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: aantoniou@ieee.org
March 7, 2008
Frame # 1
Slide # 1
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.
Frame # 2
Slide # 2
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.
Frame # 2
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.
Frame # 2
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.
Frame # 2
Slide # 5
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.
Frame # 3
Slide # 6
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.
Frame # 3
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.
Frame # 5 Slide # 15
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)
Frame # 5 Slide # 16
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 −∞
Frame # 6 Slide # 17
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.
Frame # 6 Slide # 18
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.
Frame # 6 Slide # 19
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 p̄τ (t) = lim
τ →0
Frame # 7 Slide # 20
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 p̄τ (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 .
Frame # 7 Slide # 21
A. Antoniou
Digital Signal Processing – Secs. 6.1, 6.2
Impulse Functions Cont’d
Pulse function p̄τ (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
Frame # 8 Slide # 22
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 [p̄τ (t)]e−jωt dt
=
−∞ τ →0
where
p̄τ (t) =
Frame # 9 Slide # 23
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 [p̄τ (t)]e−jωt dt
=
−∞ τ →0
where
p̄τ (t) =
1
τ
0
for |t| ≤ τ/2
otherwise
t If we now attempt to evaluate the function p̄τ (t)e−jωt at
τ = 0, we find that it becomes infinite and, therefore, the
above integral cannot be evaluated.
Frame # 9 Slide # 24
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 [p̄τ (t)]e−jωt dt
1
dt
lim
≈
−τ/2 τ →0 τ
F lim p̄τ (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 [p̄τ (t)]e−jωt dt
1
dt
lim
≈
−τ/2 τ →0 τ
F lim p̄τ (t) =
−∞ τ →0
τ/2
t Since the area of the pulse function p̄τ (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 p̄τ (t) = 1
τ →0
Frame # 10 Slide # 26
A. Antoniou
Digital Signal Processing – Secs. 6.1, 6.2
Mathematical Problem Cont’d
t The Fourier transform of p̄τ (t) for a finite τ is given by
F p̄τ (t) =
Frame # 11 Slide # 27
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 p̄τ (t) for a finite τ is given by
F p̄τ (t) =
1
2 sin ωτ/2
Fpτ (t) =
τ
ωτ
t Obviously, this is well defined and, interestingly, it has the
limit
lim F p̄τ (t) = 1
τ →0
Frame # 11 Slide # 28
A. Antoniou
Digital Signal Processing – Secs. 6.1, 6.2
Mathematical Problem Cont’d
t The Fourier transform of p̄τ (t) for a finite τ is given by
F p̄τ (t) =
1
2 sin ωτ/2
Fpτ (t) =
τ
ωτ
t Obviously, this is well defined and, interestingly, it has the
limit
lim F p̄τ (t) = 1
τ →0
t So far so good!
Frame # 11 Slide # 29
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.
Frame # 12 Slide # 30
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!
Frame # 12 Slide # 32
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.
Frame # 13 Slide # 33
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.
t Unfortunately, it is impossible to recover the impulse
function from its Fourier transform by applying the inverse
Fourier transform.
Frame # 13 Slide # 34
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 p̄τ (t) = p̄ (t) =
τ →
1
0
for |t| ≤ /2
otherwise
where is a small but finite constant.
Frame # 15 Slide # 40
A. Antoniou
Digital Signal Processing – Secs. 6.1, 6.2
Practical Approach to Impulse Functions Cont’d
···
∞
−∞
γ (t)x (t) dt x (0)
t If we let
γ (t) = lim p̄τ (t)
τ →
in the left-hand side of Eq. (C), we obtain
/2
∞
1
x (t) dt
lim [p̄τ (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
···
∞
−∞
γ (t)x (t) dt x (0)
(C)
t If we let
γ (t) = lim p̄τ (t)
τ →
in the left-hand side of Eq. (C), we obtain
/2
∞
1
x (t) dt
lim [p̄τ (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 → p̄τ (t)
behaves as an impulse function and, therefore, we can write
δ(t) = lim p̄τ (t)
τ →
Frame # 16 Slide # 42
A. Antoniou
Digital Signal Processing – Secs. 6.1, 6.2
Practical Approach to Impulse Functions Cont’d
···
δ(t) = lim p̄τ (t)
τ →
t Now if we apply the Fourier transform to the impulse
function as defined, we get
lim p̄τ (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 p̄τ (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 p̄τ (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 p̄τ (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) = p̄ (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 )]
Frame # 29 Slide # 60
A. Antoniou
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
x̃(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
x̃(t) =
∞
Xk e−jk ω0 t
k =−∞
t Hence
F x̃(t) =
∞
2π Xk Fe−jk ω0 t k =−∞
or
∞
2π Xk δ(ω − kω0 )
k =−∞
x̃(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
···
x̃(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
···
x̃(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
···
x̃(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