Chapter 6
CONTINUOUS-TIME,
IMPULSE-MODULATED, AND
DISCRETE-TIME SIGNALS
6.4 Poisson’s Summation Formula
6.5 Impulse-Modulated Signals
c 2005- Andreas Antoniou
Copyright Victoria, BC, Canada
Email: [email protected]
March 12, 2008
Frame # 1
Slide # 1
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Introduction
Various time-domain and frequency-domain relationships exist
between continuous-time and discrete-time signals.
Frame # 2
Slide # 2
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Introduction
Various time-domain and frequency-domain relationships exist
between continuous-time and discrete-time signals.
These relationships are developed by defining a special class of
signals known as impulse-modulated signals which comprise
sequences of continuous-time impulse functions.
Frame # 2
Slide # 3
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Introduction
Various time-domain and frequency-domain relationships exist
between continuous-time and discrete-time signals.
These relationships are developed by defining a special class of
signals known as impulse-modulated signals which comprise
sequences of continuous-time impulse functions.
Impulse-modulated signals are essentially continuous-time
signals but simultaneously they are also sampled signals.
Therefore, on the one hand, they have Fourier transforms and,
on the other, they can be represented by z transforms.
Consequently, impulse-modulated signals can serve as a
mathematical bridge between continuous-time and discrete-time
signals that facilitates the derivations of the various relationships
between the two classes of signals.
Frame # 2
Slide # 4
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Impulse-Modulated Signals
An impulse modulated-signal, denoted as x̂(t), can be
generated by sampling a continuous-time signal x(t) using
an ideal impulse modulator.
Impulse modulator
ˆ
x(t)
x(t)
(a)
c(t)
Frame # 3
Slide # 5
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Impulse-Modulated Signals Cont’d
An impulse modulator is characterized by the equation
x̂ (t) = c(t)x (t)
where c(t) is a carrier given by
c(t) =
∞
δ(t − nT )
n=−∞
Frame # 4
Slide # 6
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Impulse-Modulated Signals Cont’d
An impulse modulator is characterized by the equation
x̂ (t) = c(t)x (t)
where c(t) is a carrier given by
c(t) =
∞
δ(t − nT )
n=−∞
Hence
x̂(t) = x (t)
∞
δ(t − nT )
n=−∞
Frame # 4
Slide # 7
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Impulse-Modulated Signals Cont’d
An impulse modulator is characterized by the equation
x̂ (t) = c(t)x (t)
where c(t) is a carrier given by
c(t) =
∞
δ(t − nT )
n=−∞
Hence
∞
x̂(t) = x (t)
δ(t − nT )
n=−∞
From the properties of the unit impulse function, we get
x̂(t) =
∞
x (nT )δ(t − nT )
n=−∞
Frame # 4
Slide # 8
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Impulse-Modulated Signals Cont’d
The input and output of an impulse modulator are as follows:
x(t)
x(kT)
t
kT
×
(b)
c(t)
(c)
1
t
kT
=
ˆ
x(t)
x(kT)
kT
Frame # 5
Slide # 9
A. Antoniou
(d)
t
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Impulse-Modulated and
Discrete-Time Signals
Impulse-modulated signals are sequences of
continuous-time impulses.
Frame # 6 Slide # 10
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Impulse-Modulated and
Discrete-Time Signals
Impulse-modulated signals are sequences of
continuous-time impulses.
They can be converted to discrete-time signals by
replacing impulses by numbers.
Frame # 6 Slide # 11
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Impulse-Modulated and
Discrete-Time Signals
Impulse-modulated signals are sequences of
continuous-time impulses.
They can be converted to discrete-time signals by
replacing impulses by numbers.
On the other hand, impulse-modulated signals can be
obtained from discrete-time signals by replacing numbers
by impulses.
Frame # 6 Slide # 12
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Relation between Impulse-Modulated and
Discrete-Time Signals Cont’d
ˆ
x(t)
x(kT)
kT
(d)
t
x(nT)
x(kT)
kT
Frame # 7 Slide # 13
A. Antoniou
(e)
nT
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Fourier Transform and Z
Transform
An impulse-modulated signal is both a continuous-time as
well as a sampled signal, as was stated earlier, and this
dual personality will immediately prove very useful.
Frame # 8 Slide # 14
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Fourier Transform and Z
Transform
An impulse-modulated signal is both a continuous-time as
well as a sampled signal, as was stated earlier, and this
dual personality will immediately prove very useful.
