4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t (1)x rag TTt = t ole in the (b) Use Parseval's relation and the result of the previous part to determine the nu merical value of sin t dt A = TTt
4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following signal: sin t (1)x rag TTt = t ole in the (b) Use Parseval's relation and the result of the previous part to determine the nu merical value of sin t dt A = TTt
Introductory Circuit Analysis (13th Edition)
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ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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(a) Comes out ot this but new help getting there
?(??)= 1/2? { ?,−2<? <0
−?, 0<? <2
0,|?|>2
(b) comes out to ?= 1/(2?^2) but new help with that too
Transcribed Image Text:4.10. (a) Use Tables 4.1 and 4.2 to help determine the Fourier transform of the following
signal:
sin t
(1)x drao
(b) Use Parseval's relation and the result of the previous part to determine the nu-
merical value of
m We
dt
sin t
A
2oulay or
TTt
![In Table 4.2, we have assembled a list of many of the basic and important Foure
ered some of the important properties of the Fourier transform. These are summarized
transform pairs. We will encounter many of these repeatedly as we apply the tools
In the preceding sections and in the problems at the end of the chapter, we have consid
Table 4.1, in which we have also indicated the section of this chapter in which each p
4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS
The Continuous-Time Fourier Transform
The Continuous-Time Fourier Transform
Tables of Fourier Properties and of Basic Fourier Transform PairsEE
Chap
ol Sec. 4.6
328
329
TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS
Signal
Fourier transform
Fourier series coefficients
(if periodic)
prop
E ai8(w - koro)
erty has been discussed.
apply the tools
transform pairs. We will encounter many of these repeatedly as we
el
2m8(w - wo)
a, = 1
a, - 0, otherwise
TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM
Aperiodic signal
a la in
(8(w - w) + 8(a + we))
Fourier transform
cos wof
di = a-
Property
ouy ao Section
a, = 0, otherwise
i
agband lasbT
X(jw)
Yja)
()
sin wof
(8( - wo) - 8(w + ao)]
ai = -a-=
st le vansipait
a, - 0, otherwise
---
axt) + by(t)
x(t - to)
d = 1, a, - 0, k0
(this is the Fourier series representation for
any choice of T >0
aX(ja) + bY(jw)
e Julo X(jw)
X(J(@ - wo))
X'(- ja)
X(- ju)
1v ja
laa
Lincarity
Time Shifting
4.3.1
4.3.2
4.3.6
4.3.3
4.3.5
x(t) = 1
2n 6(w)
Frequency Shifting
Conjugation
Time Reversal
x'(r)
x(-1)
Periodic square wave
1, <T,
x(1) =
0. T, < |r| s }
* 2 sin kwgT
kanT
sinc
4.3.5
Time and Frequency
Scaling
x(at)
-8(w - kan)
sin kwT
and
km
4.4
X(jo)Y(ja)
Convolution
x(1) • y()
x(t + T) = x(t)
b 4.5
Multiplication
(1)y(t)
Differentiation in Time x(1)
dt
* 81 - nT)
2nk
a7 for llk
4.3.4
jwX(jw)
all k
A1maseve a 1
4.3.4
Integration
( 1, e|< T,
x(t)
0, >T
x()dr
ia Xjw) + TX(0)8(a)
2 sin wT,is
bos su 4.3.6
Differentiation in
Frequency
baa d
j X(jw)
dw
- ba
tx(1)
sin Wt
(1, l < W
(o. k) > W
reted astal
l labloun X(jw) = X'(- ju)
X(ja) =
RefX(j@)} = Re(X(- jo)
Im(X(jw)} = -Sm{X(= joll
X(j) = |X(-jaw)
*X(jw) = -X(-jas)
X(jw) real and even
pleasi T n
x(t) real
4.3.3
Conjugate Symmetry
for Real Signals
8(1)
%3D
+ 8(w)
ja
u(t)
4.3.3
Symmetry for Real and
Even Signals
Symmetry for Real and
Odd Signals
x(t) real and even
4.3.3
8(t - to)
x(f) real and odd
A e PEA
54.3.3
X(jw) purely imaginary and od
1
Even-Odd Decompo-
x,() = Evfx(1)) [x(t) real)
RefX(jw)
e"u(t), Refa} >0
a+ ja
sition for Real Sig- x,(1) = Od{x(1)} [x(t) real] jám(X(jw)}
nals
te u(t), Refa} >0
(a + jw)
-1e"u(t),
Refa) > 0
4.3.7
Parseval's Relation for Aperiodic Signals
(a + ja)
2m](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F484cbb4e-45b6-4e6e-84af-00adb2efd343%2Fba79a323-eeec-4132-87e0-998b6b3b8e39%2Futi8ac6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Table 4.2, we have assembled a list of many of the basic and important Foure
ered some of the important properties of the Fourier transform. These are summarized
transform pairs. We will encounter many of these repeatedly as we apply the tools
In the preceding sections and in the problems at the end of the chapter, we have consid
Table 4.1, in which we have also indicated the section of this chapter in which each p
4.6 TABLES OF FOURIER PROPERTIES AND OF BASIC FOURIER TRANSFORM PAIRS
The Continuous-Time Fourier Transform
The Continuous-Time Fourier Transform
Tables of Fourier Properties and of Basic Fourier Transform PairsEE
Chap
ol Sec. 4.6
328
329
TABLE 4.2 BASIC FOURIER TRANSFORM PAIRS
Signal
Fourier transform
Fourier series coefficients
(if periodic)
prop
E ai8(w - koro)
erty has been discussed.
