationships among continuous-time,impulse-modulated, and discrete-time signals must beunderstood.

� These relationships are derived by using the Fouriertransform, the Fourier series, the z transform, andPoisson’s summation formula.


Introduction Cont’d

� Impulse-modulated signals comprise sequences ofcontinuous-time impulse functions and to understand theirsignificance, the properties of impulse functions must beunderstood.

On the other hand, Poisson’s summation formula is basedon a relationship between the Fourier series and theFourier transform of periodic signals.





� This presentation begins with a review of the Fouriertransform.






� Then impulse functions are defined and their propertiesare examined.






� Then impulse functions are defined and their propertiesare examined.

� Subsequently, the application of the Fourier transform toimpulse functions and periodic signals is investigated.


Review of Fourier Transform

� The Fourier transform of a continuous-time signal x(t) is definedas

X (jω) =∫ ∞

−∞x(t)e−jωt dt (A)




X (jω) =∫ ∞


� In general, X (jω) is complex and can be written as

X (jω) = A(ω)ejφ(ω)

whereA(ω) = |X (jω)| and φ(ω) = arg X (jω)




X (jω) =∫ ∞





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




X (jω) =∫ ∞





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

� Together, the amplitude and phase spectrums or constitute thefrequency spectrum.


Review of Fourier Transform Cont’d

· · ·X (jω) =

∫ ∞


� Function x(t) is the inverse Fourier transform of X (jω) and isgiven by

x(t) = 12π

∫ ∞

−∞X (jω)ejωt dω (B)



· · ·X (jω) =

∫ ∞



x(t) = 12π

∫ ∞


� Eqs. (A) and (B) can be written in operator format as

X (jω) = Fx(t) and x(t) = F−1X (jω)

respectively.



· · ·X (jω) =

∫ ∞



x(t) = 12π

∫ ∞


� Eqs. (A) and (B) can be written in operator format as

X (jω) = Fx(t) and x(t) = F−1X (jω)

respectively.

� An alternative shorthand notation is

x(t)↔X (jω)


Convergence Theorem

� The convergence theorem of the Fourier transform states that if

limT→∞

∫ T

−T|x(t)| dt < ∞

then the Fourier transform of x(t), X (jω), exists and its inversecan be obtained by using the equation

x(t) = 12π

∫ ∞

−∞X (jω)ejωt dω


Convergence Theorem


limT→∞

∫ T

−T|x(t)| dt < ∞


x(t) = 12π

∫ ∞


� Many signals that are of considerable interest in practice violatethe above condition, for example, impulse functions,impulse-modulated signals, and periodic signals.


Convergence Theorem


limT→∞

∫ T

−T|x(t)| dt < ∞


x(t) = 12π

∫ ∞


� Many signals that are of considerable interest in practice violatethe above condition, for example, impulse functions,impulse-modulated signals, and periodic signals.

� However, convergence problems can be circumvented by payingparticular attention to the definition of impulse functions.


Impulse Functions

� The unit impulse function has been defined in the past as

δ(t) = limτ→0

p̄τ (t) = limτ→0

{1τ

for |t| ≤ τ/2

0 otherwise


Impulse Functions

� The unit impulse function has been defined in the past as

δ(t) = limτ→0

p̄τ (t) = limτ→0

{1τ

for |t| ≤ τ/2

0 otherwise

� Obviously, this is an infinitesimally thin, infinitely tall pulsewhose area is equal to unity for any finite value of τ .


