(a) Let the modulated wave s (t) in Problem 2.42 be applied to a hard limiter, whose output z (t) is defined by below, show that the limiter output may be expressed in the form of a Fourier series as follows:

(b) Suppose that the limiter output is applied to a band-pass filter with a pass-band magnitude response of one and bandwidth B_T centered about the carrier frequency ƒ_c, where B_T is the transmission bandwidth of the FM signal in the absence of amplitude modulation. Assuming that ƒ_c is much greater than B_T, show that the resulting filter output equals y (t) = 4/π cos [2πƒ_ct = Φ (t)].  By comparing this output with the original modulated signal s (t) defined in Problem 2.42, comment on the practical usefulness of the result.

z(t) = sgn[s(t)] s(t) > 0 +1, s(t) < 0 -1, 4 zit) =-2 2n + 1 (-1)" cos[2 Tf.t(2n + 1) + (2n + 1)4(t)]