4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t = t (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of A 2oulay or sin t dt TTt
4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t = t (b) Use Parseval's relation and the result of the previous part to determine the nu- merical value of A 2oulay or sin t dt TTt
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Author:Robert L. Boylestad
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Transcribed Image Text:4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following
4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the followi.
signal:
2
sin t
udeupe zlearlid/2 x(t) = t
TTt
(b) Use Parseval's relation and the result of the previous part to determine the nu-
merical value of
4.
A =
pulay or on
sin t
dt
oy se
Tt
![In the preceding sections and in the problems at the end of the chapter, we have o
4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS
D s vellmie In Table 4.2, we have assembled a list of many of the basic and important Fourier
ered some of the important properties of the Fourier transform. These are summarized in
transform pairs. We will encounter many of these repeatedly as we apply the tools of
Table 4.1, in which we have also indicated the section of this chapter in which each prop
eg01 The Continuous-Time Fourier Transform
Tables of Fourier Properties and of Basic Fourier Transform Pairs EE
mmot Sec. 4.6
328
Chap.4
SE
329
hvol od tedi wollct
BASIC FOURIER TRANSFORM PAIRS
r2o molens
libbs nl
Signal
consid
Fourier series coefficients
Fourier transform
(if periodic)
Z akeikwor
27 2 ar8(w - kwo)
on ( erty has been discussed.
k=-∞
k=-0
ak
as we apply the tools of
transform pairs. We will encounter many of these repeatedly
2T8(w – wo)
aj = 1
ak = 0, otherwise
TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM
Fourier transform
T[8(w – wo) +8(w + wo)]
aj = a-j =
Aperiodic signaloh
cos wot
SAig9mo Section
sgbaed lsabr
odj Property
ak = 0, otherwise
x(t)
y(t)
X(jw)
Y(jw)
porl
sin wot
[8(@ – wo) – 8(w + wo)]
aj = -a-j =
gino sif lo
ak = 0, otherwise
at 1o yageupant a boisv
Linearity
Time Shifting
Frequency Shifting
Conjugation
Time Reversal
ax(t) + by(t)
x(t – to)
ejao! x(t)
aX(jw) + bY(jw)
- julo X(jw)
ao = 1, a = 0, k# 0
this is the Fourier series representation for
any choice of T >0
4.3.1
x(t) = 1
2m 8(w)
%3D
e
X(j(@ – wo))
X(-jw)
X(-jw)
ja
4.3.2
4.3.6
sdt tu omua
x'(t)
x(-t)
4.3.3
Periodic square wave
1, |2| < T,
0, T, < \e| < }
4.3.5
too
2 sin kwoT,
Time and Frequency
Scaling
Convolution
(kwoT
sinc
T
4.3.5
x(at)
8(@ – kwo)
sin kwoT,
-
k
and
k=-∞
TT
4.4
x(t) * y(t)
X(jw)Y(jw)
x(t + T) = x(t)
ntt thobiaod
gailuffiedu?
9adbeis 2 8(t – nT)
wup 1sl easqans 4.5
Multiplication
x(t)y(t)
X(je)Y(j(@ – 0))do
2Tk
-8
4.3.4
Differentiation in Time
jwX(jw)
bo
for all k
T
ak
T
k=-∞
T
n= -∞
ai nwoda 1sdr
4.3.4
Integration
x(t)dt
X(jw) + mX(0)8(@)
2 sin wT,
hitlor ow jw
a bus x(jw)
gorg frotulovm | 0, > T,
x(t)-
541
4.3.6
Differentiation in
d
racod ela al
mataya ad S
Da da
X(ja) = X(-jw)
Frequency
sin Wt
1, l이 < W
X(jo) =0. lo > W
Tt
giRe{X(j@)} = Re{X(-ju)}
Im{X(j@)} = -Ia{X(-ju)
|X(j@)| = |X(- j@)I
*X(jw) = -*X(-jw)
X(jw) real and even
bonoles ei aidT
x(t) real
4.3.3
Conjugate Symmetry
for Real Signals
8(t)
1
1
+ T8(@)
jw
u(t)
4.3.3
Symmetry for Real and x(t) real and even
Even Signals
Symmetry for Real and
Odd Signals
ot ou
4.3.3
8(t- to)
e juio
x(t) real and odd
L S 4.3.3
Even-Odd Decompo-
x() = Ev{x(1)} [x(1) real]
e al u(t), Re{a} > 0
1
%3D
[x(t) real]
Re{X(jw)}
jIm{X(j@)}
a + jw
sition for Real Sig- *() = O{x(t)} [x(t) real]
nals
1
te du(t), Re{a} > 0
-at
(a + jw)?