Impulse Functions Cont’d

Pulse function p̄τ (t) for three values of τ :

−1 0 10

5

10

15

20

25

t

τ = 0.50 τ = 0.10 τ = 0.05

(a)

p τ(t

) -


Mathematical Problem

� The Fourier transform of the unit impulse function asdefined in the past should be given by the integral

X (jω) =∫ ∞

−∞x(t)e−jωt dt

=∫ ∞

−∞limτ→0

[p̄τ (t)]e−jωt dt

where

p̄τ (t) ={

1τ

for |t| ≤ τ/2

0 otherwise


Mathematical Problem

� The Fourier transform of the unit impulse function asdefined in the past should be given by the integral

X (jω) =∫ ∞

−∞x(t)e−jωt dt

=∫ ∞

−∞limτ→0


where

p̄τ (t) ={

1τ

for |t| ≤ τ/2

0 otherwise

� If we now attempt to evaluate the function p̄τ (t)e−jωt atτ = 0, we find that it becomes infinite and, therefore, theabove integral cannot be evaluated.


Mathematical Problem Cont’d

� We can write

F limτ→0

p̄τ (t) =∫ ∞

−∞limτ→0


≈∫ τ/2

−τ/2limτ→0

[1τ

]dt



� We can write

F limτ→0

p̄τ (t) =∫ ∞

−∞limτ→0


≈∫ τ/2

−τ/2limτ→0

[1τ

]dt

� Since the area of the pulse function p̄τ (t) is unity for anyfinite value of τ , we might be tempted to assume that thearea is equal to unity even for τ = 0, i.e.,

F limτ→0

p̄τ (t) = 1



� The Fourier transform of p̄τ (t) for a finite τ is given by

F p̄τ (t) = 1τFpτ (t) = 2 sin ωτ/2

ωτ





ωτ

� Obviously, this is well defined and, interestingly, it has thelimit

limτ→0

F p̄τ (t) = 1





ωτ

� Obviously, this is well defined and, interestingly, it has thelimit

limτ→0

F p̄τ (t) = 1

� So far so good!



� If we now attempt to find the inverse Fourier transform of 1,we run into certain mathematical difficulties.




� From the definition of the inverse Fourier transform, wehave

F−11 = 12π

∫ ∞

−∞ejωt dω

= 12π

[∫ ∞

−∞cosωt dω + j

∫ ∞

−∞sin ωt dω

]




� From the definition of the inverse Fourier transform, wehave

F−11 = 12π

∫ ∞

−∞ejωt dω

= 12π

[∫ ∞

−∞cosωt dω + j

∫ ∞

−∞sin ωt dω

]

� However, mathematicians will tell us that these integrals donot converge or do not exist!



� Summarizing, by cheating a little bit we can get a more orless meaningful Fourier transform for the unit impulsefunction.



� Summarizing, by cheating a little bit we can get a more orless meaningful Fourier transform for the unit impulsefunction.

� Unfortunately, it is impossible to recover the impulsefunction from its Fourier transform by applying the inverseFourier transform.



� The impulse-function problem can be circumvented in twoways, a practical and a theoretical one:




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




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

– The theoretical approach is rigorous but it is rather abstractand more difficult to understand or apply in practicalsituations.


Practical Approach to Impulse Functions

� In the practical approach to impulse functions, a function γ (t) issaid to be a unit impulse function if, for any continuous functionx(t) over the range −ε < t < ε, the following relation is satisfied:∫ ∞

−∞γ (t)x(t) dt � x(0) (C)




−∞γ (t)x(t) dt � x(0) (C)

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




−∞γ (t)x(t) dt � x(0) (C)

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

� Now consider the pulse function

limτ→ε

p̄τ (t) = p̄ε(t) ={

1ε

for |t| ≤ ε/2

0 otherwise

where ε is a small but finite constant.


Practical Approach to Impulse Functions Cont’d

· · · ∫ ∞

−∞γ (t)x(t) dt � x(0) (C)

� If we letγ (t) = lim

τ→εp̄τ (t)

in the left-hand side of Eq. (C), we obtain∫ ∞

−∞limτ→ε

[p̄τ (t)]x(t) dt =∫ ε/2

−ε/2

1εx(t) dt

� 1εx(0)

∫ ε/2

−ε/2dt � x(0)



· · · ∫ ∞

−∞γ (t)x(t) dt � x(0) (C)

