(a) A continuous-time periodic signal x(t) with period T is said to be odd harmonic if,...

(a) A continuous-time periodic signal x(t) with period T is said to be odd harmonic if, in its Fourier series representation

for every non-zero even integer k.

(i) Show that if x(t) is odd harmonic, then

(ii) Show that if x(t) satisfies eq. (P3.43-2), then it is odd harmonic.

(b) Suppose that x(t) is an odd-harmonic periodic signal with period 2 such that

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 for k 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 T is the fundamental period of x(t) in eq. (P3.43-1) if one of two things happens:

(1) Either is nonzero;

or

(2) There are two integers k and l that have no common factors and are such that both are nonzero.