As a continuous-time signal, an impulse-modulated signal
has a Fourier transform given by
X̂ (jω) = F
=
∞
x(nT )δ(t − nT ) =
n=−∞
∞
∞
x(nT )Fδ(t − nT )
n=−∞
x(nT )e−jωnT
n=−∞
Frame # 8 Slide # 15
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Relationship between Fourier Transform and Z
Transform Cont’d
···
X̂ (jω) =
∞
x (nT )e−jωnT
n=−∞
Therefore, from the definition of the z transform we note that
X̂ (jω) = XD (z) jωT
z=e
where
XD (z) = Zx (nT )
(A)
In effect, the Fourier transform of impulse-modulated signal x̂(t)
is numerically equal to the z transform of the corresponding
discrete-time signal x (nT ) evaluated on the unit circle |z| = 1.
In other words, the frequency spectrum of x̂ (t) is equal to that of
x (nT ).
Frame # 9 Slide # 16
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example
The continuous-time signal
⎧
⎪
0
⎪
⎪
⎪
⎪
⎪
⎨1
x(t) = 2
⎪
⎪
⎪
1
⎪
⎪
⎪
⎩
0
for t < −3.5 s
for −3.5 ≤ t < −2.5
for −2.5 ≤ t < 2.5
for 2.5 ≤ t ≤ 3.5
for t > 3.5
is subjected to impulse modulation.
Find the frequency spectrum of x̂(t) in closed form assuming a
sampling frequency of 2π rad/s.
Frame # 10 Slide # 17
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
Solution The frequency spectrum of an impulse-modulated signal,
x̂ (t), can be readily obtained by evaluating the z transform of x (nT )
on the unit circle of the z plane.
The impulse-modulated version of x (t) can be expressed as
x̂ (t) = δ(t + 3T ) + 2δ(t + 2T ) + 2δ(t + T ) + 2δ(0)
+2δ(t − T ) + 2δ(t − 2T ) + δ(t − 3T )
where T = 1 s.
A corresponding discrete-time signal can be obtained by replacing
impulses by numbers as
x (nT ) = δ(nT + 3T ) + 2δ(nT + 2T ) + 2δ(nT + T ) + 2δ(0)
+2δ(nT − T ) + 2δ(nT − 2T ) + δ(nT − 3T )
Hence XD (z) = Zx (t) = z 3 + 2z 2 + 2z 1 + 2 + 2z −1 + 2z −2 + z −3
Frame # 11 Slide # 18
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
···
XD (z) = Zx(t) = z 3 + 2z 2 + 2z 1 + 2 + 2z −1 + 2z −2 + z −3
Since the frequency spectrum of an impulse-modulated signal
is given by
X̂ (jω) = XD (ejωT )
we get
X̂ (jω) = XD (ejωT )
= (ej3ωT + e−j3ωT ) + 2(ej2ωT + e−j2ωT )
+ 2(ejωT + e−jωT ) + 2
= 2 cos 3ωT + 4 cos 2ωT + 4 cos ωT + 2
Frame # 12 Slide # 19
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example
The continuous-time signal
x(t) = u(t)e−t sin 2t
is subjected to impulse modulation.
Find the frequency spectrum of x̂(t) in closed form assuming a
sampling frequency of 2π rad/s.
Frame # 13 Slide # 20
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
Solution A discrete-time signal can be readily derived from x (t) by
replacing t by nT as
x (nT ) = u(nT )e−nT sin 2nT = u(nT )e−nT ×
= u(nT )
1 j2nT
e
− e−j2nT
2j
1 nT (−1+j2)
e
− enT (−1−j2)
2j
Since T = 2π/ωs = 1 s, the table of z transforms gives
1
z
z
−
XD (z) =
2j z − e−1+j2
z − e−1−j2
and after some manipulation
XD (z) =
Frame # 14 Slide # 21
z2
ze−1 sin 2
− 2ze−1 cos 2 + e−2
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
···
XD (z) =
ze−1 sin 2
z 2 − 2ze−1 cos 2 + e−2
Since the frequency spectrum of an impulse-modulated signal
is given by
X̂ (jω) = XD (ejωT )
we get
X̂ (jω) = XD (ejωT ) =
Frame # 15 Slide # 22
ejω−1 sin 2
e2jω − 2ejω−1 cos 2 + e−2
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula
As may be expected, the spectrum of a discrete-time
signal must be related to that spectrum of the
continuous-time signal from which it was derived.
Frame # 16 Slide # 23
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula
As may be expected, the spectrum of a discrete-time
signal must be related to that spectrum of the
continuous-time signal from which it was derived.
This relationship can be established by using Poisson’s
summation formula.
Frame # 16 Slide # 24
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula
Consider a signal x(t) with a Fourier transform X (jω).