4.3.7
Parseval's Relation for Aperiodic Signals
(n-1)!
e-at u(t),
1
Refa} > 0
(a + jw)"
dt =
|X(j@)fdw](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F484cbb4e-45b6-4e6e-84af-00adb2efd343%2F6c430aa8-7b36-4ad4-9aa3-67a75fb8d249%2Fm3i5ze_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In the preceding sections and in the problems at the end of the chapter, we have o
4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS
D s vellmie In Table 4.2, we have assembled a list of many of the basic and important Fourier
ered some of the important properties of the Fourier transform. These are summarized in
transform pairs. We will encounter many of these repeatedly as we apply the tools of
Table 4.1, in which we have also indicated the section of this chapter in which each prop
eg01 The Continuous-Time Fourier Transform
Tables of Fourier Properties and of Basic Fourier Transform Pairs EE
mmot Sec. 4.6
328
Chap.4
SE
329
hvol od tedi wollct
BASIC FOURIER TRANSFORM PAIRS
r2o molens
libbs nl
Signal
consid
Fourier series coefficients
Fourier transform
(if periodic)
Z akeikwor
27 2 ar8(w - kwo)
on ( erty has been discussed.
k=-∞
k=-0
ak
as we apply the tools of
transform pairs. We will encounter many of these repeatedly
2T8(w – wo)
aj = 1
ak = 0, otherwise
TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM
Fourier transform
T[8(w – wo) +8(w + wo)]
aj = a-j =
Aperiodic signaloh
cos wot
SAig9mo Section
sgbaed lsabr
odj Property
ak = 0, otherwise
x(t)
y(t)
X(jw)
Y(jw)
porl
sin wot
[8(@ – wo) – 8(w + wo)]
aj = -a-j =
gino sif lo
ak = 0, otherwise
at 1o yageupant a boisv
Linearity
Time Shifting
Frequency Shifting
Conjugation
Time Reversal
ax(t) + by(t)
x(t – to)
ejao! x(t)
aX(jw) + bY(jw)
- julo X(jw)
ao = 1, a = 0, k# 0
this is the Fourier series representation for
any choice of T >0
4.3.1
x(t) = 1
2m 8(w)
%3D
e
X(j(@ – wo))
X(-jw)
X(-jw)
ja
4.3.2
4.3.6
sdt tu omua
x'(t)
x(-t)
4.3.3
Periodic square wave
1, |2| < T,
0, T, < \e| < }
4.3.5
too
2 sin kwoT,
Time and Frequency
Scaling
Convolution
(kwoT
sinc
T
4.3.5
x(at)
8(@ – kwo)
sin kwoT,
-
k
and
k=-∞
TT
4.4
x(t) * y(t)
X(jw)Y(jw)
x(t + T) = x(t)
ntt thobiaod
gailuffiedu?
9adbeis 2 8(t – nT)
wup 1sl easqans 4.5
Multiplication
x(t)y(t)
X(je)Y(j(@ – 0))do
2Tk
-8
4.3.4
Differentiation in Time
jwX(jw)
bo
for all k
T
ak
T
k=-∞
T
n= -∞
ai nwoda 1sdr
4.3.4
Integration
x(t)dt
X(jw) + mX(0)8(@)
2 sin wT,
hitlor ow jw
a bus x(jw)
gorg frotulovm | 0, > T,
x(t)-
541
4.3.6
Differentiation in
d
racod ela al
mataya ad S
Da da
X(ja) = X(-jw)
Frequency
sin Wt
1, l이 < W
X(jo) =0. lo > W
Tt
giRe{X(j@)} = Re{X(-ju)}
Im{X(j@)} = -Ia{X(-ju)
|X(j@)| = |X(- j@)I
*X(jw) = -*X(-jw)
X(jw) real and even
bonoles ei aidT
x(t) real
4.3.3
Conjugate Symmetry
for Real Signals
8(t)
1
1
+ T8(@)
jw
u(t)
4.3.3
Symmetry for Real and x(t) real and even
Even Signals
Symmetry for Real and
Odd Signals
ot ou
4.3.3
8(t- to)
e juio
x(t) real and odd
L S 4.3.3
Even-Odd Decompo-
x() = Ev{x(1)} [x(1) real]
e al u(t), Re{a} > 0
1
%3D
[x(t) real]
Re{X(jw)}
jIm{X(j@)}
a + jw
sition for Real Sig- *() = O{x(t)} [x(t) real]
nals
1
te du(t), Re{a} > 0
-at
(a + jw)?
4.3.7
Parseval's Relation for Aperiodic Signals
(n-1)!
e-at u(t),
1
Refa} > 0
(a + jw)"
dt =
|X(j@)fdw
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