Signal Analysis & Synthesis

Analogue Modulation

Analogue signal processing

Digital signal processing

Digital modulation

Communication Systems

Frequency Modulation (FM)

Frequency modulation is the process of varying the frequency of a carrier wave in proportion to the instantaneous amplitude of the modulating signal without any variation in the amplitude of the carrier wave. Because the amplitude of the wave remains unchanged, the power associated with an FM wave is constant.

As can be seen from the figure, when the modulating signal is zero, the output frequency equals F_c (centre frequency). When the modulating signal reaches its positive peak, the frequency of the modulated signal is maximum and equals
( f_c + F_m ). At negative peaks of the modulating signal, the frequency of the FM wave becomes minimum and equal to ( f_c - F_m ). Thus, the process of frequency modulation make the frequency of the FM wave & deviate from its centre frequency
( f_c ) by an amount is termed as the frequency deviation of the system.

During this process, the total power in the wave does not change but a part of the carrier power is transferred to the side-bands.

Assume the modulating signal to be represented byand the carrier

wave being represented by

Thus,

The angular velocity may be determined by finding the rate of change of this phase angle.

i.e.

After frequency modulation takes place, angular velocity of the carrier wave varies in proportion to the instantaneous amplitude of the modulation signal. The instantaneous angular velocity w_i is given by

Where K is a constant of proportionality.

Maximum frequency shift or deviation occurs when the cosine terms become unity. Under this condition, the instantaneous angular velocity is given by

So that the maximum frequency deviation is given by

This gives,

The equation may be rewritten as

Integration gives the instantaneous phase of the frequency modulated wave.

=

Where is a constant of integration representing a constant phase angle and may be

neglected in the following analysis.

The instantaneous amplitude of the modulated waves is given by

The ratio is termed as the modulation index of the frequency modulated wave and is denoted by . It should be noted that for a given frequency deviation, the

modulation index varies with the modulating frequency f_m. A comparison of the modulation index m for the AM and for the frequency modulated wave shows that while m is given as the ratio of the change in the carrier amplitude due to amplitude modulation to the carrier amplitude, whereas m_f modulation index for FM is given as the ratio of frequency deviation to the modulating frequency i.e.,

Substituting equation for FM is given as