apply the tools
transform pairs. We will encounter many of these repeatedly as we
el
2m8(w - wo)
a, = 1
a, - 0, otherwise
TABLE 4.1 PROPERTIES OF THE FOURIER TRANSFORM
Aperiodic signal
a la in
(8(w - w) + 8(a + we))
Fourier transform
cos wof
di = a-
Property
ouy ao Section
a, = 0, otherwise
i
agband lasbT
X(jw)
Yja)
()
sin wof
(8( - wo) - 8(w + ao)]
ai = -a-=
st le vansipait
a, - 0, otherwise
---
axt) + by(t)
x(t - to)
d = 1, a, - 0, k0
(this is the Fourier series representation for
any choice of T >0
aX(ja) + bY(jw)
e Julo X(jw)
X(J(@ - wo))
X'(- ja)
X(- ju)
1v ja
laa
Lincarity
Time Shifting
4.3.1
4.3.2
4.3.6
4.3.3
4.3.5
x(t) = 1
2n 6(w)
Frequency Shifting
Conjugation
Time Reversal
x'(r)
x(-1)
Periodic square wave
1, <T,
x(1) =
0. T, < |r| s }
* 2 sin kwgT
kanT
sinc
4.3.5
Time and Frequency
Scaling
x(at)
-8(w - kan)
sin kwT
and
km
4.4
X(jo)Y(ja)
Convolution
x(1) • y()
x(t + T) = x(t)
b 4.5
Multiplication
(1)y(t)
Differentiation in Time x(1)
dt
* 81 - nT)
2nk
a7 for llk
4.3.4
jwX(jw)
all k
A1maseve a 1
4.3.4
Integration
( 1, e|< T,
x(t)
0, >T
x()dr
ia Xjw) + TX(0)8(a)
2 sin wT,is
bos su 4.3.6
Differentiation in
Frequency
baa d
j X(jw)
dw
- ba
tx(1)
sin Wt
(1, l < W
(o. k) > W
reted astal
l labloun X(jw) = X'(- ju)
X(ja) =
RefX(j@)} = Re(X(- jo)
Im(X(jw)} = -Sm{X(= joll
X(j) = |X(-jaw)
*X(jw) = -X(-jas)
X(jw) real and even
pleasi T n
x(t) real
4.3.3
Conjugate Symmetry
for Real Signals
8(1)
%3D
+ 8(w)
ja
u(t)
4.3.3
Symmetry for Real and
Even Signals
Symmetry for Real and
Odd Signals
x(t) real and even
4.3.3
8(t - to)
x(f) real and odd
A e PEA
54.3.3
X(jw) purely imaginary and od
1
Even-Odd Decompo-
x,() = Evfx(1)) [x(t) real)
RefX(jw)
e"u(t), Refa} >0
a+ ja
sition for Real Sig- x,(1) = Od{x(1)} [x(t) real] jám(X(jw)}
nals
te u(t), Refa} >0
(a + jw)
-1e"u(t),
Refa) > 0
4.3.7
Parseval's Relation for Aperiodic Signals
(a + ja)
2m
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