Correct Answer - Option 1 : β

_{1}f

_{1} + β

_{2}f

_{2}
**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 f_{c}t + β_{1} sin 2π f_{1}t + β_{2} sin 2π f_{2}t]

The instantaneous angle of the above signal is:

θi(t) = 2π f_{c}t + β_{1} sin 2π f_{1}t + β_{2} sin 2π f_{2} t

\({f_i}\left( t \right) = \frac{1}{{2\pi }}\) [2π f_{c}t + β_{1} cos 2π f_{1}t(f_{1}) + β_{2} cos 2π f_{2}t + (f_{2})]

f_{i}(t) = f_{c} + β_{1}f_{1} cos 2π f_{1}t + β_{2}f_{2} cos 2π f_{2}t

The maximum deviation from the carrier frequency f_{c} will be:

Δf = max (β_{1}f_{1} cos 2π f_{1}t + β_{2}f_{2} cos 2π f_{2}t)

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

Δf = β_{1}f_{1} + β_{2}f_{2}