� If we letγ (t) = lim

τ→εp̄τ (t)

in the left-hand side of Eq. (C), we obtain∫ ∞

−∞limτ→ε

[p̄τ (t)]x(t) dt =∫ ε/2

−ε/2

1εx(t) dt

� 1εx(0)

∫ ε/2

−ε/2dt � x(0)

� Thus we conclude that the very thin pulse function limτ→ε p̄τ (t)behaves as an impulse function and, therefore, we can write

δ(t) = limτ→ε

p̄τ (t)



· · ·δ(t) = lim

τ→εp̄τ (t)

� Now if we apply the Fourier transform to the impulsefunction as defined, we get

limτ→ε

p̄τ (t) ↔ limτ→ε

2 sin ωτ/2ωτ



· · ·limτ→ε


2 sin ωτ/2ωτ

� As τ is reduced, the pulse function at the left tends tobecome thinner and taller whereas the sinc function at theright tends to be flattened out.

−40 −30 −20 −10 0 10 20 30 40−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

sin

(ωε/

2)/(ωε/

2)

ε = 0.50 ε = 0.10 ε = 0.05

−1 0 10

5

10

15

20

25

t

ε = 0.50 ε = 0.10 ε = 0.05

(a) (b)

δω∞

ω

p ε(t

) -

ω∞2− ω∞

2



� For some small but finite ε, the sinc function will be equalto unity to within an error δω∞ over some frequency range−ω∞/2 < ω < ω∞/2.

−40 −30 −20 −10 0 10 20 30 40−0.4

−0.2

0

0.2

0.4

0.6

0.8

1.0

1.2si

n (ω

ε/2)

/(ωε/

2)

ε = 0.50 ε = 0.10 ε = 0.05

(b)

δω∞

ω

ω∞2− ω∞

2



� Therefore, we can write

δ(t) = limτ→ε


2 sin ωτ/2ωτ

= i(ω)

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



� Summarizing,

– the Fourier transform of a time-domain impulse function is afrequency-domain unity function, and

– the inverse Fourier transform of a frequency-domain unityfunction is a time-domain impulse function,



� Summarizing,



i.e., δ(t) ↔ i(ω)



� Summarizing,



i.e., δ(t) ↔ i(ω)

� 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 isapproximate with the understanding that it can be made as exactas desired by making ε sufficiently small.



The impulse and unity functions can be represented by theidealized graphs:

δ(t)

t ω

1i(ω)

(a)

1


Properties of Impulse Functions

� Assuming that x(t) is a continuous function of t over therange −ε < t < ε, 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.)


Frequency-Domain Impulse Functions

� Given a transform pairδ(t) ↔ i(ω)

where δ(t) = limτ→ε

p̄τ (t)

i(ω) = limτ→ε

2 sin ωτ/2ωτ

� 1 for |ω| < ω∞

the corresponding transform pair

i(t) ↔ 2πδ(ω)

where i(t) = 2 sin tε/2tε

� 1 for |t| < t∞

δ(ω) = p̄ε(ω)

can be generated by applying the symmetry theorem of the Fouriertransform.


Frequency-Domain Impulse Functions Cont’d

· · ·i(t) ↔ 2πδ(ω)

� Function i(t) is a time-domain unity function whereas δ(ω) is afrequency-domain unit impulse function.



· · ·i(t) ↔ 2πδ(ω)

� Function i(t) is a time-domain unity function whereas δ(ω) is afrequency-domain unit impulse function.

� Since i(t) � 1, we have

1 � 2πδ(ω)



· · ·i(t) ↔ 2πδ(ω) or 1 � 2πδ(ω)

This transform pair can be represented by the idealized graphsshown.

i(t)1

t ω

δ(ω)

(b)

2π


Properties of Frequency-Domain Impulse Functions

� Assuming that X (jω) is a continuous function of ω over therange −ε < ω < ε, 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.)


