eece 359 - signals and communications: part 1 spring 2014...
Post on 01-Aug-2021
4 Views
Preview:
TRANSCRIPT
EECE 359 - Signals and Communications: Part 1 Spring 2014
Fourier transform representation of CT
aperiodic signals – Section 4.1
A large class of aperiodic CT signals can
be represented by the CT Fourier transform
(CTFT).
The (CT) Fourier transform (or spectrum) of
x(t) is
X(jω) =
∫ ∞
−∞
x(t)e−jωt dt.
x(t) can be reconstructed from its spectrum
using the inverse Fourier transform
x(t) =1
2π
∫ ∞
−∞
X(jω)ejωt dω.
The above two equations are referred to as
the Fourier transform pair with the first one
SM 107
EECE 359 - Signals and Communications: Part 1 Spring 2014
being the analysis equation and the second
being the synthesis equation.
Notation:
X(jω) = F{x(t)}x(t) = F−1{X(jω)}
x(t) and X(jω) form a Fourier transform
pair, denoted by
x(t)F←→ X(jω)
Often, we will use the simpler notation
x(t)←→ X(jω)
Example:
rect(t) or Π(t) =
1, |t| < 1/2
0, |t| > 1/2
1/2, |t| = 1/2
SM 108
EECE 359 - Signals and Communications: Part 1 Spring 2014
t
Π t( )
1
1
2---
1
2---– 0
rect(t)
SM 109
EECE 359 - Signals and Communications: Part 1 Spring 2014
ω
F Π t( ){ }
1
0 2π2– π4– π6– π 4π 6π
Fourier transform of rect(t)
Example: x(t) = e−atu(t), a > 0. We
want to show that
e−atu(t)←→ 1
a+ jω, a > 0 .
SM 110
EECE 359 - Signals and Communications: Part 1 Spring 2014
Since the above FT is complex-valued, it is
customary to plot its magnitude and phase,
i.e.,
| 1
a+ jω| =
1√a2 + ω2
∠1
a+ jω= − tan−1
(ω
a
)
.
ω
1
a jω+---------------
1 a⁄
0 aa–
Plot of spectrum magnitude
ω
1
a jω+---------------∠
π 2⁄
0 aa–
π– 2⁄
Plot of spectrum phase
SM 111
EECE 359 - Signals and Communications: Part 1 Spring 2014
Example: triangle function
∧(t) =
{
1− |t|, −1 ≤ t ≤ 10, elsewhere
t
Λ t( )
1
11– 0
triangle function
Exercise: Show that
∧(t)F←→ sinc2(
ω
2π)
SM 112
EECE 359 - Signals and Communications: Part 1 Spring 2014
Properties of CT Fourier transform –
Section 4.3
Table 4.1 on p. 328 summarizes many CTFT
properties.
1. Linearity – Section 4.3.1
Let x(t)←→ X(jω), y(t)←→ Y (jω).
Then,
z(t) = ax(t) + by(t)
←→ Z(jω) = aX(jω) + bY (jω) .
Proof:
SM 113
EECE 359 - Signals and Communications: Part 1 Spring 2014
Example: What is the CTFT of the
signal, z(t), shown in the figure below?
t
z t( )
1
11– 0
1.5
0.5
0.50.5–
Signal z(t)
SM 114
EECE 359 - Signals and Communications: Part 1 Spring 2014
2. Time shift – Section 4.3.2
If x(t)←→ X(jω), then
x(t− t0)←→ e−jωt0X(jω) .
Proof:
Note that when a signal is delayed by t0, the
spectrum amplitude is unchanged whereas
the spectrum phase is changed by −ωt0.
SM 115
EECE 359 - Signals and Communications: Part 1 Spring 2014
Example: What are the CTFT’s of the
signals shown below?
t
Π t 0.5+( )
1
1–0
t
Π t 0.5–( )
1
10
Shifted rect signals
SM 116
EECE 359 - Signals and Communications: Part 1 Spring 2014
3. Time scaling – Section 4.3.5
If x(t)↔ X(jω), then
x(at)↔1
|a|X
(
jω
a
)
where a is a non-zero real constant.
The time scaling property states that if a
signal is compressed in time by a factor a,
then its spectrum is expanded in frequency
by the same factor a and vice-versa.
This property is an example of the
inverse relationship between the time and
frequency domains.
Example: Since
Π(t)↔ sinc( ω
2π
)
,
then
Π(2t)↔1
2sinc
( ω
4π
)
.
