Let message signal m(t) = cos (4π 103t) and carrier signal

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Let message signal m(t) = cos (4π 10³t) and carrier signal c(t) = 5 cos (2π 10⁶)t are used to generate a FM signal. If the peak frequency deviation of the generated FM signal is three times the transmission bandwidth of the AM signal, then the coefficient of the term, cos [2π(1008 × 10³t)t] in the FM signal would be—

1. 5J4 (3)
2. 5 J₈
  2
  (3)
3. 5 J₈
  2
  (4)
4. 5 J₄(6)

Correct Option: D

Given that, m(t) = cos (2π × 10³t)
c(t) = 5 cos (2π × 10⁶t)
and
Δf = 3 (transmission bandwidth)
Δf = 3(2f_m) = 6f_m
Now, modulation index

β =	Δf	=	6f_m	= 6
	f_m		f_m

The FM signal is represented in terms of Bessel function as
S_FM(t)=A_c ∑^∞ _{n = – ∞} Jn (β) cos (ω_c + nω_mt)
given that
S_FM (t) = [cos 2π (10008 × 10³t)] or
S_FM (t) = cos[(2π × 10⁶ + 8π × 10³)t]
= A_c ∑^∞_{n = – ∞} J_n (β) cos(ω_c + ω_m)t
n = 4
Thus coefficient, A_c J_n (β) = 5j₄ (6)

5	J₈
2

5	J₈
2