A carrier wave of 1 MHz and amplitude 3V is frequency modulated

A carrier wave of 1 MHz and amplitude 3V is frequency modulated by a sinusoidal modulating signal frequency of 500 Hz and peak amplitude of 1V. The frequency deviation is 1kHz. The peak level of the modulating waveform is changed to 2kHz. The expression for the new modulation waveform is—

1. cos [2π × 10⁶t + 2·5 cos (4π × 10³t)]
2. cos [2π × 10⁶t + 5 cos (4π × 10³t)]
3. 3 cos [2π × 10⁶t + 2·5 sin (4π × 10³t)]
4. 3 cos [2π × 10⁶t + 5 cos (4π × 10³t)]

Since for FM modulated wave, frequency sensitivity constant Kf remains constant. So first we will calculate,

K_f =	Δf	=	1 × 10³	= 1kHz/V
	A_m		1

Now for the second case,
A_m = 5V and f_m = 2 kHz
Δf = K_f × A_m = 1 × 5 = 5 kHz
Modulation index,

β =	Δf	=	5 kHz	= 2.5
	f_m		2 kHz

The FM signal say S(t),
s(t) = 3 cos [2π × 10⁶ + β sin (2π·2 × 10³t)]
= 3 cos [2π × 10⁶t + 2·5 sin (4π × 10³t)]