SM 117
EECE 359 - Signals and Communications: Part 1 Spring 2014
Note also that if a = −1 (corresponding to
a time reversal) in the time scaling property,
we have
x(−t)↔ X(−jω)
i.e. the spectrum is also reversed.
Combining the time shift and scaling
properties
What is the Fourier transform of x(at− b)?
SM 118
EECE 359 - Signals and Communications: Part 1 Spring 2014
4. Conjugation – Section 4.3.5
If x(t)↔ X(jω), then
x∗(t)↔ X∗ (−jω) .
As a result, if x(t) is real, we have
X (−jω) = X∗ (jω)
i.e. the spectrum magnitude is an even
function of ω and the spectrum phase is an
odd function of ω.
More generally, we can summarize the
relationship between a signal and its
spectrum as follows:
SM 119
EECE 359 - Signals and Communications: Part 1 Spring 2014
5. Convolution – Section 4.4y(t) = x(t) ∗ h(t)↔ Y (jω) = X (jω)H (jω) .
Proof:
Application: Since ∧(t) = Π(t) ∗Π(t),
F{∧(t)} = F{Π(t)}F{Π(t)}
= sinc2( ω
2π
)
SM 120
EECE 359 - Signals and Communications: Part 1 Spring 2014
6. Differentiation & Integration – Section 4.3.4
If x(t)↔ X(jω), then
d
dtx(t)↔ jωX(jω) .
Proof:
This result indicates that differentiation
accentuates the high frequency components
in the signal.
Application: Consider y(t) as shown below
t
y t( )
1
1
1– 0
SM 121
EECE 359 - Signals and Communications: Part 1 Spring 2014
Since y(t) = ddt∧ (t), we have
F{y(t)} = jω F{∧(t)}
= jω sinc2( ω
2π
)
If x(t)↔ X(jω), then
∫ t
−∞
x(τ)dτ ↔1
jωX (jω) + πX(0)δ(ω) .
We see that in contrast to differentiation,
integration attenuates the high frequency
components in the signal.
7. Area property
If x(t)↔ X(jω), then
∫ ∞
−∞
x(t)dt = X (0) .
SM 122
EECE 359 - Signals and Communications: Part 1 Spring 2014
This result follows directly from the
definition of the Fourier transform, i.e.
X(jω) =
∫ ∞
−∞
x(t)e−jωt dt.
Also,∫ ∞
−∞
X(jω)dω = 2π x (0)
since
x(t) =1
2π
∫ ∞
−∞
X(jω)ejωt dω.
Example: Evaluate∫∞
−∞sinc(x)dx.
This can be obtained as follows:
SM 123
EECE 359 - Signals and Communications: Part 1 Spring 2014
8. Duality
Due to the similarity between the FT
analysis and synthesis equations, we have a
“duality” relationship.
If x(t)↔ X(jω), then
X (jt)↔ 2πx(−ω) .
Proof:
SM 124
EECE 359 - Signals and Communications: Part 1 Spring 2014
Example: Determine the inverse FT of
Π(ω).
Since
Π(t)↔ sinc( ω
2π
)
,
then
sinc
(
t
2π
)
↔ 2πΠ(−ω) = 2πΠ(ω)
or equivalently
1
2πsinc
(
t
2π
)
↔ Π(ω).
Duality can also be useful in suggesting
new properties of the FT.
SM 125
EECE 359 - Signals and Communications: Part 1 Spring 2014
Example:
9. Parseval’s relation
If x(t)↔ X(jω), then
∫ ∞
−∞
|x(t)|2 dt =1
2π
∫ ∞
−∞
|X(jω)|2 dω .
Remarks:
(a) The LHS is the total energy in the signal
x(t).
(b) |X(jω)|2 describes how the energy in
x(t) is distributed as a function of
frequency. It is commonly called the
energy density spectrum of x(t).
SM 126
EECE 359 - Signals and Communications: Part 1 Spring 2014
Proof:
Example: Evaluate∫∞
−∞sinc2x dx.
SM 127
EECE 359 - Signals and Communications: Part 1 Spring 2014
CT unit impulse function – pp. 32–38,
pp. 92-93
The CT unit impulse function is also known
as the Dirac delta function, δ(t). Contrast
with the Kronecker delta function, δij. The
introduction of δ(t) allows us to look at the
FT of periodic signals.
δ(t) is defined by the sifting property, namely
∫ ∞
−∞
δ(t)f(t) dt = f(0)
if f(t) is continuous at t = 0.
Properties of the Dirac delta function
1.