Poisson’s summation formula states that
∞
∞
1 x(t + nT ) =
X (jnωs )ejnωs t
T
n=−∞
n=−∞
where ωs = 2π/T .
Frame # 17 Slide # 25
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula
Consider a signal x(t) with a Fourier transform X (jω).
Poisson’s summation formula states that
∞
∞
1 x(t + nT ) =
X (jnωs )ejnωs t
T
n=−∞
n=−∞
where ωs = 2π/T .
If t = 0 and x(t) is a two-sided signal, we have
∞
∞
1 x(nT ) =
X (jnωs )
T n=−∞
n=−∞
Frame # 17 Slide # 26
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
(B)
Poisson’s Summation Formula Cont’d
···
∞
x(t + nT ) =
n=−∞
∞
1 X (jnωs )ejnωs t
T n=−∞
If t = 0 and x(t) is a right-sided signal, i.e., x(t) = 0 for
t < 0, then
∞
x(nT ) =
n=0
where
lim x(t) =
t →0
Frame # 18 Slide # 27
∞
1 x(0+)
+
X (jnωs )
2
T n=−∞
x(0+)
x(0−) + x(0+)
=
2
2
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula Cont’d
···
∞
x(t + nT ) =
n=−∞
∞
1 X (jnωs )ejnωs t
T n=−∞
If t = 0 and x(t) is a right-sided signal, i.e., x(t) = 0 for
t < 0, then
∞
x(nT ) =
n=0
∞
1 x(0+)
+
X (jnωs )
2
T n=−∞
where
x(0+)
x(0−) + x(0+)
=
t →0
2
2
Note: In Fourier analysis, the value of a time function at a
discontinuity is always taken to be the average of the left
and right limits (see textbook).
lim x(t) =
Frame # 18 Slide # 28
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Poisson’s Summation Formula Cont’d
1.0
x(t)
0.5
0
−0.5
−1.0
0
2T
T
3T
4T
5T
(a)
1.5
|X(jω)|
1.0
0.5
0
ω
−3ωs
−2ωs
−ωs
0
ωs
−2ωs
−ωs
0
(b)
ωs
2ωs
3ωs
2ωs
3ωs
arg x(jω), rad/s
4.0
2.0
0
−2.0
−4.0
−3ωs
Frame # 19 Slide # 29
ω
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship between Discrete-Time and
Continuous-Time Signals
Given a continuous-time signal x(t) with a Fourier
transform X (jω), then from the frequency-shifting theorem
we have
x(t)e−jω0 t ↔ X (jω0 + jω)
Frame # 20 Slide # 30
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship between Discrete-Time and
Continuous-Time Signals
Given a continuous-time signal x(t) with a Fourier
transform X (jω), then from the frequency-shifting theorem
we have
x(t)e−jω0 t ↔ X (jω0 + jω)
From Poisson’s summation formula, we get
∞
x(nT )e−jω0 nT =
n=−∞
∞
1 X (jω0 + jnωs )
T n=−∞
where ωs = 2π/T and if we now replace ω0 by ω, we
deduce the important relationship
∞
x(nT )e
−jωnT
n=−∞
Frame # 20 Slide # 31
A. Antoniou
∞
1 =
X (jω + jnωs )
T n=−∞
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
···
∞
x (nT )e−jωnT =
n=−∞
∞
1 X (jω + jnωs )
T n=−∞
It was shown earlier that
X̂ (jω) = XD (ejωT ) =
∞
x (nT )e−jωnT
n=−∞
and hence X̂ (jω) = XD (e
jωT
∞
1 )=
X (jω + jnωs )
T n=−∞
Therefore, the frequency spectrum of the impulse-modulated
signal x̂(t) is numerically equal to the frequency spectrum of
discrete-time signal x (nT ) and the two can be uniquely
determined from the frequency spectrum of the continuous-time
signal x (t), namely, X (jω).
Frame # 21 Slide # 32
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
As is to be expected, X̂ (jω) is a periodic function of ω with
period ωs since the frequency spectrum of discrete-time
signals is periodic.
Frame # 22 Slide # 33
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
As is to be expected, X̂ (jω) is a periodic function of ω with
period ωs since the frequency spectrum of discrete-time
signals is periodic.
