5.7. For each of the following Fourier transforms, use Fourier transform properties (Table 5.1) to determine whether the corresponding time-domain signal is (i) real, imagi- nary, or neither and (ii) even, odd, or neither. Do this without evaluating the inverse of any of the given transforms. (a) X₁(ejw) = e¯jw 10 = e¯jw(sin kw) = (b) X2(ej) = j sin(w) cos(5w) (c) X3(ej) = A(w) + ej B(w) where A(w) =========== 3w || < |W| ≤ πT and B(w) === +. 2 PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Aperiodic Signal TABLE 5.1 Section Property Fourier Transform x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n-no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation x*[n] X(e) periodic with Y(ej) period 2π aX(e)+bY(ej) e-jono X(ejw) X(ej(w-wp)) X* (e¯jw) 5.3.6 Time Reversal x[−n] X(e jw) 5.3.7 Time Expansion X(k)[n] = { x[n/k], if n = multiple of k X(ejkw) 0, if n multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n 1] - 2πT 2πT (1 - e¯jw)X(ejw) |_ Xe³ Ye³ (0) 5.3.5 Accumulation n Σ *[k] 1 1 e-jw ·X(ejw) k = -x 5.3.8 Differentiation in Frequency nx[n] 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition of Real Signals x [n] = &{x[n]} [x[n] real] x,[n] = Od{x[n]} [x[n] real] 5.3.9 Parseval's Relation for Aperiodic Signals +x Σ|x[n]? 11=-0 1 = 2π 12T +x + TX (e³) (w – 2πk) ·dX (ejw) k = -x dw X(eju ) = X*(ei) (e¯jw) Che{X(ei )} = Che{X(e-j)} Im{X(e)} = -Im{X(e¯jw)} == |X(ejw)| = |X(e¯jw)|
5.7. For each of the following Fourier transforms, use Fourier transform properties (Table 5.1) to determine whether the corresponding time-domain signal is (i) real, imagi- nary, or neither and (ii) even, odd, or neither. Do this without evaluating the inverse of any of the given transforms. (a) X₁(ejw) = e¯jw 10 = e¯jw(sin kw) = (b) X2(ej) = j sin(w) cos(5w) (c) X3(ej) = A(w) + ej B(w) where A(w) =========== 3w || < |W| ≤ πT and B(w) === +. 2 PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Aperiodic Signal TABLE 5.1 Section Property Fourier Transform x[n] y[n] 5.3.2 Linearity ax[n] + by[n] 5.3.3 Time Shifting x[n-no] 5.3.3 Frequency Shifting ejwon x[n] 5.3.4 Conjugation x*[n] X(e) periodic with Y(ej) period 2π aX(e)+bY(ej) e-jono X(ejw) X(ej(w-wp)) X* (e¯jw) 5.3.6 Time Reversal x[−n] X(e jw) 5.3.7 Time Expansion X(k)[n] = { x[n/k], if n = multiple of k X(ejkw) 0, if n multiple of k 5.4 Convolution 5.5 Multiplication x[n] * y[n] x[n]y[n] X(ejw)Y(ejw) 1 5.3.5 Differencing in Time - x[n] x[n 1] - 2πT 2πT (1 - e¯jw)X(ejw) |_ Xe³ Ye³ (0) 5.3.5 Accumulation n Σ *[k] 1 1 e-jw ·X(ejw) k = -x 5.3.8 Differentiation in Frequency nx[n] 5.3.4 Conjugate Symmetry for Real Signals x[n] real 5.3.4 Symmetry for Real, Even Signals x[n] real an even 5.3.4 Symmetry for Real, Odd Signals x[n] real and odd 5.3.4 Even-odd Decomposition of Real Signals x [n] = &{x[n]} [x[n] real] x,[n] = Od{x[n]} [x[n] real] 5.3.9 Parseval's Relation for Aperiodic Signals +x Σ|x[n]? 11=-0 1 = 2π 12T +x + TX (e³) (w – 2πk) ·dX (ejw) k = -x dw X(eju ) = X*(ei) (e¯jw) Che{X(ei )} = Che{X(e-j)} Im{X(e)} = -Im{X(e¯jw)} == |X(ejw)| = |X(e¯jw)|
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This is a practice question from the Signal and Systems course of my Electrical Engineering Program.
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![5.7. For each of the following Fourier transforms, use Fourier transform properties (Table
5.1) to determine whether the corresponding time-domain signal is (i) real, imagi-
nary, or neither and (ii) even, odd, or neither. Do this without evaluating the inverse
of any of the given transforms.
(a) X₁(ejw) = e¯jw 10
= e¯jw(sin kw)
=
(b) X2(ej) = j sin(w) cos(5w)
(c) X3(ej) = A(w) + ej B(w) where
A(w)
===========
3w
|| < |W| ≤ πT
and B(w)
===
+.