Fourier Transforms of Exponentials

� Sinceδ(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


Fourier Transforms of Exponentials

� Sinceδ(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

� Now applying the frequency-shifting theorem to thefrequency-domain impulse function, we obtain

i(t)ejω0t ↔ 2πδ(ω − ω0)

and since i(t) � 1, we get

ejω0 t � 2πδ(ω − ω0)


Fourier Transforms of Sinusoidal Signals

� We know thati(t)ejω0t ↔ 2πδ(ω − ω0)

andi(t)e−jω0 t ↔ 2πδ(ω + ω0)


Fourier Transforms of Sinusoidal Signals

� We know thati(t)ejω0t ↔ 2πδ(ω − ω0)


� If we add the two equations, we get

i(t)(ejω0 t + e−jω0t) = 2i(t) · cos ω0t ↔ 2π [δ(ω + ω0) + δ(ω − ω0)]

and since i(t) � 1, we have

cos ω0t � π [δ(ω + ω0) + δ(ω − ω0)]


Fourier Transforms of Sinusoidal Signals Cont’d

· · ·cos ω0t � π [δ(ω + ω0) + δ(ω − ω0)]

x(t)

t

X( jω)

−ω0 ω0 ω

π



� As before,i(t)ejω0t ↔ 2πδ(ω − ω0)




� As before,i(t)ejω0t ↔ 2πδ(ω − ω0)


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

i(t)(e−jω0 t −ejω0t) = −2ji(t) · sin ω0t ↔ 2π [δ(ω+ω0)− δ(ω −ω0)]

and since i(t) � 1, we can write

sin ω0t � jπ [δ(ω + ω0) − δ(ω − ω0)]


Fourier Transforms of Arbitrary Periodic Signals

� An arbitrary periodic signal can be represented by the Fourierseries

x̃(t) =∞∑

k=−∞Xke−jkω0t


Fourier Transforms of Arbitrary Periodic Signals

� An arbitrary periodic signal can be represented by the Fourierseries

x̃(t) =∞∑

k=−∞Xke−jkω0t

� Hence

F x̃(t) =∞∑

k=−∞2πXkFe−jkω0t �

∞∑k=−∞

2πXkδ(ω − kω0)

or x̃(t) � 2π

∞∑k=−∞

Xkδ(ω − kω0)


Fourier Transforms of Periodic Signals Cont’d

· · ·x̃(t) � 2π

∞∑k=−∞

Xkδ(ω − kω0)

� Summarizing, the frequency spectrum of a periodic signal canbe represented by an infinite sequence of numbers Xk for−∞ < k < ∞, i.e., the Fourier-series coefficients as shown inChap. 2



· · ·x̃(t) � 2π

∞∑k=−∞

Xkδ(ω − kω0)


or



· · ·x̃(t) � 2π

∞∑k=−∞

Xkδ(ω − kω0)


or

� by an infinite sequence of frequency-domain impulse functionsof strength 2πXk for −∞ < k < ∞ as shown in the previousslide.


Theoretical Approach to Impulse Functions

� The Fourier transform pairs generated through the practicalapproach are approximate since the pulse width ε cannot bereduced to absolute zero.




� However, by defining impulse functions in terms of generalizedfunctions, analogous, but exact, Fourier transform pairs can begenerated.





� Unfortunately, impulse functions so defined are ratherimpractical and difficult to implement in a laboratory.





� Unfortunately, impulse functions so defined are ratherimpractical and difficult to implement in a laboratory.

� See textbook for more details and references on generalizedfunctions.


Summary of Fourier Transforms Derived

x(t) X (jω)

δ(t) 1

1 2πδ(ω)

δ(t − t0) e−jωt0

ejω0t 2πδ(ω − ω0)

cos ω0t π [δ(ω + ω0) + δ(ω − ω0)]

sin ω0t jπ [δ(ω + ω0) − δ(ω − ω0)]


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


Documents

Chapter 6 CONTINUOUS-TIME, IMPULSE-MODULATED, AND DISCRETE ...2.pdf · Introduction Digital ﬁlters are often used to process discrete-time signals that have been generated by sampling