To check this out, we can replace jω by jω + jmωs in
∞
1 X (jω + jnωs )
X̂ (jω) =
T n=−∞
to obtain
∞
1 X [jω + j(m + n)ωs ]
X̂ (jω + jmωs ) =
T n=−∞
∞
1 X (jω + jn ωs ) = X̂ (jω)
=
T n =−∞
Frame # 22 Slide # 34
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
For a right-sided signal, x (t) = 0 for t ≤ 0−, and hence the
impulse-modulated signal assumes the form
x̂(t) =
∞
x (nT )δ(t − nT )
n=0
where x (0) ≡ x (0+).
Frame # 23 Slide # 35
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
For a right-sided signal, x (t) = 0 for t ≤ 0−, and hence the
impulse-modulated signal assumes the form
x̂(t) =
∞
x (nT )δ(t − nT )
n=0
where x (0) ≡ x (0+).
The Fourier transform of the signal is given by
X̂ (jω) =
∞
x (nT )e−jωnT = XD (ejωT )
n=0
Frame # 23 Slide # 36
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
For a right-sided signal, x (t) = 0 for t ≤ 0−, and hence the
impulse-modulated signal assumes the form
x̂(t) =
∞
x (nT )δ(t − nT )
n=0
where x (0) ≡ x (0+).
The Fourier transform of the signal is given by
X̂ (jω) =
∞
x (nT )e−jωnT = XD (ejωT )
n=0
Thus Poisson’s summation formula gives
X̂ (jω) = XD (ejωT ) =
Frame # 23 Slide # 37
∞
1 x (0+)
+
X (jω + jnωs )
2
T n=−∞
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
(C)
Spectral Relationship Cont’d
···
X̂ (jω) = XD (ejωT ) =
∞
1 x(0+)
+
X (jω + jnωs )
2
T n=−∞
(C)
By letting jω = s and esT = z, Eq. (C) can be expressed in
the s domain as
X̂ (s) = XD (z) =
∞
1 x(0+)
+
X (s + jnωs )
2
T n=−∞
where X (s) and X̂ (s) are the Laplace transforms of x(t)
and x̂(t), respectively.
Frame # 24 Slide # 38
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Spectral Relationship Cont’d
···
X̂ (jω) = XD (ejωT ) =
∞
1 x(0+)
+
X (jω + jnωs )
2
T n=−∞
(C)
By letting jω = s and esT = z, Eq. (C) can be expressed in
the s domain as
X̂ (s) = XD (z) =
∞
1 x(0+)
+
X (s + jnωs )
2
T n=−∞
where X (s) and X̂ (s) are the Laplace transforms of x(t)
and x̂(t), respectively.
This relationship will be used in Chap. 11 to design digital
filters based on analog filters.
Frame # 24 Slide # 39
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example
Find X̂ (jω) if x(t) = cos ω0 t.
Solution From the table of Fourier transforms (Table 6.2), we
have
X (jω) = F cos ω0 t = π [δ(ω + ω0 ) + δ(ω − ω0 )]
Hence Poisson’s summation formula, i.e.,
X̂ (jω) = XD (e
jωT
∞
1 )=
X (jω + jnωs )
T n=−∞
gives
∞
π [δ(ω + nωs + ω0 ) + δ(ω + nωs − ω0 )]
X̂ (jω) =
T n=−∞
Frame # 25 Slide # 40
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
^
| X( jω)|
−2ωs
−ωs
−ω0
ω0
ωs
(a)
Frame # 26 Slide # 41
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
2ωs
Example
Find X̂ (jω) if x (t) = u(t)e−t .
Solution From the table of Fourier transforms (Table 6.2),
X (jω) = F [u(t)e−t ] =
1
1 + jω
Since x (t) = 0 for t < 0 in this case, we need to use the second form
of Poisson’s summation formula, i.e.,
X̂ (jω) = XD (ejωT ) =
∞
x (0+)
1 X (jω + jnωs )
+
2
T n=−∞
The initial-value theorem of the Laplace transform gives
x (0+) = lim [sX (s)] = lim
s→∞
and hence
Frame # 27 Slide # 42
X̂ (jω) =
s→∞
s
=1
1+s
∞
1
1 1
+
2 T n=−∞ 1 + j(ω + nωs )
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
Example Cont’d
1.0
|X( jω)|
0.8
0.6
0.4
0.2
0
−30
−20
−10
0
10
20
30
^
3
|X( jω)|
1
T |X( jω+ jωs)|
1 |X( jω)|
T
1 |X( jω− jω )|
s
T
2
1
0
−30
−20
−10
−ωs
−
ωs
0
2
ωs
2
10
20
ω
30
ωs
(b)
Frame # 28 Slide # 43
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5
This slide concludes the presentation.
Thank you for your attention.
Frame # 29 Slide # 44
A. Antoniou
Digital Signal Processing – Secs. 6.4, 6.5