# Consider an FM signal f(t) = cos [2πfct + β1 sin 2πf1t + β2 sin 2πf2t]. The maximum deviation of the instantaneous frequency from the carrier frequenc

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Consider an FM signal f(t) = cos [2πfct + β1 sin 2πf1t + β2 sin 2πf2t]. The maximum deviation of the instantaneous frequency from the carrier frequency fc is
1. β1f1 + β2f2
2. β1f2 + β2f1
3. β1 + β2
4. f1 + f2

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Correct Answer - Option 1 : β1f1 + β2f2

Concept:

The instantaneous frequency is nothing but the rate of change of phase, i.e.

${f_i}\left( t \right) = \frac{1}{2\pi} \frac{d\theta_i}{dt}$

Application:

Given FM signal is:

f(t) = cos [2π fct + β1 sin 2π f1t + β2 sin 2π f2t]

The instantaneous angle of the above signal is:

θi(t) = 2π fct + β1 sin 2π f1t + β2 sin 2π f2 t

${f_i}\left( t \right) = \frac{1}{{2\pi }}$ [2π fct + β1 cos 2π f1t(f1) + β2 cos 2π f2t + (f2)]

fi(t) = fc + β1f1 cos 2π f1t + β2f2 cos 2π f2t

The maximum deviation from the carrier frequency fc will be:

Δf = max (β1f1 cos 2π f1t + β2f2 cos 2π f2t)

Since, the maximum value of cosine function is 1, we get the maximum frequency deviation as:

Δf = β1f1 + β2f2