(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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.5: Applications Of Inner Product Spaces

Problem 91E

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