2
PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM
Aperiodic Signal
TABLE 5.1
Section
Property
Fourier Transform
x[n]
y[n]
5.3.2
Linearity
ax[n] + by[n]
5.3.3
Time Shifting
x[n-no]
5.3.3
Frequency Shifting
ejwon x[n]
5.3.4
Conjugation
x*[n]
X(e) periodic with
Y(ej) period 2π
aX(e)+bY(ej)
e-jono X(ejw)
X(ej(w-wp))
X* (e¯jw)
5.3.6
Time Reversal
x[−n]
X(e jw)
5.3.7
Time Expansion
X(k)[n] =
{
x[n/k], if n = multiple of k
X(ejkw)
0,
if n
multiple of k
5.4
Convolution
5.5
Multiplication
x[n] * y[n]
x[n]y[n]
X(ejw)Y(ejw)
1
5.3.5
Differencing in Time
-
x[n] x[n 1]
-
2πT 2πT
(1 - e¯jw)X(ejw)
|_ Xe³ Ye³ (0)
5.3.5
Accumulation
n
Σ *[k]
1
1
e-jw
·X(ejw)
k = -x
5.3.8
Differentiation in Frequency
nx[n]
5.3.4
Conjugate Symmetry for
Real Signals
x[n] real
5.3.4
Symmetry for Real, Even
Signals
x[n] real an even
5.3.4
Symmetry for Real, Odd
Signals
x[n] real and odd
5.3.4
Even-odd Decomposition
of Real Signals
x [n] = &{x[n]} [x[n] real]
x,[n] = Od{x[n]} [x[n] real]
5.3.9
Parseval's Relation for Aperiodic Signals
+x
Σ|x[n]?
11=-0
1
=
2π 12T
+x
+ TX (e³) (w – 2πk)
·dX (ejw)
k = -x
dw
X(eju ) = X*(ei)
(e¯jw)
Che{X(ei )} = Che{X(e-j)}
Im{X(e)} = -Im{X(e¯jw)}
==
|X(ejw)| = |X(e¯jw)|
<X(e) = -XX(e-j)
XX(ejw)
X(e) real and even
X(e) purely imaginary and
odd
Re{X(ej")}
jIm{X(eja)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b9e8ebd-ceba-4fff-be05-4ffc155e08e3%2Fbac4e48e-f10c-4e99-9b4c-1d0891aaa9de%2F133d9kj_processed.png&w=3840&q=75)
Transcribed Image Text:5.7. For each of the following Fourier transforms, use Fourier transform properties (Table
5.1) to determine whether the corresponding time-domain signal is (i) real, imagi-
nary, or neither and (ii) even, odd, or neither. Do this without evaluating the inverse
of any of the given transforms.
(a) X₁(ejw) = e¯jw 10
= e¯jw(sin kw)
=
(b) X2(ej) = j sin(w) cos(5w)
(c) X3(ej) = A(w) + ej B(w) where
A(w)
===========
3w
|| < |W| ≤ πT
and B(w)
===
+.
2
PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM
Aperiodic Signal
TABLE 5.1
Section
Property
Fourier Transform
x[n]
y[n]
5.3.2
Linearity
ax[n] + by[n]
5.3.3
Time Shifting
x[n-no]
5.3.3
Frequency Shifting
ejwon x[n]
5.3.4
Conjugation
x*[n]
X(e) periodic with
Y(ej) period 2π
aX(e)+bY(ej)
e-jono X(ejw)
X(ej(w-wp))
X* (e¯jw)
5.3.6
Time Reversal
x[−n]
X(e jw)
5.3.7
Time Expansion
X(k)[n] =
{
x[n/k], if n = multiple of k
X(ejkw)
0,
if n
multiple of k
5.4
Convolution
5.5
Multiplication
x[n] * y[n]
x[n]y[n]
X(ejw)Y(ejw)
1
5.3.5
Differencing in Time
-
x[n] x[n 1]
-
2πT 2πT
(1 - e¯jw)X(ejw)
|_ Xe³ Ye³ (0)
5.3.5
Accumulation
n
Σ *[k]
1
1
e-jw
·X(ejw)
k = -x
5.3.8
Differentiation in Frequency
nx[n]
5.3.4
Conjugate Symmetry for
Real Signals
x[n] real
5.3.4
Symmetry for Real, Even
Signals
x[n] real an even
5.3.4
Symmetry for Real, Odd
Signals
x[n] real and odd
5.3.4
Even-odd Decomposition
of Real Signals
x [n] = &{x[n]} [x[n] real]
x,[n] = Od{x[n]} [x[n] real]
5.3.9
Parseval's Relation for Aperiodic Signals
+x
Σ|x[n]?
11=-0
1
=
2π 12T
+x
+ TX (e³) (w – 2πk)
·dX (ejw)
k = -x
dw
X(eju ) = X*(ei)
(e¯jw)
Che{X(ei )} = Che{X(e-j)}
Im{X(e)} = -Im{X(e¯jw)}
==
|X(ejw)| = |X(e¯jw)|
<X(e) = -XX(e-j)
XX(ejw)
X(e) real and even
X(e) purely imaginary and
odd
Re{X(ej")}
jIm{X(eja)}
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