(a) A continuous-time periodic signal x(t) with period Tis said to be odd hannonic if

Chapter 3, Problem 3.43

(choose chapter or problem)

(a) A continuous-time periodic signal x(t) with period Tis said to be odd hannonic if, in its Fourier series representation +x x(t) = L akejk(27TIT)t, k= -CG ak = 0 for every non -zero even integer k. (i) Show that if x(t) is odd harmonic, then x(t) = -x~ + ~). (ii) Show that if x(t) satisfies eq. (P3.43-2), then it is odd harmonic. (P3.43-1) (P3.43-2) (b) Suppose that x(t) is an odd-harmonic periodic signal with period 2 such that x(t) = t for 0 < t < 1. Sketch x(t) and find its Fourier series coefficients. (c) Analogously, to an odd-harmonic signal, we could define an even-harmonic signal as a signal for which ak = 0 fork odd in the representation in eq. (P3.43- 1). Could T be the fundamental period for such a signal? Explain your answer. (d) More generally, show that Tis the fundamental period of x(t) in eq. (P3.43-1) if one of two things happens: (1) Either a 1 or a-1 is nonzero; or (2) There are two integers k and l that have no common factors and are such that both ak and a, are nonzero.