(structural properties of Fourier series): A real-valued signal x(t) has funda- mental period To and Fourier series coefficients {X[m]}. That is, x(t) = Σ X[m]e³²mt/To m=-∞ (a) A signal u(t) with period To has Fourier series coefficients U[m] = 2X[m](−j)m, m ‡ 0, and U[0] = 0. How is u(t) related to x(t)? Be as explicit as possible, and specify the relationship in its simplest possible form. (b) True or False The signal w(t) = x(t) − x(t – T) has no even harmonics. Give reasons for your answer. (c) Suppose that z(t) = x(t) − x(−t). Find the Fourier series coefficients {Z[m]} as a function of {X[m]}. Comment on any special features that you notice. = (d) True or False If {X[m]} are purely imaginary, then x(t) is anti-symmetric (i.e., x(t) = -x(-t)). Give reasons for your answer. (e) True or False If {X[m]} are purely real-valued, then x(t) is symmetric (i.e., x(t) = x(−t)). Give reasons for your answer.
(structural properties of Fourier series): A real-valued signal x(t) has funda- mental period To and Fourier series coefficients {X[m]}. That is, x(t) = Σ X[m]e³²mt/To m=-∞ (a) A signal u(t) with period To has Fourier series coefficients U[m] = 2X[m](−j)m, m ‡ 0, and U[0] = 0. How is u(t) related to x(t)? Be as explicit as possible, and specify the relationship in its simplest possible form. (b) True or False The signal w(t) = x(t) − x(t – T) has no even harmonics. Give reasons for your answer. (c) Suppose that z(t) = x(t) − x(−t). Find the Fourier series coefficients {Z[m]} as a function of {X[m]}. Comment on any special features that you notice. = (d) True or False If {X[m]} are purely imaginary, then x(t) is anti-symmetric (i.e., x(t) = -x(-t)). Give reasons for your answer. (e) True or False If {X[m]} are purely real-valued, then x(t) is symmetric (i.e., x(t) = x(−t)). Give reasons for your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(structural properties of Fourier series): real-valued signal x(t) has funda-
mental period To and Fourier series coefficients {X[m]}. That is,
x(t) = X[m]e127mt/To
X[m]e™n
Σ
m=-∞
(a) A signal u(t) with period To has Fourier series coefficients U[m] = 2X[m](−j)m, m ‡ 0, and
U[0] = 0. How is u(t) related to x(t)? Be as explicit as possible, and specify the relationship in
its simplest possible form.
(b) True or False The signal w(t) = x(t) − x(t – T) has no even harmonics. Give reasons for
2
your answer.
(c) Suppose that z(t) = x(t) − x(−t). Find the Fourier series coefficients {Z[m]} as a function
of {X[m]}. Comment on any special features that you notice.
(d) True or False If {X[m]} are purely imaginary, then x(t) is anti-symmetric (i.e., x(t)
-x(-t)). Give reasons for your answer.
(e) True or False If {X[m]} are purely real-valued, then x(t) is symmetric (i.e., x(t) = x(−t)).
Give reasons for your answer.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F584ffa08-94ad-4eb4-a87d-fd258db9f802%2F2cb9a763-c782-41aa-853a-f1089811130c%2F0qirnz4_processed.png&w=3840&q=75)
Transcribed Image Text:(structural properties of Fourier series): real-valued signal x(t) has funda-
mental period To and Fourier series coefficients {X[m]}. That is,
x(t) = X[m]e127mt/To
X[m]e™n
Σ
m=-∞
(a) A signal u(t) with period To has Fourier series coefficients U[m] = 2X[m](−j)m, m ‡ 0, and
U[0] = 0. How is u(t) related to x(t)? Be as explicit as possible, and specify the relationship in
its simplest possible form.
(b) True or False The signal w(t) = x(t) − x(t – T) has no even harmonics. Give reasons for
2
your answer.
(c) Suppose that z(t) = x(t) − x(−t). Find the Fourier series coefficients {Z[m]} as a function
of {X[m]}. Comment on any special features that you notice.
(d) True or False If {X[m]} are purely imaginary, then x(t) is anti-symmetric (i.e., x(t)
-x(-t)). Give reasons for your answer.
(e) True or False If {X[m]} are purely real-valued, then x(t) is symmetric (i.e., x(t) = x(−t)).
Give reasons for your answer.
=
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