Question

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

Answer 1

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 f_ct + β₁ sin 2π f₁t + β₂ sin 2π f₂t]

The instantaneous angle of the above signal is:

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

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

f_i(t) = f_c + β₁f₁ cos 2π f₁t + β₂f₂ cos 2π f₂t

The maximum deviation from the carrier frequency f_c 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