A periodic rectangular pulse train f(t) shown below has amplitude A = 1 V, pulse width ? = (1/6) ms, and period T0 = 1 ms. The pulse is centered at t = 0.
(a) Find the fundamental frequency f0 of this signal.
(b) Find the duty cycle d of this signal.
(c) Find the expression of the Fourier coefficients a0, an, bn, Fn. (d) Find the numerical values of a0, and an for 1 ? n ? 6.
(e) Plot the one-sided magnitude spectrum for 0 ? f ? 6f0.
(f) Plot the one-sided phase spectrum for 0 ? f ? 6f0.
(g) Find the expression of approximation ˆ( ) f t consisting of the dc component, the fundamental component, the second harmonic component, and the third harmonic component of its trigonometric Fourier series (sum of cosines). The coefficients must be numbers.
(a) f0 = 1 kHz
(b) d = 1/6
(c) a0 = Ad = 1/6, an = 2Ad sinc(nd) = (1/3) sinc(n/6), bn = 0 for all n, Fn = Ad sinc(nd) = (1/6) sinc(n/6)
(d) a0 = 1/6 n 0 1 2 3 4 5 6 an 0.1667 0.3183 0.2757 0.2122 0.1378 0.0637 0.0000
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