5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) 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 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) 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] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 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 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}
5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the following signals in terms of X(ejw). You may use the Fourier transform properties listed in Table 5.1. (a) x₁[n] = x[1 − n] + x[−1 – n] (b) x2[n] = x*[−n]+x[n] - 2 (c) x3[n] = (n − 1)² x[n] TABLE 5.1 Section PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM Property Aperiodic Signal x*[n] x[−n] Fourier Transform X(e) periodic with Y(ej) period 2π aX(ej)+bY(ej) e- juno X (ejw) X(ej(w-w₁)) X* (e¯jw) X(e jw) 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 5.3.6 Time Reversal 5.3.7 Time Expansion X(k)[n] = { x[n/k], 0, if n = multiple of k if n X(ejkw) 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] n 5.3.5 Accumulation Σ x[k] k = -x ·X(ejw) 2πT 2TT (1 - e-ju)X(ejw) 1 e-jw +x ')de +πX(e³) Σ (w - 2πk) dX(ejw) k=-x 5.3.8 Differentiation in Frequency nx[n] ¡ dw 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 5.3.9 of Real Signals Parseval's Relation for Aperiodic Signals +x x,[n] = x,[n] = &{x[n]} [x[n] real] Od{x[n]} [x[n] real] Σ|x[n] == 2π 12T 2 | | _ \ X ( e ³ ³ ³ d w 11=-0 X(eju ) = X*(e-in) (e¯jw) Re{X(e)} = Re{X(e¯jw)} Im{X(e)} = −Im{X(e¯jw)} |X(e)| = |X(e¯jw)| XX(ej) = -*X(e¯jw) X(ej) real and even X(e) purely imaginary and odd Re{X(ej")} jIm{X(e³w)}
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
Related questions
Question
100%
This is a practice question from the Signal and Systems course of my Electrical Engineering Program.
Could you please walk me through the steps for solving this?
Thank you for your assistance.
![5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the
following signals in terms of X(ejw). You may use the Fourier transform properties
listed in Table 5.1.
(a) x₁[n] = x[1 − n] + x[−1 – n]
(b) x2[n]
=
x*[−n]+x[n]
-
2
(c) x3[n] = (n − 1)² x[n]
TABLE 5.1
Section
PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM
Property
Aperiodic Signal
x*[n]
x[−n]
Fourier Transform
X(e) periodic with
Y(ej) period 2π
aX(ej)+bY(ej)
e- juno X (ejw)
X(ej(w-w₁))
X* (e¯jw)
X(e jw)
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
5.3.6
Time Reversal
5.3.7
Time Expansion
X(k)[n]
=
{
x[n/k],
0,
if n = multiple of k
if n
X(ejkw)
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]
n
5.3.5
Accumulation
Σ x[k]
k = -x
·X(ejw)
2πT 2TT
(1 - e-ju)X(ejw)
1
e-jw
+x
')de
+πX(e³) Σ (w - 2πk)
dX(ejw)
k=-x
5.3.8
Differentiation in Frequency nx[n]
¡
dw
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
5.3.9
of Real Signals
Parseval's Relation for Aperiodic Signals
+x
x,[n]
=
x,[n] = &{x[n]} [x[n] real]
Od{x[n]} [x[n] real]
Σ|x[n]
==
2π 12T
2 | | _ \ X ( e ³ ³ ³ d w
11=-0
X(eju ) = X*(e-in)
(e¯jw)
Re{X(e)} = Re{X(e¯jw)}
Im{X(e)} = −Im{X(e¯jw)}
|X(e)| = |X(e¯jw)|
XX(ej) = -*X(e¯jw)
X(ej) real and even
X(e) purely imaginary and
odd
Re{X(ej")}
jIm{X(e³w)}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3b9e8ebd-ceba-4fff-be05-4ffc155e08e3%2F75c8e6be-2c1e-4985-8c9f-aa8277606bbf%2F7ul066x_processed.png&w=3840&q=75)
Transcribed Image Text:5.6. Given that x[n] has Fourier transform X(ejw), express the Fourier transforms of the
following signals in terms of X(ejw). You may use the Fourier transform properties
listed in Table 5.1.
(a) x₁[n] = x[1 − n] + x[−1 – n]
(b) x2[n]
=
x*[−n]+x[n]
-
2
(c) x3[n] = (n − 1)² x[n]
TABLE 5.1
Section
PROPERTIES OF THE DISCRETE-TIME FOURIER TRANSFORM
Property
Aperiodic Signal
x*[n]
x[−n]
Fourier Transform
X(e) periodic with
Y(ej) period 2π
aX(ej)+bY(ej)
e- juno X (ejw)
X(ej(w-w₁))
X* (e¯jw)
X(e jw)
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
5.3.6
Time Reversal
5.3.7
Time Expansion
X(k)[n]
=
{
x[n/k],
0,
if n = multiple of k
if n
X(ejkw)
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]
n
5.3.5
Accumulation
Σ x[k]
k = -x
·X(ejw)
2πT 2TT
(1 - e-ju)X(ejw)
1
e-jw
+x
')de
+πX(e³) Σ (w - 2πk)
dX(ejw)
k=-x
5.3.8
Differentiation in Frequency nx[n]
¡
dw
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
5.3.9
of Real Signals
Parseval's Relation for Aperiodic Signals
+x
x,[n]
=
x,[n] = &{x[n]} [x[n] real]
Od{x[n]} [x[n] real]
Σ|x[n]
==
2π 12T
2 | | _ \ X ( e ³ ³ ³ d w
11=-0
X(eju ) = X*(e-in)
(e¯jw)
Re{X(e)} = Re{X(e¯jw)}
Im{X(e)} = −Im{X(e¯jw)}
|X(e)| = |X(e¯jw)|
XX(ej) = -*X(e¯jw)
X(ej) real and even
X(e) purely imaginary and
odd
Re{X(ej")}
jIm{X(e³w)}
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,