Homework Help is Here – Start Your Trial Now!
Engineering
Electrical Engineering
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)|

Introductory Circuit Analysis (13th Edition)
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...
See similar textbooks
icon
Related questions
Question

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.

5.7. 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)}
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)}
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
Electrical Engineering
ISBN:
9780133923605
Author:
Robert L. Boylestad
Publisher:
PEARSON
Delmar's Standard Textbook Of Electricity
Delmar's Standard Textbook Of Electricity
Electrical Engineering
ISBN:
9781337900348
Author:
Stephen L. Herman
Publisher:
Cengage Learning
Programmable Logic Controllers
Programmable Logic Controllers
Electrical Engineering
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education
Fundamentals of Electric Circuits
Fundamentals of Electric Circuits
Electrical Engineering
ISBN:
9780078028229
Author:
Charles K Alexander, Matthew Sadiku
Publisher:
McGraw-Hill Education
Electric Circuits. (11th Edition)
Electric Circuits. (11th Edition)
Electrical Engineering
ISBN:
9780134746968
Author:
James W. Nilsson, Susan Riedel
Publisher:
PEARSON
Engineering Electromagnetics
Engineering Electromagnetics
Electrical Engineering
ISBN:
9780078028151
Author:
Hayt, William H. (william Hart), Jr, BUCK, John A.
Publisher:
Mcgraw-hill Education,
SEE MORE TEXTBOOKS