∫ ∞
−∞
δ(t) dt = 1
SM 128
EECE 359 - Signals and Communications: Part 1 Spring 2014
t
1
0
δ t( )represents the strength of
the impulse function
2.
∫ ∞
−∞
δ(t− t0)f(t) dt =
∫ ∞
−∞
δ(τ)f(τ + t0) dτ
= f(t0)
The first line is obtained using τ = t− t0.
3.
f(t) ∗ δ(t) =
∫ ∞
−∞
δ(τ)f(t− τ) dτ
= f(t)
The above is referred to as the replication
property of δ(t).
SM 129
EECE 359 - Signals and Communications: Part 1 Spring 2014
4.
δ(at) =1
|a|δ(t)
Proof:
5. What is the Fourier transform of δ(t)?
F{δ(t)} =
SM 130
EECE 359 - Signals and Communications: Part 1 Spring 2014
6. What is the inverse FT of δ(ω)?
F−1{δ(ω)}∆=
1
2π
∫ ∞
−∞
δ(ω)ejωt dω
=1
2π
Thus, 1↔ 2πδ(ω).
7. It follows from the last result that∫ ∞
−∞
e−jωt dt = 2πδ(ω)
Therefore,
2πδ(ω − ωc) =
∫ ∞
−∞
e−j(ω−ωc)t dt
=
∫ ∞
−∞
ejωcte−jωt dt
= F{
ejωct}
SM 131
EECE 359 - Signals and Communications: Part 1 Spring 2014
We thus have ejωct ↔ 2πδ(ω − ωc). This
result is useful in examining the FT of a
periodic signal.
8. Using the last result, we can write
cosωct =1
2
[
ejωct + e−jωct]
↔1
2[2πδ(ω − ωc) + 2πδ(ω + ωc)]
= πδ(ω − ωc) + πδ(ω + ωc)
0
� ωctcos{ }
ω–c ω
c
π
ω
SM 132
EECE 359 - Signals and Communications: Part 1 Spring 2014
Similarly, we have
sinωct =1
2j
[
ejωct − e−jωct]
↔π
j[δ(ω − ωc)− δ(ω + ωc)]
0
� ωc
sin t{ }
ω–c
ωc
π
j---
π
j---–
ω
SM 133
EECE 359 - Signals and Communications: Part 1 Spring 2014
Fourier transform of periodic signals –
Section 4.2
Recall from our discussion of FS
representation: A periodic signal, x̃(t), with
fundamental period T can be represented by
x̃(t) =∞∑
k=−∞
ak ejk(2πT )t
where the (possibly complex) Fourier
coefficients {ak} are given by
ak =1
T
∫ T2
−T2
x̃(t) e−jk(2πT )t dt,
k = 0,±1,±2, . . .
Let x(t) =
{
x̃(t), −T2 ≤ t ≤ T
2
0, otherwise.
SM 134
EECE 359 - Signals and Communications: Part 1 Spring 2014
Thus, x(t) is simply one basic period of x̃(t).
Then, with ω0∆= 2π
T, we can write
ak =1
T
∫ ∞
−∞
x(t) e−jkω0t dt
=1
TX (jkω0) .
The last line follows since
X(jω) =
∫ ∞
−∞
x(t)e−jωt dt.
In other words, ak is equal to 1Tmultiplied by
the FT of x(t) evaluated at ω = kω0.
Therefore,
x̃(t) =
∞∑
k=−∞
1
TX (jkω0) ejkω0t
↔1
T
∞∑
k=−∞
X (jkω0) 2πδ (ω − kω0)
SM 135
EECE 359 - Signals and Communications: Part 1 Spring 2014
In summary, the FT of a periodic signal
consists of a series of impulses located
at frequencies which are multiples of the
fundamental frequency ω0. The strength of
the impulse at the kth harmonic frequency
kω0 is 2πak.
Example: What is the FT of the impulse train
or comb function?
Recall that the comb function is given by
δT (t) =∞∑
k=−∞
δ(t− kT )
and is shown in the figure below.
t
1
0
δTt( )
T 2T 3TT–2T–3T–
SM 136
EECE 359 - Signals and Communications: Part 1 Spring 2014
In this example, “x(t)” = δ(t) so that
X(jω) = 1.
Therefore,
δT (t)↔2π
T
∞∑
k=−∞
δ
(
ω − k2π
T
)
.
2π
T------
0
� δTt( ){ }
ω2π
T------
4π
T------
6π
T------2π
T------–
4π
T------–
6π
T------–
We will see that the comb function is very
useful in discussing sampling .
SM 137
top related