An FM broadcasting radio station transmits signals of frequency 1

SOLUTION

Concept:

The general form of frequency modulated signal is represented by:

\(s\left( t \right) = {A_C}\cos \left( {2\pi {f_c}t + \beta sin2\pi {f_m}t} \right)\)

A_C: carrier signal amplitude

f_c: carrier signal frequency

f_m: message signal frequency

β: modulation index

\(\beta = \frac{{{\rm{\Delta f}}}}{{{f_m}}}\)

\(\beta = \frac{{frequency\;deviation}}{{message\;signal\;frequency}}\;\)

If β < 1 it is called a Narrowband FM

If β > 1it is called as Wideband FM

Calculation:

Given message signal frequency (f_m) range is 100 Hz to 1.5 KHz, and deviation = 75 KHz

The modulation index is defined as:

\(\beta = \frac{Δ f}{f_m}\)

Δf : frequency deviation

For f_m = 100 Hz

\(\beta = \frac{{75 \times {{10}^3}}}{{100}}\)

= 750

For f_m = 1.5 KHz

\(\beta = \frac{75 \times {10}^{3}}{1.5 \times {10}^{3}}\)

= 50

So the range of modulation index is 50 to 750

Extra Concept:

Instantaneous frequency

The instantaneous frequency of the angle modulated signal ACCos[θ (t)] is defined as:

\({f_i} = \frac{1}{{2\pi }}\frac{{d\left[ {\theta \left( t \right)} \right]}}{{dt}}\)

For FM fi = fc +kf m(t)

Kf: Frequency sensitivity (Hz/volt)